Copyright 1995 to 2007 Terry Ritter. All Rights Reserved.
For a basic introduction to cryptography, see "Learning About Cryptography" @: http://www.ciphersbyritter.com/LEARNING.HTM. Please feel free to send comments and suggestions for improvement to: ritter@ciphersbyritter.com (you may need to copy and paste the address into a web email reader). You may wish to help support this work by patronizing "Ritter's Crypto Bookshop" at: http://www.ciphersbyritter.com/BOOKSHOP.HTM.
Or use the browser facility "Edit / Find on this page" to search for particular terms.
This Glossary started as a way to explain the terms on my cryptography web pages describing my:
The value of a definition is insight. But:
Consider the idea that cryptography is used to keep secrets: We expect a cipher to win each and every contest brought by anyone who wishes to expose secrets. We call those people opponents, but who are they really, and what can they do? In practice, we cannot know. Opponents operate in secret: We do not know their names, nor how many they are, nor where they work. We do not know what they know, nor their level of experience or resources, nor anything else about them. Because we do not know our opponents, we also do not know what they can do, including whether they can break our ciphers. Unless we know these things that cannot be known, we cannot tell whether a particular cipher design will prevail in battle. We cannot expect to know when our cipher has failed.
Even though the entire reason for using cryptography is to protect secret information, it is by definition impossible to know whether a cipher can do that. Nobody can know whether a cipher is strong enough, no matter how well educated they are, or how experienced, or how well connected, because they would have to know the opponents best of all. The definition of cryptography implies a contest between a cipher design and unknown opponents, and that means a successful outcome cannot be guaranteed by anyone.
Consider the cryptographer who says: "My cipher is strong," and the cryptanalyst who says: "I think your cipher is weak." Here we have two competing claims with different sets of possibilities: First, the cryptographer has the great disadvantage of not being able to prove cipher strength, nor to even list every possible attack so they can be checked. In contrast, the cryptanalyst might be able to actually demonstrate weakness, but only by dint of massive effort which may not succeed, and will not be compensated even if it does. Consequently, most criticisms will be extrapolations, possibly based on experience, and also possibly wrong.
The situation is inherently unbalanced, with a bias against the cryptographer's detailed and thought-out claims, and for mere handwave first-thoughts from anyone who deigns to comment. This is the ultimate conservative bias against anything new, and for the status quo. Supposedly the bias exists because if the cryptographer's claim is wrong user secrets might be exposed. But the old status-quo ciphers are in that same position. Nothing about an old cipher makes it necessarily strong.
Unfortunately, for users to benefit from cryptography they have to accept some strength argument. Even more unfortunately:
In modern society we purchase things to help us in some way. We go to the store, buy things, and they work. Or we notice the things do not work, and take them back. We know to take things back because we can see the results. Manufactured things work specifically because design and production groups can test which designs work better or worse or not at all. In contrast, if the goal of cryptography is to keep secrets, we generally cannot expect to know whether our cipher has succeeded or failed. Cryptography cannot test the fundamental property of interest: whether or not a secret has been kept.
The inability to test for the property we need is an extraordinary situation; perhaps no other manufactured thing is like that. Because the situation is unique, few understand the consequences. Cryptography is not like other manufactured things: nobody can trust it because nobody can test it. Nobody, anywhere, no matter how well educated or experienced, can test the ability of an unbroken cipher to keep a secret in practice. Thus we see how mere definitions allow us to deduce fundamental limitations on cryptography and cryptanalysis by simple reasoning from a few basic facts.
The desire to expose relationships between ideas meant expanding the Glossary beyond cryptography per se to cover terms from related areas like electronics, math, statistics, logic and argumentation. Logic and argumentation are especially important in cryptography, where measures are few and math proofs may not apply in practice.
This Crypto Glossary is directed toward anyone who wants a better understanding of what cryptography can and cannot do. It is intended to address basic cryptographic principles in ways that allow them to be related, argued, and deeply understood. It is particularly concerned with fundamental limits on cryptography, and contradictions between rational thought and the current cryptographic wisdom. Some of these results may be controversial.
The Glossary is intended to build the fundamental understandings which lie at the base of all cryptographic reasoning, from novice to professional and beyond. It is particularly intended for users who wish to avoid being taken in by attacker propaganda. (Propaganda is an expected part of cryptography, since it can cause users to take actions which make things vastly easier for opponents.) The Glossary is also for academics who wish to see and avoid the logic errors so casually accepted by previous generations. One goal of the Glossary is to clarify the usual casual claims that confuse both novices and professionals. Another is to provide some of the historical technical background developed before the modern mathematical approach.
The way we understand reality is to follow logical arguments. All of us can do this, not just professors or math experts. Even new learners can follow a cryptographic argument, provided it is presented clearly. So, in this Glossary, one is occasionally expected to actually follow an argument and come to a personal conclusion. That can be scary when the result contradicts the conventional wisdom; then one then starts to question both the argument and the reasoning, as I very well know. But that scary feeling is just an expected consequence of a field which has allowed various unsupported claims and unquestioned beliefs to wrongly persist (see old wives' tales).
Unfortunately, real cryptography is not well-modeled by current math (for example, see proof and cryptanalysis). It is normally expected that the link between theory and reality is provided by the assumptions the math requires. (Obviously, proof conclusions only apply in practice when every assumed quality actually occurs in practice.) In math, each of these assumptions has equal value (since the lack of any one will void the conclusion), but in practice some assumptions are more equal than others. Certain assumptions conceivably can be guaranteed by the user, but other assumptions may be impossible to guarantee. When a model requires assumptions that cannot be verified in practice, that model cannot predict reality.
Current mathematical models almost never allow situations where the user can control every necessary assumption, making most proof results meaningless in practice. In my view, mathematical cryptography needs practical models. Of course, one might expect more realistic models to be less able to support the current plethora of mathematical results. Due to the use of more realistic models, some results in the Crypto Glossary do contradict well-known math results.
By carrying the arguments of conventional cryptographic wisdom to their extremes, it is possible to see two opposing groups, which some might call theoretical versus practical. While this simplistic model is far too coarse to take very seriously, it does have some basis in reality.
The Crypto Theorists supposedly argue that no cryptosystem can be trusted unless it has a mathematical proof, since anything less is mere wishes and hope. Unfortunately, there is no such cryptosystem. No cipher can be guaranteed strong in practice, and that is the real meaning of the one time pad. As long as even one unbreakable system existed, there was at least a possibility of others, but now there is no reason for such hope. The OTP is secure only in simplistic theory, and strength cannot be guaranteed in practice for users. This group seems most irritating when they imply that math proofs are most important, even when in practice those proofs provide no benefits to the user.
The Crypto Practitioners supposedly argue that systems should be designed to oppose the most likely reasonable threats, as in physical threat model analysis. In the physical world it is possible to make statements about limitations of opponents and attacks; unfortunately, few such statements can be made in cryptography. In cryptography, we know neither the opponents nor their attacks nor what they can do in combination. Successful attack programs can be reproduced and then applied by the most naive user, who up to that time had posed only the most laughable threat.
Both groups are wrong: There will be no proof in practice, and speculating on the abilities of the opponents is both delusional and hopeless. Moreover, no correct compromise seems possible. Taking a little proof from one side and some threat analysis from the other simply is not a valid recipe for making secure ciphers.
There is a valid recipe for security and that is a growing, competitive industry of cipher development. Society needs more than just a few people developing a handful of ciphers, but actual design groups who continually innovate, design, develop, measure, attack and improve new ciphers in a continuing flow. That is expensive work, as the NSA budget clearly shows. Open society will get such results only if open society will pay for them. Since payment is the issue, it is clear that "free" ciphers act to oppose exactly the sort of open cryptographic development society needs.
Absent an industry of cipher design, perhaps the best we can do is to design systems in ways such that a cipher actually can fail, while the overall system retains security. That is redundancy, and is a major part of engineering most forms of life-critical systems (e.g., airliners), except for cryptography. The obvious start is multiple encryption.
The practical worth of all this should be a serious regard for cryptographic risk. The possibility of cryptographic failure exists despite all claims and proofs to the contrary. Users who have something to protect must understand that cryptography has risks, and there is a real possibility of failure. If a possibility of information exposure is acceptable, one might well question the use of cryptography in the first place.
Even if users only want their information probably to be secure, they still have a problem: Only our opponents know our cipher failures, because they occur in secret. Our opponents do not expose our failures because they want those ciphers to continue in use. Few if any users will know when there is a problem, so we cannot count how many ciphers fail, and so cannot know that probability. Since there can be no expertise about what unknown opponents do, looking for an "expert opinion" on cipher failure probabilities or strength is just nonsense.
Conventional cryptographic expertise is based on the open literature. Unfortunately, unknown attacks can exist, and even the best informed cannot predict strength against them. While defending against known attacks may seem better than nothing, that actually may be nothing to opponents who have another approach. In the end, cipher and cryptosystem designers vigorously defend against attacks from academics who will not be their opponents.
On the other hand, even opponents read the open literature, and may make academic attacks their own. But surprisingly few academic attacks actually recover key or plaintext and so can be said to be real, practical threats. Much of the academic literature is based on strength assumptions which cannot be guaranteed or vulnerability assumptions which need not exist, making the literature less valuable in practice than it may appear.
Math cannot prove that a cipher is strong in practice, so we are forced to accept that any cipher may fail. We do not, and probably can not know the likelihood of that. But we do know that a single cipher is a single point of failure which just begs disaster. (Also see standard cipher.)
It is possible to design in ways which reduce risk. Systems can be designed with redundancy to eliminate the single point of failure (see multiple encryption). This is often the done in safety-critical fields, but rarely in cryptography. Why? Presumably, people have been far too credulous in accepting math proofs which rarely apply in practice. Thus we see the background for my emphasis on basics, reasoning, proof, and realistic math models.
To protect against fire, flood or other disaster, most software developers should store their current work off-site. The obvious solution is to first encrypt the files and then upload an archive to a web site. The straightforward use of cryptography to protect archives is an example of the pristine technical situation often seen as normal. Then we think of cipher strength and key protection, which seem to be all there is. But most cryptography is not that simple.
Climate of Secrecy. For any sort of cryptography to work, those who use it must not give away the secrets. Most times keeping secrets is as easy, or as hard, as just not talking or writing about them. Issues like minimizing paper output and controlling and destroying copies seem fairly obvious, although hardly business as usual. But secrets are almost always composed in plaintext, and the computers doing that may have plaintext secrets saved in various hidden operating system files. And opponents may introduce programs to compromise computers which handle secrets. It is thus necessary to control all forms of access to equipment which holds secrets, despite that being awkward and difficult. It is especially difficult to control access on the net.
Network Security. Computers only can do what they are told to do. When network designers decide to include features which allow attacks, that decision is as much a part of the problem as an attack itself. It seems a bit much to complain about insecurity when insecurity is part of the design. Design decisions have made the web insecure. Until web systems only implement features which maintain security, there can be none.
It is possible to design computing systems more secure than the ones we have now. If we provide no internal support for external attack, no attacks can prevail. The entire system must be designed to limit and control external web access and prevent surprises that slip by unnoticed. We can decompose the system into relatively small modules, and then test those modules in a much stronger way than trying to test a complex program. A possible improvement might be some form of restricted intermediate or quarantine store between the OS and the net. Better security design may mean that some things now supported insecurely no longer can be supported at all.
Current practice identifies two environments: The local computer, which is "fully" trusted, and the Internet, which is not trusted. This verges on a misuse of the concept of trust, which requires substantial consequences for misuse or betrayal. Absent consequences, trust is mere unsupported belief and provides no basis for reasoning. We do not trust a machine per se, since it only does what the designer made it do. And when there are no consequences for bad design, there really is no reason to trust the designer either.
A better approach would be fine OS control over individual programs, including individual scripts, providing validation and detailed limits on what each program can do, on a per-program basis. This would expand the firewall concept from just net access to every resource, including processor time, memory, all forms of I/O, plus the ability to invoke, or be invoked by, other programs. For example, most programs do not need, and so would not be allowed, net access, even if invoked by a program or running under a process which has such access. Programs received from the net would by default start out in quarantine, not have access to normal store, and could run only under strong limitations. A human would have to explicitly elevate them to a selected higher status, with the change logged. Program operation exceeding limitations would be prevented, logged, and accumulated in a control which supported validation, fine tuning, selective responses and serious quarantine.
Security is Off-The-Net. The best way to avoid web insecurity has nothing to do with cryptography. The way to avoid web insecurity is to not connect to the web, ever. Use a separate computer for secrets, and do not connect it to the net, or even a LAN, since computers on the LAN probably will be on the net. Carefully move information to and from the secrets computer with a USB flash drive. Protect access to that equipment.
For most users, the Crypto Glossary will have many underlined (or perhaps colored) words. Usually, those are hypertext "links" to other text in the Glossary; just click on the desired link.
Links to my other pages generally offer a choice between a "local" link or a full web link. The user working from a downloaded copy of the Glossary only would normally use the full web links. The user working from a CD or disk-based copy of all my pages would normally use the local links.
Links to my other pages also generally open and use another window. (Hopefully that will avoid the need to reload the Glossary after a reference to another article.) Similarly, links from my other pages to terms in the Glossary also generally open a window specifically for the Glossary. (In many cases, that will avoid reloading the Glossary for every new term encountered on those pages.)
In cryptography, as in much of language in general, the exact same word or phrase often is used to describe two or more distinct ideas. Naturally, this leads to confused, irreconcilable argumentation until the distinction is exposed (and often thereafter). Usually I handle this in the Crypto Glossary by having multiple numbered definitions, with the most common usage (not necessarily the best usage) being number 1.
The worth of this Glossary goes beyond mere definitions. Much of the worth is the relationships between ideas: Hopefully, looking up one term leads to other ideas which are similar or opposed or which support the first. The Glossary is a big file, but breaking it into many small files would ruin much of the advantage of related ideas, because then most related terms would be in some other part. And although the Glossary could be compressed, that would generally not reduce download time, because most modems automatically compress data during transmission anyway. Dial-up users typically should download the Glossary onto local storage, then use it locally, updating periodically.
I have obviously spent a lot of personal time constructing this Crypto Glossary, with the hope that it would be more than just background to my work. Hopefully, the Glossary and the associated introduction: "Learning About Cryptography" (see locally, or @: http://www.ciphersbyritter.com/LEARNING.HTM) will be of some wider benefit to the crypto community. So, if you have used this Glossary lately, why not drop me a short email and tell me so? Feel free to tell me how much it helped or even how it failed you; perhaps I can make it better for the next guy. If you use web email, just copy and paste my email address: ritter@ciphersbyritter.com
Resistor excess noise is a
The especially large amount of
In a single-crystal
semiconductor,
Generally used for power distribution because the changing
current supports the use of
transformers. Utilities can thus
transport power at high
voltage and low
current, which minimize
"ohmic"
or I2R losses. The high voltages are then reduced
at power substations and again by pole transformers for delivery
to the consumer.
One example is byte addition modulo 256, which simply adds two byte values, each in the range 0..255, and produces the remainder after division by 256, again a value in the byte range of 0..255. The modulo is automatic in an addition of two bytes which produces a single byte result. Subtraction is also an "additive" combiner.
Another example is bit-level exclusive-OR which is addition mod 2. A byte-level exclusive-OR is a polynomial addition.
Additive combiners are linear, in contrast to nonlinear combiners such as:
Knuth, D. 1981. The Art of Computer Programming, Vol. 2, Seminumerical Algorithms. 2nd ed. 26-31. Addison-Wesley: Reading, Massachusetts.
Marsaglia, G. and L. Tsay. 1985. Matrices and the Structure of Random Number Sequences. Linear Algebra and its Applications.67:147-156.
Advantages include:
In addition, a vast multiplicity of independent cycles has the potential of confusing even a quantum computer, should such a thing become realistic.
For Degree-n Primitive, and Bit Width w Total States: 2nw Non-Init States: 2n(w-1) Number of Cycles: 2(n-1)(w-1) Length Each Cycle: (2n-1)2(w-1) Period of lsb: 2n-1
The binary addition of two bits with no carry input is just XOR, so the lsb of an Additive RNG has the usual maximal length period.
A degree-127 Additive RNG using 127 elements of 32 bits each has 24064 unique states. Of these, 23937 are disallowed by initialization (the lsb's are all "0") but this is just one unusable state out of 2127. There are still 23906 cycles which each have almost 2158 steps. (The Cloak2 stream cipher uses an Additive RNG with 9689 elements of 32 bits, and so has 2310048 unique states. These are mainly distributed among 2300328 different cycles with almost 29720 steps each.)
Like any other
LFSR, and like any other
RNG, and like any other
FSM, an Additive RNG is very weak when
standing alone.
But when steps are taken to hide the sequence (such as using a
jitterizer nonlinear filter and
Dynamic Substitution
combining), the resulting cipher can have significant strength.
The mechanics of AES are widely available elsewhere. Here I note how one particular issue common to modern block ciphers is reflected in the realized AES design. That issue is the size of the implemented keyspace compared to the size of the potential keyspace for blocks of a given size.
A common academic model for conventional block ciphers is a "family of permutations." The "permutation" part of this means that every plaintext block value is found as ciphertext, but generally in a different position. The "family" part of this can mean every possible permutation. However, modern block ciphers key-select only an infinitesimal fraction of those possibilities.
Suppose we have a block which may take on any of n
different values.
How many ways can those n block values be rearranged as
in a block cipher?
Well, the first value can be placed in any of the n possible
positions, but that fills one position so the second value has only
A
A
For
The obvious conclusion is that almost none of the keyspace implicit in the theoretical model of a conventional block cipher is actually implemented in AES, and that is consistent with other modern designs. Is that important? Apparently not, but nobody really knows. It does seem to imply that just a few known plaintext blocks should be sufficient to identify the correct key from a set of possibilities, which might make known plaintext more of an issue than normally claimed. Does it lead to a known break? No, or at least not yet. But having only a tiny set of keyed permutations should lead to questions about patterns and relationships within the selected set.
The real issue here is not the exposure of a particular weakness in AES, since no such weakness is shown. Instead, the issue is that conventional cryptographic wisdom does not force models to correspond to reality, and poor models lead to errors in reasoning. The distinction between theory and practice is pronounced in cryptography. For other examples of failure in the current cryptographic wisdom, see one time pad, BB&S, DES, and, of course, old wives' tale.
AES is said to be certified for SECRET and TOP SECRET classified material. That might have us believe that AES is trusted by NSA, but it may mean less than it seems.
No cipher, by itself, can guarantee security.
Any cryptographic system will have to be certified by NSA before
protecting classified information.
In practice, cryptosystems will be provided by NSA to contractors,
those systems may or may not use AES, and they may not use AES in
the expected form.
That does not imply that AES is bad, it just means that we cannot
really know what NSA will allow, despite general claims.
Technically, function f : G -> G of the form:
f(x) = ax + bwith non-zero constant "b".
anxn + an-1xn-1 + ... + a1x1 + a0where the operations are mod 2: addition is Exclusive-OR, and multiplication is AND.
Note that all of the variables xi are to the first power only, and each coefficient ai simply enables or disables its associated variable. The result is a single Boolean value, but the constant term a0 can produce either possible output polarity.
Here are all possible 3-variable affine Boolean functions (each of which may be inverted by complementing the constant term):
affine truth table
c 0 0 0 0 0 0 0 0
x0 0 1 0 1 0 1 0 1
x1 0 0 1 1 0 0 1 1
x1+x0 0 1 1 0 0 1 1 0
x2 0 0 0 0 1 1 1 1
x2+ x0 0 1 0 1 1 0 1 0
x2+x1 0 0 1 1 1 1 0 0
x2+x1+x0 0 1 1 0 1 0 0 1
See also:
Boolean function
nonlinearity.
F(x) = ax + b (mod n)
where the non-zero
term makes the
equation
affine.
Most of the classic hand-ciphers can be seen as
simple substitution
stream ciphers. Each
plaintext letter selects an entry in the
substitution table (for that
cipher), and the contents of that entry becomes the
ciphertext letter.
The affine equation thus represents one way to set
up the table, as a particular simple
permutation of the letters in the table.
(Of course, by using the equation we need no explicit table, but
we also constrain ourselves to the simplicity of the equation.)
To assure that we have a permutation, we require that a and n
be
relatively prime, that is, the
In modern terms, the
strength of the classic substitution
ciphers is essentially nil. In modern
cryptanalysis, we generally assume
that the
opponent has a substantial amount of
known plaintext.
Since the table does not change, every known-plaintext character
has the potential to fill in another entry in the table.
Very soon the table is almost completely exposed, which ends all
strength.
These simple substitution ciphers with small, fixed tables (or even
just equations for such tables) are also extremely vulnerable to
attacks using
ciphertext only.
"The first combining operation is called the product operation and corresponds to enciphering the message with the first secrecy system R and enciphering the resulting cryptogram with the second system S, the keys for R and S being chosen independently."
"The second combining operation is 'weighted addition.'
It corresponds to making a preliminary choice as to whether system R or S is to be used with probabilities p and q, respectively. When this is done or R or S is used as originally defined." [p.658]S = pR + qS p + q = 1.
More specifically (and with a change of notation):
"If we have two secrecy systems T and R we can often combine them in various ways to form a new secrecy system S. If T and R have the same domain (message space) we may form a kind of 'weighted sum,'
S = pT + qRwherep + q = 1. This operation consists of first making a preliminary choice with probabilities p and q determining which of T and R is used. This choice is part of the key of S. After this is determined T or R is used as originally defined. The total key of S must specify which of T and R is used, and which key of T (or R) is used.""More generally we can form the sum of a number of systems.
S = p1T + p2R + . . . + pmU Sum( pi ) = 1We note that any system T can be written as a sum of fixed operationsT = p1T1 + p2T2 + . . . + pmTmTi being a definite enciphering operation of T corresponding to key choice i, which has probability p.""A second way of combining two secrecy systems is taking the 'product,' . . . . Suppose T and R are two systems and the domain (language space) of R can be identified with the range (cryptogram space) of T. Then we can apply first T to our language and then R to the result of this enciphering process. This gives a resultant operation S which we write as a product
S = RTThe key for S consists of both keys of T and R which are assumed chosen according to their original probabilities and independently. Thus if the m keys of T are chosen with probabilitiesp1p2 . . . pmand the n keys of R have probabilitiesp'1p'2 . . . p'n ,then S has at most mn keys with probabilities pi p'j. In many cases some of the product transformations Ri Tj will be the same and can be grouped together, adding their probabilities."Product encipherment is often used; for example, one follows a substitution by a transposition or a transposition by a Vigenere, or applies a code to the text and enciphers the result by substitution, transposition, fractionation, etc."
"It should be emphasized that these combining operations of addition and multiplication apply to secrecy systems as a whole. The product of two systems TR should not be confused with the product of the transformations in secrecy systems Ti Rj . . . ."
-- Shannon, C. E. 1949. Communication Theory of Secrecy Systems. Bell System Technical Journal.28:656-715.
It is easy to dismiss this as being of historical interest only, but there are advantages here which are well beyond our current usage.
For the keyed selection among ciphers, there would be some sort of simple protocol (i.e., not cryptographic per se), for communicating cipher selections to the deciphering end. (Perhaps there would be some sort of simple handshake for email use.) The result would be to have (potentially) a new selection from a set of ciphers on a message-by-message basis.
With respect to multiple encryption or ciphering "stacks" (as in "protocol stacks"), there are various security advantages:
Also see:
Perfect Secrecy and
Ideal Secrecy.
An algorithm intended to execute reliably as a computer program necessarily must handle, or in some way at least deal with, absolutely every error condition which can possibly occur in operation. (We do assume functional hardware, and thus avoid programming around the possibility of actual hardware faults, such as memory or CPU failure.) These "error conditions" normally include Operating System errors (e.g., bad parameters passed to an OS operation, resource not available, various I/O failures, etc.), and arithmetic issues (e.g., division by zero, overflow, etc.) which may halt execution when they occur.
Other possibilities include errors the OS will not know about, including the misuse of programmer-defined data structures, such as buffer overrun.
A practical algorithm must recognize various things which
validly may occur, even if such things are exceedingly rare.
One example might be in assuming that two floating-point variables
which represent the same value will be equal.
Another example might be to assume that a floating-point variable
will "never" have some particular value (which might lead to a
divide-by-zero fault).
Yet another example would be to assume that an arbitrary selection
of x will lead to a sufficiently long cycle in
BB&S, even if the alternative is very,
very unlikely.
In particular, my Cloak2 and Penknife ciphers implemented encrypted alias files of text lines of arbitrary length, each of which included name, start date, and key. New keys were made available only as secure ciphertext, but the alias files were arranged so they could consist of multiple ciphertext files simply concatenated as ciphertext. Thus, new keys could be added to the start of the alias file just using a simple and secure file copy operation. When searching for a particular alias, the date was also checked, and that key used only when the correct date had arrived. This allowed an entire office of users to change to a new key automatically, at the same time, without even knowing they were using a different key. Appropriate functions allowed access to old keys so that email traffic could be archived in ciphertext form.
Obviously, an alias file must be encrypted. The single key or keyphrase decrypting an alias file thus provides access to all the keys in the file. But each alias file contains only a subset of the keys in use within an organization, and even those are only valid over a subset of time. An organization security officer could archive old alias files, strip out the old keys and add new ones, then encipher the new alias file under a new pass phrase. In this way, the contents of old encrypted email would not be hidden from the authorizing organization. Alias file maintenance could be either as complex or as simple as one might like.
See, for example,
Allan Variance is useful in analysis of residual noise in precision frequency measurement. Five different types of noise are defined: white noise phase modulation, flicker noise phase modulation, white noise frequency modulation, flicker noise frequency modulation, and random walk frequency modulation. A log-log plot of Allan variance versus sample period produces approximate straight line values of different slopes in four of the five possible cases. A different (more complex) form called "modified Allan deviation" can distinguish between the remaining two cases.
Also see
"Definition. A transformation f mapping a message sequence m1,m2,...,ms into a pseudo-message sequence m1',m2',...,ms' is said to be an all-or-nothing transform if:
- The transform f is reversible: given the pseudo-message sequence, one can obtain the original message sequence.
- Both the transformation f and its inverse are efficiently computable (that is, computable in polynomial time).
- It is computationally infeasible to compute any function of any message block if any one of the pseudo-message blocks is unknown."
-- Rivest, R. 1997. All or nothing encryption and the package transform. Fast Software Encryption 1997. 210-218.
When used with a conventional block cipher, an AONT appears to increase the cost of a brute-force attack by a factor which is the number of blocks in the message. Rivest also notes that the large effective block size can avoid ciphertext expanding chaining modes by using ECB mode on the large block. Also see huge block cipher advantages.
The
Balanced Block Mixing (BBM)
which I introduced to cryptography in my article:
"Keyed Balanced Size-Preserving Block Mixing Transforms" (
locally, or @:
http://www.ciphersbyritter.com/NEWS/94031301.HTM)
in early 1994 (three years before the Rivest publication), and then
developed in a series of subsequent articles, apparently can be an
especially fine example of an all-or-nothing transform.
The alternative hypothesis H1 is also called the
research hypothesis,
and is logically identical to
"NOT-H0" or "H0
is not true."
Transistors are analog amplifiers which are basically linear over a reasonable range and so require DC power. In contrast, relays are classically mechanical devices with direct metal-to-metal moving connections, and so can handle generally higher power and AC current. The classic analog amplifier is an operational amplifier.
Unexpected oscillation can be indicated by:
Oscillation occurs when:
To stop undesired oscillation:
To Increase Isolation
To decrease gain:
To change phase:
When two things are related by appropriate similarity in
structure or function, we can infer that what is known about one
thing also may apply to the other.
Such an inference may or may not be true, but it can be examined
and tested.
In On Sophistical Refutations (350 B.C.E.), Aristotle (384-322 B.C.E.) lists five goals for countering arguments:
Refutation can occur in various ways. Disputing the evidence being used to support a claim can be considered a new claim and different evidence presented. However, disputing the reasoning itself requires only logic and typically no further evidence at all. See extraordinary claims.
Like cryptography, argumentation is war, and tricks abound when winning is the ultimate goal. But arguing to win is fundamentally unscientific, since learning occurs mainly when an error is found and recognized.
The first requirement of successful argumentation is to have a stated topic or thesis. Without a stated topic, an unscrupulous opponent can lead the argument to some apparently similar but more vulnerable issue, and few in the audience will notice. That is especially true when a topic is introduced casually, and then changed by the opponent in the very first response. Another approach for the opponent is to indignantly bring up and discuss in detail some supposed error on an irrelevant but apparently related topic. A clever topic change also may cause awkward repetition and babbling in the attempt to expose the change and reverse it. The correct response is to be aware enough to recognize the topic change immediately, and return to the original topic; to argue that the comments are off-topic is to introduce a new topic.
There is no way to make an opponent stay on-topic, and if they know they will lose on-topic, that actually may be impossible. Moreover, the opponent may pose various questions (on some new topic), and claim you are not being responsive, the discussion of that claim itself being a new topic. But if you want to take your topic to conclusion, you cannot follow an opponent who wants anything but that. (Also see spin.)
The second requirement of successful argumentation is to force the discussion to remain on the material content. If the original argument might be successful, an unscrupulous opponent may seek to divert the discussion to the appropriateness of, or bias in, the symbols or names used for the concept. Or the opponent may find and protest premises stated without mathematical precision. But a conventional argument need be neither mathematically complete nor mathematically precise to be valid. (This is the fallacy of accident.) The correct response is to point out that the comments are irrelevant and return to the material issue; to argue that the comments are wrong is to argue a changed topic.
The goal of scientific argumentation is to improve knowledge and insight, not to anoint a "superior" contestant. Sadly, those willing to "win" with dishonesty generally do find an easily mislead audience.
Almost all on-line arguments are technically informal in the sense of depending upon context and definitions. The need for particular context generally leaves ample room to confuse the issue, even for someone who knows almost nothing about the topic.
If the proposed argument is basically unsound, that case can be won on its merits.
If the proposed argument is basically sound, but based on analogy, we need to realize that there are few really good analogies. Examine the analogy in detail and try various cases until one is found that is good in the analogy but bad in the proposed argument.
If the proposed argument is basically sound, one can win anyway by changing the topic and doing so in a smooth way the audience will not notice.
Most responses carry a least a thin patina of respectability.
However, many times a response is actually just the first sad shot
in a verbal combat that seeks defeat and winning by deception.
Unfortunately, it may be difficult to distinguish between mere
ignorance and actual attack.
It is thus important to actually examine the logic of any response.
DEC HEX CTRL CMD DEC HEX CHAR DEC HEX CHAR DEC HEX CHAR
0 00 ^@ NUL 32 20 SPC 64 40 @ 96 60 '
1 01 ^A SOH 33 21 ! 65 41 A 97 61 a
2 02 ^B STX 34 22 " 66 42 B 98 62 b
3 03 ^C ETX 35 23 # 67 43 C 99 63 c
4 04 ^D EOT 36 24 $ 68 44 D 100 64 d
5 05 ^E ENQ 37 25 % 69 45 E 101 65 e
6 06 ^F ACK 38 26 & 70 46 F 102 66 f
7 07 ^G BEL 39 27 ' 71 47 G 103 67 g
8 08 ^H BS 40 28 ( 72 48 H 104 68 h
9 09 ^I HT 41 29 ) 73 49 I 105 69 i
10 0a ^J LF 42 2a * 74 4a J 106 6a j
11 0b ^K VT 43 2b + 75 4b K 107 6b k
12 0c ^L FF 44 2c , 76 4c L 108 6c l
13 0d ^M CR 45 2d - 77 4d M 109 6d m
14 0e ^N SO 46 2e . 78 4e N 110 6e n
15 0f ^O SI 47 2f / 79 4f O 111 6f o
16 10 ^P DLE 48 30 0 80 50 P 112 70 p
17 11 ^Q DC1 49 31 1 81 51 Q 113 71 q
18 12 ^R DC2 50 32 2 82 52 R 114 72 r
19 13 ^S DC3 51 33 3 83 53 S 115 73 s
20 14 ^T DC4 52 34 4 84 54 T 116 74 t
21 15 ^U NAK 53 35 5 85 55 U 117 75 u
22 16 ^V SYN 54 36 6 86 56 V 118 76 v
23 17 ^W ETB 55 37 7 87 57 W 119 77 w
24 18 ^X CAN 56 38 8 88 58 X 120 78 x
25 19 ^Y EM 57 39 9 89 59 Y 121 79 y
26 1a ^Z SUB 58 3a : 90 5a Z 122 7a z
27 1b ^[ ESC 59 3b ; 91 5b [ 123 7b {
28 1c ^\ FS 60 3c < 92 5c \ 124 7c |
29 1d ^] GS 61 3d = 93 5d ] 125 7d }
30 1e ^^ RS 62 3e > 94 5e ^ 126 7e
31 1f ^_ US 63 3f ? 95 5f _ 127 7f DEL
a + (b + c) = (a + b) + c a * (b * c) = (a * b) * c
Also see:
commutative and
distributive.
In a mathematical proof, each and every assumption must be true for the proof result to be true. If the truth of any assumption is unknown, the proof is formally incomplete and the result has no meaning.
In practice, proofs have meaning only to the extent that each
and every required assumption can be assured, including
assumptions which may not be immediately apparent.
In practical cryptography, while some assumptions possibly could
be assured by the user, others could only be assured by the cipher
designer, who must then be
trusted, along with his company, the entire
distribution path and so on.
Even worse, still other assumptions may be impossible to
assure in practice by any means at all, which makes any such
proof useless for practical cryptography.
Also RS-232 and similar "serial port" signals, in which
byte or character values are transferred
bit-by-bit in bit-serial format.
Since digital signals require both proper
logic levels and proper timing to sense
those levels, timing is established by the leading edge of a
"start bit" sent at the start of each data byte. See
asynchronous transmission.
Transmit: The line rests "high." When a character is to be sent, a start bit or "low" level is sent for one bit-time. Then each data bit is sent, for one bit-time each, as are one or two stop or "high" level bit-times. Then, if no more data are ready for sending, the line just rests "high."
Receive: The line is normally "high." The instant the line goes "low" is the beginning of a start bit, and that establishes an origin for bit timing. Exactly 1.5 bit-times later, hopefully in the middle of the first data-bit time, the line level is sampled to record the first incoming bit. The second bit is recorded one bit-time later, and so on. When all bits have been recorded, the receiver sends the resulting character, all bits simultaneously, to a local register or FIFO queue for pickup. Note that all this implies that we know the format of the character with respect to bit time and number of bits.
Timing Accuracy: Everything depends upon both transmit
and receive ends having approximately the same bit timing.
The leading edge of the start bit temporarily
synchronizes the receiver, even though
the transmit and receive clock rates may be somewhat different.
With 8-bit characters, the last data bit is sampled exactly 8.5
bit-times from the detected leading edge of the start bit.
If the receive timing varies as much as +/- 0.5 bit in 8.5, the
last bit will be sampled outside the correct bit time.
So the total timing accuracy must be within +/- 5.8 percent, for
all sources transmit and receive clock variation, including
sampling delay in detecting the start bit.
Nowadays this is easily achieved with cheap
crystal oscillator
clock modules
and digital count logic.
In normal cryptanalysis we start out knowing plaintext, ciphertext, and cipher construction. The only thing left unknown is the key. A practical attack must recover the key. (Or perhaps we just know the ciphertext and the cipher, in which case a practical attack would recover plaintext.) Simply finding a distinguisher (showing that the cipher differs from the chosen model) is not, in itself, an attack. If an attack does not recover the key (or perhaps the particular key-selected internal state used in ciphering), it is not a real attack.
In cryptography, when someone says they have "an attack," the implication is that they have a successful attack (a break) and not just another failed attempt. It is obviously much easier to simply claim to have an attack than to actually analyze, innovate, build and test a working attack, which makes it necessary to back up such claims with evidence. Arrogant claims, with "proof left as an exercise for the student" or "read the literature" responses, deserve jeers instead of the cowed respect they often get.
A claim to have an attack can be justified by:
It is not sufficient to say: "My interpretation of the theory is that there must be a break, so the cipher is broken"; it is instead necessary to actually devise a process which recovers key or plaintext. Furthermore, there are many attacks which work against scaled-down tiny ciphers, but which do not scale up as valid attacks against the original large cipher: Just because we can solve newspaper-amusement ciphers (tiny versions of conventional block ciphers) does not imply that any real-size block ciphers are "broken." The process used to solve newspaper ciphers is not "an attack" on block ciphers in general.
Classically, attacks were neither named nor classified; there was just: "here is a cipher, and here is 'the' attack." (Many different attacks may be possible, but even one practical attack is sufficient to cause us to avoid that cipher.) And while this gradually developed into named attacks, there is no overall attack taxonomy. Currently, attacks are often classified by the information available to the attacker or constraints on the attack, and then by strategies which use the available information. Not only ciphers, but also cryptographic hash functions can be attacked, generally with very different strategies.
We are to attack a cipher which enciphers plaintext into ciphertext or deciphers the opposite way, under control of a key. The available information necessarily constrains our attack strategies.
The goal of an attack is to reveal some unknown plaintext, or the key (which will reveal the plaintext). An attack which succeeds with less effort than a brute-force search we call a break. An "academic" ("theoretical," "certificational") break may involve impractically large amounts of data or resources, yet still be called a "break" if the attack would be easier than brute force. (It is thus possible for a "broken" cipher to be much stronger than a cipher with a short key.) Sometimes the attack strategy is thought to be obvious, given a particular informational constraint, and is not further classified.
Many attacks try to isolate unknown small components or aspects
so they can be solved separately, a process known as
divide and conquer. Also see:
security.
A network in the form of a tree is used, with goals represented as nodes. Various possible ways to achieve a particular goal are represented as branches, which then can be taken as goals with their own branch nodes.
In cryptographic analysis, the idea is that the root node will represent the ultimate security we seek. Each path to the root then represents the accumulated effort needed to break that security. The problem is that it is typically impossible to assure that every alternative attack has been considered. And if some unconsidered approach is cheaper than any other, that becomes the true limit on security, despite not being present in the analysis.
Attack tree analysis does not tend to expose unconsidered attacks. Yet those are exactly the issues which carry the greatest cryptographic risk, because we can at least generally quantify the risk from known attacks. Since an attack tree cannot do what most needs to be done, it would seem to be a strange choice for cryptographic risk analysis. One could even argue that an attack tree is most useful as a formal aid in deluding naive executives and users.
Threat models basically concern what is to be protected, from whom, and for how long. But with ciphers, we seek to protect all our data, from everyone, forever. The extreme nature of these expectations is only part of what makes a conventional threat model unhelpful in understanding ciphering risks.
Cipher failure and exploitation happens in secret, so we cannot know how often it occurs and cannot develop a probability for it. Absent a probability of cipher failure, any attempt to understand ciphering risk is necessarily limited.
A more effective approach to system security is to build with understandable components. In component design, we can define exactly what each component permits. In component analysis, we can consider the security effects and expose the precise range of things each component allows. If none of the allowed things can cause a security problem, we will have no security problems. Components essentially become a custom language of system design which has no way of expressing security faults.
A component-based security design is far more restrictive and so is far more demanding than the conventional mode of hacking through a design and implementation. However, this design process provides a road map for real security, as opposed to belief in results from flawed analytical tools (like attack trees) and ad hoc analysis that simply cannot deliver the assurances we need.
The ability to analyze security must be designed into a system;
it cannot be just added on to finished systems.
For a known population, the number of repetitions expected at each level has long been understood to be a binomial expression. But if we are sampling in an attempt to establish the effective size of an unknown population, we have two problems:
Fortunately, there is an unexpected and apparently previously unknown combinatoric relationship between the population and the number of combinations of occurrences of repeated values. This allows us to convert any number of triples and higher n-reps to the number of 2-reps which have the same probability. So if we have a double, and then get another of the same value, we have a triple, which we can convert into three 2-reps. The total number of 2-reps from all repetitions (the augmented 2-reps value) is then used to predict population.
We can relate the number of samples s to the population N through the expected number of augmented doubles Ead:
Ead(N,s) = s(s-1) / 2N .
This equation is exact, provided we interpret all
the exact n-reps in terms of 2-reps. For example, a triple is
interpreted as three doubles; the augmentation from 3-reps to 2-reps
is (3 C 2) or 3. The augmented result is the sum of the
contributions from all higher repetition levels:
n i
ad = SUM ( ) r[i] .
i=2 2
where ad is the number of augmented doubles, and r[i]
is the exact repetition count at the i-th level.
And this leads to an equation for predicting population:
Nad(s,ad) = s(s-1) / 2 ad .
This predicts the population Nad as based on a mean value
of augmented doubles ad. (For an example and comparison to
various other methods, see the
conversation:
However, since the trials should have approximately a simple Poisson distribution (which has only a single parameter), we could be a bit more clever and fit the results to the expected distribution, thus perhaps developing a bit more accuracy. Also see population estimation, birthday attack, birthday paradox and entropy.
Also see
It is possible to authenticate individual blocks, provided they are large enough to minimize the impact of adding extra authentication data in each block (see block code). One advantage lies in avoiding the alternative of buffering an entire message before it can be authenticated. That can be especially important for real-time (e.g., voice) communications.
Other forms of cryptographic authentication include key authentication for public keys, and source or user authentication, for the authorization to send the message in the first place.
Also see
certification and
certification authority.
Science does not recognize mere authority as sufficient basis for a conclusion, but instead requires that facts and reasoning be exposed for review. The simple use of a name does not automatically create an ad verecundiam fallacy ("Appeal to Awe"). A name can identify a body of work giving the needed facts and the reasoning supporting a scientific conclusion.
Authority tends to hide the basis for drawing conclusions. Authority tends to avoid addressing complaints of false reasoning. Authority tends to hide reasoning and insists that a statement is correct simply because of who made it. A person repeating a conclusion from an authority often has no idea of the reasoning behind it, or what it really means with respect to limits or context.
In contrast, scientific thought exposes the factual basis and
the reasoning, which tells us what the conclusion really means.
Scientific thought is democratic and informs, and ideally gives
everyone the same materials from which to draw factual conclusions,
some of which may be new, strange and disconcerting, but nevertheless
correct.
"As the input moves through successive layers the pattern of 1's generated is amplified and results in an unpredictable avalanche. In the end the final output will have, on average, half 0's and half 1's . . . ." [p.22]-- Feistel, H. 1973. Cryptography and Computer Privacy. Scientific American.228(5):15-23.
Also see mixing, diffusion, overall diffusion, strict avalanche criterion, complete, S-box. Also see the bit changes section of the "Binomial and Poisson Statistics Functions in JavaScript," locally, or @: http://www.ciphersbyritter.com/JAVASCRP/BINOMPOI.HTM#BitChanges.
"For a given transformation to exhibit the avalanche effect, an average of one half of the output bits should change whenever a single input bit is complemented." [p.523]-- Webster, A. and S. Tavares. 1985. On the Design of S-Boxes. Advances in Cryptology-- CRYPTO '85.523-534.
Also see the bit changes section of the "Binomial and Poisson Statistics Functions in JavaScript" page (locally, or @: http://www.ciphersbyritter.com/JAVASCRP/BINOMPOI.HTM#BitChanges).
In normal junctions, the space-charge region (depletion region) between P and N materials is fairly broad, so the extreme fields found in Zener breakdown do not occur. However, a combination of applied voltage, temperature, and random motion may cause a covalent bond to break anyway, in a manner similar to normal diode leakage. When a breakdown does occur, the charge carrier is attracted by the opposing potential and drops through the space-charge region, periodically interacting with covalent bonds there. When the field is sufficiently high, a falling charge carrier may build up enough energy to break another carrier free when it hits. Then both the original and resulting carriers continue to accelerate through the space-charge region, each possibly hitting and breaking many other bonds. The result is a growing avalanche of carriers produced by each single breakdown. The avalanche effect can be seen as a form of amplification and can be huge, for example, 10**8.
In a series of almost forgotten semiconductor physics research papers from the 1950's and 1960's, avalanching breakdown was shown to consist of a multitude of "microplasma" events of perhaps 20uA each. These events are not completely independent, but instead interact, but also have some apparently random component, probably thermal. At least some of the microplasma events seem to have negative dynamic resistance and function like tiny like neon bulbs (and may even emit light). One implication if this is an ability of some avalanching "zener" diodes to directly support small, unsuspected oscillations. A series of very extensive discussions on sci.electronics.design in 1997 (search: "zener oscillation") give experimental details. Both LC tank oscillation and RC relaxation oscillation were demonstrated in practice. Thus, avalanche multiplication, often assumed to be unquestionably "quantum random," actually may have a disturbing amount of predictable structure. True Zener breakdown does not appear to have the same problems, nor does thermal noise, as far as we know. Unfortunately, these "purer" sources may be much smaller than noise from avalanche multiplication.
In contrast to Zener breakdown, which has a negative
temperature
coefficient, avalanche multiplication has a positive temperature
coefficient, like most resistances or
conductors.
Presumably this is due to heat causing increased activity in the
crystal lattice, thus preventing electrons from falling as far
before interacting, thus reducing the probability of breaking
another bond, and reducing the amplification.
In junctions that break down at about 6 volts the temperature
effects tend to cancel.
Also see: "Random Electrical Noise: A Literature Survey"
(locally, or @:
http://www.ciphersbyritter.com/RES/NOISE.HTM).
"A function is balanced if, when all input vectors are equally likely, then all output vectors are equally likely."-- Lloyd, S. 1990. Properties of binary functions. Advances in Cryptology-- EUROCRYPT '90.124-139.
There is some desire to generalize this definition to describe multiple-input functions. (Is a dyadic function "balanced" if, for one value on the first input, all output values can be produced, but for another value on the first input, only some output values are possible?) Presumably a two-input balanced function would be balanced for either input fixed at any value, which would essentially be a Latin square or a Latin square combiner. Also see Balanced Block Mixing. As opposed to bias. Also see Ideal Secrecy and Perfect Secrecy.
Balance is a pervasive requirement in many areas of cryptography; for example:
A mechanism for mixing large block values like those used in block ciphers. A BBM is balanced to avoid leaking information, and is effective in just a single pass, thus avoiding the need for repeated rounds and added hardware. A BBM has no data expansion. A BBM supports the construction of scalable ciphers with large blocks, and can be more efficient, more flexible, and more useful than conventional fixed and smaller designs. (See: ideal mixing, Mixing Cipher, Mixing Cipher design strategy and also the BBM articles, locally, or @: http://www.ciphersbyritter.com/index.html#BBMTech).
Technically, a Balanced Block Mixer is an m-input-port m-output-port mechanism with various properties:
The inverse mixing behaves similarly. Say, for example, we are mixing 64 bytes of message into 64 bytes of result: If we know 63 of the result bytes, we can step through the values of the 64th byte, and get 256 different messages, each of which will produce the 63 bytes we know (a homophonic sort of situation). If the actual messages are random-like and evenly distributed, it will be difficult to know which particular message is implied. The amount of uncertainty we have in the result is reflected in the amount of uncertainty we have about the message.
The basic Balanced Block Mixer is a pair of orthogonal Latin squares. The two input ports affect the rows and columns of both squares, with the selected result in each square being the two output ports. For example, here is a tiny nonlinear "2-bit" or "order 4" BBM:
3 1 2 0 0 3 2 1 30 13 22 01 0 2 1 3 2 1 0 3 = 02 21 10 33 1 3 0 2 1 2 3 0 11 32 03 20 2 0 3 1 3 0 1 2 23 00 31 12
Suppose we wish to mix (1,3); 1 selects the second row up in both squares, and 3 selects the rightmost column, thus selecting (2,0) as the output. Since there is only one occurrence of (2,0) among all entry pairs, this discrete mixing function is reversible, as well as being balanced on both inputs.
In practice, we would probably want to use at least order 16, which can be efficiently stored as an ordinary 256-byte "8-bit" substitution table, one with a particular oLs structure in the data.
One way to use the BBM mixing concept is to develop linear equations for oLs mixing for scaling to various sizes (see my article: Fencing and Mixing Ciphers from 1996 Jan 16). We can do that in the finite field of mod-2 polynomials with an irreducible modulus. So we can easily have similar mixers of 16, 32, 64, 128 and 256 bit port widths, and so on. By using multiple mixers of different size in various connections, we can easily mix blocks of size compatible to existing ciphers, and much larger.
A usually better way to use the BBM mixing concept is to develop small, nonlinear and keyed oLs's for use in FFT-like patterns with 2n ports. It is easy to construct keyed nonlinear orthogonal pairs of Latin squares of arbitrary 4n order as I describe in my articles:
In any FFT-style structure, there is exactly one "path" from any input to any output, and "cancellation" cannot occur. Thus, we can guarantee that any change to any one input must "affect" each and every output. Similarly, each input is equally represented in each output, which is ideal mixing. The resulting wide ideal mixing structure, using small BBM tables as each butterfly operation, is itself a BBM, and is dynamically scalable to virtually arbitrary size.
Mixing has long been a problem in block ciphers. The difficulty of mixing wide block values is one reason most conventional block ciphers are small. But having a small block means that there is not much room to add features like:
Large blocks also have room to hold sufficient uniqueness to support electronic codebook mode, which is not normally appropriate for block ciphers. Large blocks in ECB mode can support secure ciphering without ciphertext expansion, a goal which is very hard to reach in other ways.
When a BBM is implemented in software, the exact same unchanged routine can handle both wide mixing for real operation and narrow "toy" mixing for thorough experimental testing. This supports both scalable operation, and exhaustive testing of the exact code used in actual operation.
In hardware, BBM block throughput or block rate can be independent of block size. Wide blocks can be mixed in the same time as narrow blocks by pipelining each sub-layer of the mixing. That of course makes large blocks far faster per byte than small ones.
Also see All or Nothing Transform, Mixing Cipher, Dynamic Substitution Combiner, and Variable Size Block Cipher.
Also see some of the development sequence:
In a statically-balanced combiner, any particular result
value can be produced by any value on one input, simply by
selecting some appropriate value for the other input. In this way,
knowledge of only the output value provides no information
The common examples of cryptographic combiner, including byte exclusive-OR (mod 2 polynomial addition), byte addition (integer addition mod 256), or other "additive" combining, are perfectly balanced. Unfortunately, these simple combiners are also very weak, being inherently linear and without internal state.
A Latin square combiner
is an example of a statically-balanced
reversible nonlinear combiner with massive internal state.
A Dynamic Substitution
Combiner is an example of a dynamically or
statistically-balanced reversible nonlinear combiner with
substantial internal state.
Each conductor of a balanced line systems should have similar driver output impedances (ideally low), similar wire effects, and similar receiver termination impedances (ideally high). At audio frequencies cables are not transmission lines, so "cable impedance" is not an issue, and the differential receiver need not match either the cable or the driver. When each wire has a similar impedance to ground, external magnetic and electrostatic fields should act on them similarly, producing a common effect on each wire which can "cancel out."
A transformer winding makes a good balanced line driver. In contrast, operational amplifier circuits with direct outputs probably will have only roughly-similar output impedances. Output resistors (e.g., 100 ohms) typically isolate each op amp output from the cable, and any difference will represent driver imbalance to external noise. After being transported, the differential mode signal is taken between the two conductors, thus ignoring common mode noise. A transformer winding makes a good differential receiver and also provides ground loop isolation. Operational amplifier receivers need a common-mode-rejection null adjustment for best performance.
At audio frequencies, the main advantage of balanced line is rejection of AC hum and related power noises. This can be achieved by driving only one line with the desired audio signal, provided both lines are terminated similarly both in the driver and receiver.
At radio frequencies, balanced line also minimizes undesired
signal radiation.
When the current changes in each wire are equal but opposite, they
radiate "out of phase," resulting in cancellation.
This is especially useful in
TEMPEST, but does require that both lines
be actively driven.
0 1 2 3 4 5 6 7 8 9 a b c d e f
0 A B C D E F G H I J K L M N O P
1 Q R S T U V W X Y Z a b c d e f
2 g h i j k l m n o p q r s t u v
3 w x y z 0 1 2 3 4 5 6 7 8 9 + /
use "=" for padding
A bipolar transistor is made by diffusing impurities into
a thin slice of extremely pure single-crystal
semiconductor, such as silicon.
Typically, the collector contact is made at the top surface, and
the emitter contact is made on the bottom.
The base element is essentially a thin film situated between the
collector and emitter plates.
The base current must flow on the film, which is naturally more
resistive than the other thicker elements.
Blum, L., M. Blum and M. Shub. 1983. Comparison of Two Pseudo-Random Number Generators. Advances in Cryptology: CRYPTO '82 Proceedings. Plenum Press: New York.61-78. Blum, L., M. Blum and M. Shub. 1986. A Simple Unpredictable Pseudo-Random Number Generator. SIAM Journal on Computing.
15:364-383.
The BB&S RNG is basically a simple squaring of the current seed (x) modulo a composite (N) composed of two primes (P,Q) of public key size. Primes P and Q must both be congruent to 3 mod 4, but the BB&S articles say that P and Q also must be special primes. The special primes construction apparently has the advantage of controlling the cycle structure of the system, and is part of the BB&S design in the original articles. Unfortunately, the special primes construction generally is not presented in current texts. Instead the texts deceptively describe a simplified version which they nevertheless call BB&S. Readers who do not study referenced articles will assume they know what BB&S said, but they are only partly correct.
Unlike more common RNG's, the BB&S construction is not maximal length, but instead defines systems with multiple cycles, including degenerate, short and long cycles. With large integer factors, state values on short cycles are very rare, but do exist. Short cycles are dangerous with any RNG, because when a RNG sequence begins to repeat, it has just become predictable, despite any theorems to the contrary. Consequently, if we key BB&S by choosing x[0] at random, we may unknowingly select a weak short cycle (a weak key), which would make the sequence predictable as soon as the cycle starts to repeat.
The original BB&S articles lay out the technology to compute the exact length of a long-enough cycle in the BB&S system. Since it can be much easier to verify cycle length than to actually traverse the cycle, this is a practical way to verify that x[0] selects a long-enough cycle. Values of x[0] can be chosen and checked until a long cycle is selected. Modern cryptography insists, to the point of strident intimidation, that such verification is unnecessary. However, the original authors apparently thought it was important enough to include in their work.
The real issue here is not the exposure of a particular weakness in BB&S, since choosing x[0] on a short cycle is very unlikely. But "unlikely" is not the same as "impossible." And if the design goal is to eliminate every known weakness, even extensive math which concludes "that particular weakness is too unlikely to worry about" is beside the point: "unlikely" does not satisfy the goal. Mathematics does not get to impose goals on designers or users.
BB&S is said to be "proven secure" in the sense that if
factoring is hard, then the sequence
is unpredictable. And many people do think that factoring large
composites of public key size is hard.
Yet when a short cycle is selected and used, BB&S is obviously
insecure, and that is a direct contradiction for anyone who imagines
that "proven secure" applies to them.
Just knowing the length of a cycle (by finding sequence repetition)
should be enough to expose the factors.
This is also
evidence that the
assumption
that factoring is hard is not universally true.
Of course, we already know that factoring is not hard
The advantage of the special primes construction apparently is that all "short" (but not degenerate) cycles are "long enough" for use. Thus, we can simply choose x[0] at random, and then easily test that it is not on a degenerate cycle. (Just get some x[0], step x[0] to x[1], save x[1], step x[1] to x[2], then compare x[2] to x[1] and if they are the same, start over.) The result is a guarantee that the selected cycle is "long enough" for use. See the sci.crypt discussion:
It is sometimes said that the special primes construction adds nothing to BB&S, but that really depends more on the goals of the cipher designer than the math. Since BB&S is very slow in comparison to other RNG's, someone selecting BB&S clearly has decided to pay a heavy toll with the expectation of getting an RNG which is "proven secure" in practice. (That actually misrepresents the BB&S proof, which apparently allows weakness to exist provided it is not an easy way to factor N.) The obvious goal is to get a practical RNG which has no known weakness at all.
No mere proof can protect us when we ourselves choose and use a weak key, even if doing that is shown to be statistically very unlikely. And if we do use a weak key, the "proven secure" RNG is clearly insecure, which surely contradicts the motive for using BB&S in the first place. In contrast, simply by using the special primes construction and checking for degenerate cycles, weak keys can be eliminated, at modest expense. Eliminating a known possibility of weakness, even if that possibility is very small, seems entirely consistent with the goal of achieving a practical RNG with no known weakness, even if the result is not an RNG proven to have absolutely no weakness at all.
Some would say that even the special primes construction is
overkill, but without it the so-called "proof of strength" becomes
a mere wish or hope that a short cycle is not being used, and
I see that as a contradiction.
It also might be a cautionary tale as to what mathematical
cryptography currently accepts as
proof, and as to what such "proof" means
in practical use.
For other examples of failure in the current cryptographic wisdom,
see
one time pad, and
AES (as an example of the size of the
permutation family in real conventional
block ciphers), and, of course,
old wives' tale.
Also see
algorithm.
Ordinarily we distinguish mere belief from proven truth, belief thus implying something less than conclusive evidence. In this sense, to believe is to be willing to accept unproven or even unprovable assumptions, such as having faith, or trusting in some machine or property. One issue is whether such assumptions or trust is reasonable in the real world.
Limiting what one can or should believe seems intertwined with freedom of speech and individual rights: Surely, anyone can believe what they want. However, to the extent that we have real responsibilities to others and society at large, unfounded belief can not uphold those obligations. In a seminal essay called "The Ethics of Belief" (circa 1877 and reprinted on the web), William Clifford shows how unfounded belief is insufficient support for decisions of life and death and reputation. Many of us would extend that to business planning (the recent Waltzing with Bears by DeMarco and Lister (see risk management) reprints the first section of "The Ethics of Belief" as an appendix), as well as scientific discussions and claims.
In that point of view, claiming something is true, when one has not investigated the topic and does not know, is ethically wrong, even if the claim it turns out (by pure dumb luck) to be correct. It is not enough to claim something and hope it works out; it is instead necessary to know that the claim is correct before making the claim. The ethical requirement is to have performed an investigation sufficient to expect to know one way or another, and come to a rationally supportable conclusion. While not rising to the level of known fact, belief is something on which reputation rests. Being wrong thus has consequences to reputation, provided the error is in the essence and not mere correctable detail.
This idea of requiring substantial investigation to come to a belief may seem to conflict with the scientific method, in that a scientist seemingly makes a mere claim, which generally stands until shown false. But in reality we expect that claim to be something beyond "mere." We demand that a scientific investigator have put sufficient professional effort into a conclusion before using a scientific podium to spout off. The investigation is what provides an ethical basis for belief, which still may be wrong or (more likely) incomplete.
For example, scientific publication does not mean that all of science supports the described conclusions, which are still just claims made by particular scientists. Showing (not necessarily proving) a claim to be wrong is part of the process of science, not unwarranted intrusion. Showing someone wrong in this context naturally affects reputation, but rarely results in absolute ruin.
The process of experimentation involves making "claims," often to be disproven, but those are clearly labeled hypotheses for experiment, not conclusions for use by others.
In contrast, when we have conclusive evidence of truth
we have knowledge and fact instead of belief.
Facts do not require belief, nor do they respond to voting
or authority.
Clearly, science depends upon knowledge and fact, not personal
beliefs, and it is crucial to know the difference. Also see:
scientific method,
extraordinary claims and
rhetoric.
We can do FWT's in "the bottom panel" at the end of my: "Active Boolean Function Nonlinearity Measurement in JavaScript" page, locally, or @: http://www.ciphersbyritter.com/JAVASCRP/NONLMEAS.HTM.
Here is every bent sequence of length 4, first in {0,1} notation, then in {1,-1} notation, with their FWT results:
bent {0,1} FWT bent {1,-1} FWT
0 0 0 1 1 -1 -1 1 1 1 1 -1 2 2 2 -2
0 0 1 0 1 1 -1 -1 1 1 -1 1 2 -2 2 2
0 1 0 0 1 -1 1 -1 1 -1 1 1 2 2 -2 2
1 0 0 0 1 1 1 1 -1 1 1 1 2 -2 -2 -2
1 1 1 0 3 1 1 -1 -1 -1 -1 1 -2 -2 -2 2
1 1 0 1 3 -1 1 1 -1 -1 1 -1 -2 2 -2 2
1 0 1 1 3 1 -1 1 -1 1 -1 -1 -2 -2 2 -2
0 1 1 1 3 -1 -1 -1 1 -1 -1 -1 -2 2 2 2
These sequences, like all true bent sequences, are not
balanced.
Literature references on this point include:
"Let Qn = {1,-1}n. The defining property of a bent sequence x in Qn is that the Hadamard transform of x has constant magnitude."
"Let y be a bent sequence over {0,-1}n.. . . "The Hamming weight of y is 22k-1 (+ or -) 2k-1."
-- Adams, C. and S. Tavares. 1990. Generating and counting binary bent sequences. IEEE Transactions on Information Theory.IT-36(5):1170-1173.
"Bent functions, except for the fact that they are never balanced, exhibit ideal cryptographic properties."
-- Chee, S., S. Lee, K. Kim. 1994. Semi-bent functions. Advances in Cryptology -- ASIACRYPT '94107-118.
". . . it has often to be considered as a defect from a cryptographic point of view that bent functions are necessarily non-balanced."
-- Dobbertin, H. 1994. Construction of Bent Functions and Balanced Boolean Functions with High Nonlinearity. K.U. Leuven Workshop on Cryptographic Algorithms (Fast Software Encryption).61-74.
"Example 23 (Bent Functions Are Not Balanced). . . ."
-- Seberry, J. and X. Zhang. "Hadamard Matrices, Bent Functions and Cryptography." The University of Wollongong. November 23, 1995.
The zeroth element of the {0,1} FWT is the number of 1's in the sequence.
Here are some bent sequences of length 16:
bent {0,1} 0 1 0 0 0 1 0 0 1 1 0 1 0 0 1 0
FWT 6,-2,2,-2,2,-2,2,2,-2,-2,2,-2,-2,2,-2,-2
bent {1,-1} 1 -1 1 1 1 -1 1 1 -1 -1 1 -1 1 1 -1 1
FWT 4,4,-4,4,-4,4,-4,-4,4,4,-4,4,4,-4,4,4
bent {0,1} 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 0
FWT 6,2,2,-2,-2,2,-2,2,-2,-2,-2,-2,2,2,-2,-2
bent {1,-1} 1 1 -1 1 1 -1 1 1 -1 1 1 1 -1 -1 -1 1
FWT 4,-4,-4,4,4,-4,4,-4,4,4,4,4,-4,-4,4,4
Bent sequences are said to have the highest possible uniform nonlinearity. But, to put this in perspective, recall that we expect a random sequence of 16 bits to have 8 bits different from any particular sequence, linear or otherwise. That is also the maximum possible nonlinearity, and here we actually get a nonlinearity of 6.
There are various more or less complex constructions for these
sequences. In most cryptographic uses, bent sequences are modified
slightly to achieve balance.
Massey, J. 1969. Shift-Register Synthesis and BCH Decoding. IEEE Transactions on Information Theory.IT-15(1):122-127.
Bernoulli trials have a
Binomial distribution.
Transistor biasing is trickier than it might seem from knowing the simple purpose of keeping the device "partly on":
One common biasing approach is to place a particular DC voltage
on the base or gate, and a resistor in the emitter or source lead.
Transistor action then tends to increase current until the emitter
or source has a voltage related to the base or gate, a form of
negative
feedback.
This sets the output bias current, which with a particular pull-up
resistor sets a desired output voltage.
One difficulty with this approach is that it demands an input signal
with lower impedance than the biasing, so that the AC signal will
dominate.
Another issue is that the emitter or source resistor will use some
of the available voltage simply to establish bias, voltage which
then is not available across the device for AC signals. (Also see
transistor self-bias.)
ifWith a bijection an inverse always exists. (Contrast with: involution.)f(x) = y thenf-1(y) = x .
A bijection on bit-strings,
Making random data decompress into language text (necessarily also random in some way) would seem to be difficult. Different classes of plaintext, such as language, database files, program code, or whatever, probably require different compressors or at least different compression models. With respect to language text, such a compressor should decompress random strings into spaced correct words or "word salad." That should complicate attempts to automatically distinguish the original message or block from among other possibilities.
Should bijective compression actually be possible and practical,
the significance would be massive.
Computerized
attacks can succeed only if a correct
deciphering can be recognized automatically.
When incorrect decipherings have structure which is close to
plaintext, a computer may not be able to distinguish them from
success.
If humans skill is needed to read and judge the result of thousands
or millions of brute-force attempts, traversing a keyspace may take
tens of millions of times longer than simple computer scanning.
Making an attack millions of times harder than it was before could
be the difference between complete practical security and almost
no security at all.
Holding an attack loop down to human reading speeds could produce
a massive increase in practical
strength.
n k n-k
P(k,n,p) = ( ) p (1-p)
k
This ideal distribution is produced by evaluating the probability function for all possible k, from 0 to n.
If we have an experiment which we think should produce a binomial distribution, and then repeatedly and systematically find very improbable test values, we may choose to reject the null hypothesis that the experimental distribution is in fact binomial.
Also see the binomial section of my JavaScript page:
"Binomial and Poisson Statistics Functions in JavaScript,"
locally, or @:
http://www.ciphersbyritter.com/JAVASCRP/BINOMPOI.HTM#Binomial,
and my early message on randomness testing
(locally, or @:
http://www.ciphersbyritter.com/NEWS2/94080601.HTM).
The "paradox" is resolved by noting that we have a 1/365 chance
of success for each possible pairing of students, and there
are 253 possible pairs or
combinations of 23 things taken 2 at
a time. (To count the number of pairs, we can choose any of the 23
students as part of the pair, then any of the 22 remaining students
as the other part. But this counts each pair twice, so we have
This problem seems to beg confusion between probability and expected counts, since the correct expectation is often fractional. We can relate the probability of finding a "double" of some birthday (Pd) to the expected number of doubles (Ed) as approximately (equations (5.4) and (5.5) from my article):
Pd = 1 - e-Ed , so Ed = -Ln( 1 - Pd ) .For a success probability of 0.5, the expected doubles are
Ed = -Ln( 1 - 0.5 ) = 0.693147 .
One way to model the overall probability of success is from the
probability of failure
A different model addresses the probability of success for each sample, instead of each pair. For population (N) and samples (s) (equation (1.2) from my article):
Pd(N,s) = 1 - (1-1/N)(1-2/N)..(1-(s-1)/N) ,which gives a success probability for 23 samples of 0.5073.
Sometimes the problem is to find the number of samples (s) needed for a given probability of success in finding doubles (Pd) from a given population (N). Starting with equation (2.5) from my article and substituting (5.5), we get:
s(N,p) = (1 + SQRT(1 - 8N Ln( 1 - Pd ))) / 2 .For the birthday case the number of samples needed from a population of 365 for an even chance of success is:
s(365,0.5) = (1 + SQRT(1 - (8 * 365 * -0.693))) / 2
= (1 + SQRT( 2024.56 )) / 2
= 45.995 / 2
= 22.997 .
This result means that 23 samples should meet with success just a
little more often than the 1 time in 2 demanded by
Also see:
birthday attack,
population estimation,
augmented repetitions,
my Cryptologia "birthday" article: "Estimating Population
from Repetitions in Accumulated Random Samples,"
locally, or @:
http://www.ciphersbyritter.com/ARTS/BIRTHDAY.HTM,
and an example and comparison to various other methods in
the conversation "Birthday Attack Calculations,"
locally, or @:
http://www.ciphersbyritter.com/NEWS4/BIRTHDAY.HTM.
In
digital
electronics,
bits generally are represented by
voltage levels on connected
wires, at a given time.
When the bit-value on a wire changes, some time will elapse
until the wire reaches the new voltage level.
Until that happens, the wire voltage is not a valid digital
level and should not be interpreted as having a particular bit
value. Also see:
logic level.
There are various ways this might be achieved:
"Exact bit-balance can be achieved by accumulating data to a block byte-by-byte, only as long as the block can be balanced by adding appropriate bits at the end."
"We will always add at least one byte of 'balance data' at the end of the data, a byte which will contain both 1's and 0's. Subsequent balance bytes will be either all-1's or all-0's, except for trailing 'padding' bytes, of some balanced particular value. We can thus transparently remove the balance data by stepping from the end of the block, past any padding bytes, past any all-1's or all-0's bytes, and past the first byte containing both 1's and 0's. Padding is needed both to allow balance in special cases, and when the last of the data does not completely fill the last block."
"This method has a minimum expansion of one byte per block, given perfectly balanced binary data. ASCII text may expand by as much as 1/3, which could be greatly reduced with a pre-processing data compression step."
(My article "A Keyed Shuffling System for Block Cipher Cryptography," illustrates key hashing, a nonlinearized RNG, and byte shuffling. We would do a similar thing for bit-permutation, but with a larger and wider RNG and shuffling bits instead of bytes. See either locally, or @: http://www.ciphersbyritter.com/KEYSHUF.HTM).
Ciphering by bit-transposition has unusual resistance to known plaintext attack because many, many different bit-permutations of the plaintext data will each produce exactly the same ciphertext result. Consequently, even knowing both the plaintext and the associated ciphertext does not reveal the shuffling sequence. Bit-permutation thus joins double-shuffling in hiding the shuffling sequence, which is important when we cannot guarantee the strength of that sequence (as we generally cannot).
A transposition cipher is "dynamic" when it "never" permutes two blocks in the same way. Dynamic bit-permutation ciphers can be a very competitive practical alternative to both stream ciphers and conventional block ciphers. Although bit-shuffling may be slower, it has a clearer and more believable source of strength than the other alternatives.
Also see my article: "Dynamic Transposition Revisited Again" (40K)
(locally, or @:
http://www.ciphersbyritter.com/ARTS/DYNTRAGN.HTM),
the Dynamic Transposition Ciphering conversation (730K)
(locally, or @:
http://www.ciphersbyritter.com/NEWS5/REDYNTRN.HTM).
Digital logic IC's are wildly successful examples of hardware black box components. Externally, they perform useful digital functions, and in most cases, digital designers need not think about the internal construction. Internally, however, the "digital" devices use analog transistors to effect digital operation.
An example of black box software design is a subroutine or Structured Programming module, where all interaction with the caller is in the form of parameters. The module uses the given resources, does what it needs, completes, and returns to the caller. As long as the module does what we want, there is no need to know how the module works, so we can avoid dealing with internal complexity at the lower level. And when the module does not work, it can be debugged in a minimal environment which avoids most of the complexity of the larger system, thus making debugging far easier.
In a discussion of block cipher concepts, cryptography implicitly uses definition (2), because it is the accumulation of multiple characters (and the resulting larger ciphering alphabet) which is characteristic of conventional block ciphers. A one-element "block" simply cannot exhibit the various block issues (such as mixing, diffusion, padding and expansion) that we see in a real block cipher, and so fails to model both the innovation and the resulting problems. Similar effects occur when any scalable model is simplified beyond reason. (See: scientific method.) It is also possible to cipher blocks of dynamically selectable size, or even fine-grained variable size.
All real block ciphers are in fact streamed to handle more than one block of data. The actual ciphering might be seen as a stream meta-ciphering using a block cipher transformation. The point of this is not to provide a convenient academic way to contradict any possible response to a question of "stream or block," but instead to identify the origin of various ciphering properties and problems (see: a cipher taxonomy).
It is not possible to block-cipher just a single bit or byte of a block. (When that is possible, we may be dealing with a stream cipher.) If individual bytes really must be block-ciphered, it will be necessary to fill out each block with padding in some way that allows the padding to be distinguished from the actual plaintext data after deciphering.
Partitioning an arbitrary stream of into fixed-size blocks generally means the ciphertext handling must support data expansion, if only by one block. But handling even minimal data expansion may be difficult in some systems.
The distinction between "block" and "stream" corresponds to the common distinction between "block" and "character" device drivers in operating systems. This is the need to accumulate multiple elements and/or pad to a full block before a single operation, versus the ability to operate without delay but requiring multiple operations. This is a common, practical distinction in data processing and data communications.
A competing interpretation of block versus stream operation
seems to be based on transformation "re-use":
In that interpretation, block ciphering is about having a complex
transformation, which thus directly supports re-use (providing each
plaintext block "never" re-occurs).
In that same interpretation, stream ciphering is about supporting
transformation re-use by changing the transformation itself.
These effects do of course exist (although in my view they are not
the most fundamental issues for analysis or design).
But that interpretation also allows both qualities to exist
simultaneously at the same level of design, and so does not provide
the full
analytical benefits of a true
logical
dichotomy.
There is some background:
A conventional block cipher is a transformation between all possible plaintext block values and all possible ciphertext block values, and is thus an emulated simple substitution on huge block-wide values. Within a particular block size, both plaintext and ciphertext have the same set of possible values, and when the ciphertext values have the same ordering as the plaintext, ciphering is obviously ineffective. So effective ciphering depends upon re-arranging the ciphertext values from the plaintext ordering, and this is a permutation of the plaintext values. A conventional block cipher is keyed by constructing a particular permutation of ciphertext values for each key.
The mathematical model of a conventional block cipher is bijection, and the set of all possible block values is the alphabet. In cryptography, the bijection model corresponds to an invertible table having a storage element associated with each possible alphabet value. Since each different table represents a different permutation of the alphabet, the number of possible tables is the factorial of the alphabet size.
In particular, a conventional block cipher with a
In an ideal conventional block cipher, changing even a single bit of the input block will change all bits of the ciphertext result, each with independent probability 0.5. This means that about half of the bits in the output will change for any different input block, even for differences of just one bit. This is overall diffusion and is present in a block cipher, but usually not in a stream cipher. Data diffusion is a simple consequence of the keyed invertible simple substitution nature of the ideal block cipher.
Improper diffusion of data throughout a block cipher can have serious strength implications. One of the functions of data diffusion is to hide the different effects of different internal components. If these effects are not in fact hidden, it may be possible to attack each component separately, and break the whole cipher fairly easily.
A large message can be ciphered by partitioning the plaintext into blocks of a size which can be ciphered. This essentially creates a stream meta-cipher which repeatedly uses the same block cipher transformation. Of course, it is also possible to re-key the block cipher for each and every block ciphered, but this is usually expensive in terms of computation and normally unnecessary.
A message of arbitrary size can always be partitioned into some number of whole blocks, with possibly some space remaining in the final block. Since partial blocks cannot be ciphered, some random padding can be introduced to fill out the last block, and this naturally expands the ciphertext. In this case it may also be necessary to introduce some sort of structure which will indicate the number of valid bytes in the last block.
Proposals for using a block cipher supposedly without data expansion may involve creating a tiny stream cipher for the last block. One scheme is to re-encipher the ciphertext of the preceding block, and use the result as the confusion sequence. Of course, the cipher designer still needs to address the situation of files which are so short that they have no preceding block. Because the one-block version is in fact a stream cipher, we must be very careful to never re-use a confusion sequence. But when we only have one block, there is no prior block to change as a result of the data. In this case, ciphering several very short files could expose those files quickly. Furthermore, it is dangerous to encipher a CRC value in such a block, because exclusive-OR enciphering is transparent to the field of mod 2 polynomials in which the CRC operates. Doing this could allow an opponent to adjust the message CRC in a known way, thus avoiding authentication exposure.
Another proposal for eliminating data expansion consists of ciphering blocks until the last short block, then re-positioning the ciphering window to end at the last of the data, thus re-ciphering part of the prior block. This is a form of chaining and establishes a sequentiality requirement which requires that the last block be deciphered before the next-to-the-last block. Or we can make enciphering inconvenient and deciphering easy, but one way will be a problem. And this approach cannot handle very short messages: its minimum size is one block. Yet any general-purpose ciphering routine will encounter short messages. Even worse, if we have a short message, we still need to somehow indicate the correct length of the message, and this must expand the message, as we saw before. Thus, overall, this seems a somewhat dubious technique.
On the other hand, it does show a way to chain blocks for authentication in a large-block cipher: We start out by enciphering the data in the first block. Then we position the next ciphering to start inside the ciphertext of the previous block. Of course this would mean that we would have to decipher the message in reverse order, but it would also propagate any ciphertext changes through the end of the message. So if we add an authentication field at the end of the message (a keyed value known on both ends), and that value is recovered upon deciphering (this will be the first block deciphered) we can authenticate the whole message. But we still need to handle the last block padding problem and possibly also the short message problem.
Ciphering raw plaintext data can be dangerous when the cipher has a relatively small block size. Language plaintext has a strong, biased distribution of symbols and ciphering raw plaintext would effectively reduce the number of possible plaintext blocks. Worse, some plaintexts would be vastly more probable than others, and if some known plaintext were available, the most-frequent blocks might already be known. In this way, small blocks can be vulnerable to classic codebook attacks which build up the ciphertext equivalents for many of the plaintext phrases. This sort of attack confronts a particular block size, and for these attacks Triple-DES is no stronger than simple DES, because they both have the same block size.
The usual way of avoiding these problems is to randomize the plaintext block with an operating mode such as CBC. This can ensure that the plaintext data which is actually ciphered is evenly distributed across all possible block values. However, this also requires an IV which thus expands the ciphertext.
Worse, a block scrambling or randomization function like CBC is public, not private. It is easily reversed to check overall language statistics and thus distinguish the tiny fraction of brute force results which produce potentially valid plaintext blocks. This directly supports brute force attack, as well as any attack in which brute force is a final part. One alternative is to use a preliminary cipher to randomize the data instead of an exposed function. Pre-ciphering prevents easy plaintext discrimination; this is multiple ciphering, leading in the direction Shannon's Ideal Secrecy.
Another approach (to using the full block data space) is to apply data compression to the plaintext before enciphering. If this is to be used instead of plaintext randomization, the designer must be very careful that the data compression does not contain regular features which could be exploited by the opponents.
An alternate approach is to use blocks of sufficient size
for them to be expected to have a substantial amount of uniqueness or
entropy.
If we expect plaintext to have about one bit of
entropy per byte of text, we might want a block size of at
least 64 bytes before we stop worrying about an uneven
distribution of plaintext blocks. This is now a practical
block size.
It may be helpful to recall a range of published distinctions between "stream cipher" and "block cipher" (and if anyone has any earlier references, please send them along). Note that open discussion was notably muted during the Cold War, especially during the 50's, 60's and 70's. I see the earlier definitions as attempts at describing an existing codification of knowledge, which was at the time tightly held but nevertheless still well-developed.
The intent of classification is understanding and use. Accordingly, it is up to the analyst or student to "see" a cipher in the appropriate context, and it is often useful to consider a cipher to be a hierarchy of ciphering techniques. For example, it is extremely rare for a block cipher to encipher exactly one block. But when that same cipher is re-used again that seems a lot like repeated substitution, which is the basis for stream ciphering. (Of course repeatedly using the same small substitution would be ineffective, but if we attempt to classify ciphers by their effectiveness, we start out assuming what we are trying to understand or prove.) So an alternate way to "see" the re-use of a block cipher is as a higher-level stream "meta-cipher" which uses a block cipher component. But that is exactly what we call "block ciphering."
Some academics insist upon distinguishing stream versus block ciphering by saying that block ciphers have no retained state between blocks, while stream ciphers do. Simply saying that, however, does not make it true, and only one example is needed to expose the distinction as false and misleading. A good example for that is my own Dynamic Transposition cipher, which is a block cipher in that it requires a full block of data before processing can begin, yet also retains state between blocks. So if DT is not a block cipher, what is it? We would hope to define only two categories, not four or more. Note that Lempel (1979, above) explicitly says that transposition is a block cipher. Again, see: a cipher taxonomy to see one approach on how ciphers relate.
Another issue is that stream ciphers can be implemented in ways that accumulate a block of data before ciphering. Internally, such systems generally have a streaming system which traverse the block element-by-element, perhaps multiple times. It is important to see beyond an apparent block requirement stemming from data manipulation only, which thus contributes no strength, to the internal ciphers which (hopefully) do provide strength.
It is also possible to have multiple stream ciphers work on the
same "block," and then we do have a legitimate "block cipher"
(or perhaps a "block meta-cipher") formed by
multiple encryption of stream
ciphers.
(Although multiple ciphering with
additive stream ciphers is
usually unhelpful, most conventional block ciphers are in fact
multiple encryptions internally, so internal multiple ciphering is
hardly a crazy approach.)
But if we want to understand strength, we still need to consider the
fundamental ciphering operations which, here, are streams.
Simply making something work like a block cipher does not give it
the same
model as a conventional
block cipher, and so does not provide for analysis at that level.
In the end, we might see such a construction as a block meta-cipher
composed of internal stream ciphers.
Currently, there are three main block cipher models:
The common academic model of a block cipher is the mathematical bijection, which cryptography calls simple substitution. In practice, such a cipher requires a table far too large to instantiate, and so the actual cipher only emulates a huge, keyed table.
One advantage of the bijection model is that specific, measurable mathematical things can be said about a bijection. Of course exactly the same things also can be said about simple substitution, and the field of ciphering is cryptography, not mathematics.
One problem with the bijection model is that it does not attempt to establish a dichotomy. In the bijection model, "block cipher" just another label in a presumably endless sequence of such labels, each representing a distinct ciphering approach. Consequently, the bijection model makes a poor contribution toward an overall cipher taxonomy useful in the analysis of arbitrary cipher designs.
Another problem with the bijection model is that it establishes yet another term of art: The word block is well known, understood, and rarely disputed. The word cipher is also widely agreed upon. The phrase "block cipher" obviously includes nothing about bijections. So to define "block cipher" in terms of bijections is to take the phrase far beyond the simple meaning of the terms. We could scarcely describe this as anything other than misleading.
Yet another problem with the bijection model, is that, since it presumes to define "block cipher" as a particular type of cipher, what are we to do with ciphers which operate on blocks and yet do not function as bijections (e.g., transposition cipher)? No longer are ciphers related by their proper description. This is even more misleading.
Ultimately, the problem with the bijection model is not the model itself: The model is what it is because substitution is what it is. The problem is the insistence by some academics that this is the only valid model for a "block cipher." A much better choice for the bijection model is the phrase: "conventional block cipher."
The static state model puts forth the proposition that stream ciphers dynamically change their internal state, whereas block ciphers do not. Typically, there is also an understanding that the bijective block cipher model applies.
One problem with the static state definition is again in the name itself: The phrase "block cipher" does not include the word "state." To use the phrase "block cipher" for a property of state is to create yet another term of art, preempting the obvious meaning of the phrase "block cipher," and preventing related block-like ciphers from having similar descriptions, thus misleading both instructor and student.
Another problem with the static state model is that we can build stream-like ciphers which do not change their internal state (in fact, I claim we stream a substitution table when we repeatedly use it across a message, just like we stream DES). Similarly, we can build block-like ciphers which do change their internal state (I usually offer my Dynamic Transposition cipher as an example, but so is a block cipher built from multiple internal stream operations). So if we accept the static state model, what do we call those ciphers which function on blocks, and yet do change state? Why preempt the well-known terms "block" or "stream" for the fundamentally different properties of internal state?
Ultimately, what insight does state classification provide that warrants usurping the obvious descriptive phrases "block cipher" and "stream cipher" instead of thinking up something appropriate?
The original mechanized ciphers were stream ciphers, starting with the Vernam cipher of 1919. The term "block cipher" may have been introduced in the secret world of government security to draw a practical distinction between the well-known stream concept, and the newer designs that operated on a block. (That would have been in the 50's, 60's or even 70's; hopefully, someone will either confirm this or correct it.) In the multiple-element model, a block cipher requires the accumulation of more than one data element before ciphering can begin.
One advantage of the multiple-element definition is that it forms an easy dichotomy with the definition of a stream cipher as a cipher which does not require such accumulation. Also note that this is no mere semantic issue, but is instead just one representation of a broader concept of "one versus many" which rises repeatedly in computing practice, including:
The various consequences of the single-element versus multiple-element dichotomy are well known: When blocks are accumulated from individual elements, storage is required for that accumulation, and time is required as well, which can imply latency. In contrast, when elements need not be accumulated, there need be neither storage nor latency, but the total overhead may be greater. While latency probably is not much of an issue for email ciphering, latency can be significant for real-time streams like music or video, or interactive handshake protocols. Overhead is, of course, a significant issue in system design.
To see how the multiple-element block cipher definition works, consider the following:
Strangely, a degenerate block is exactly the same as a
degenerate sequence: just one element.
In neither case does that element teach about the larger object: a
one-element block does not have diffusion between elements, and a
one-element stream does not have correlation between elements.
(Similarly, is a single electronic wire with a fixed voltage
one-wire "parallel" or one-value "serial"?)
From this we conclude that the most important aspects of
cryptographic (and electronic) design and analysis simply
do not exist as a single element, so it is inappropriate
to either use or judge a model at that level.
A block size of n bits typically means that 2n different codewords or "block values" can occur. An (n,k) block code uses those 2n codewords to represent the equal or smaller count of 2k different messages. Thus, a 64-bit block cipher normally encodes 64 plaintext bits into 64 ciphertext bits as a simple (64,64) code. But if 16 input bits are reserved for other use, the coding expands 48 plaintext bits into 64 ciphertext bits, so we have a (64,48) code.
The normal use for extra codewords is to implement some form of error detection and/or error correction. This overhead is not normally called "inefficient coding," but is instead a simple cost of providing improved quality. In cryptography, the extra code words may be used to add security or improve performance by implementing:
Also see
8b10b and
huge block cipher advantages.
NOT as complementation, indicated by ' (single quote) 0' = 1 1' = 0 OR as addition, denoted "+" 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1 AND as multiplication, denoted "*" as usual 0 * 0 = 0 0 * 1 = 0 1 * 0 = 0 1 * 1 = 1 XOR a useful but not essential operation 0 XOR 0 = 0 0 XOR 1 = 1 1 XOR 0 = 1 1 XOR 1 = 0
1. Addition is Commutative: x + y = y + x 2. Addition is Associative: x + (y + z) = (x + y) + z 3. The Additive Identity: 0 + x = x 4. The Additive Inverse: x + x' = 1 5. Multiplication is Associative: x(yz) = (xy)z 6. Multiplication is Distributive: x(y + z) = xy + xz 7. Multiplication is Commutative: xy = yx 8. The Multiplicative Identity: 1 * x = x Other: 1 + x = x x + x = x x * x = x (x')' = x DeMorgan's Laws: (x + y)' = x'y' (xy)' = x' + y' XOR: x XOR y = xy' + x'y (x XOR y)' = xy + x'y'
Typically computed as the fast Walsh-Hadamard transform (FWT) of the function being measured. For more details, see the topic unexpected distance and the "Active Boolean Function Nonlinearity Measurement in JavaScript" page (locally, or @: http://www.ciphersbyritter.com/JAVASCRP/NONLMEAS.HTM).
Note that the FWT computation is done for efficiency only. It is wholly practical to compute the nonlinearity of short sequences by hand. It is only necessary to manually compare each bit of the measured sequence to each bit of an affine Boolean function. That gives us the distance from that particular function, and we repeat that process for every possible affine Boolean function of the measurement length.
Especially useful in S-box analysis, where the nonlinearity for the table is often taken to be the minimum of the nonlinearity values computed for each output bit.
Also see my articles:
The original definition is for linear mapping theta,
"The minimum total Hamming weight [wh] of (a,theta(a)) is a measure for the minimum amount of diffusion that is realized by a linear mapping."Definition 6.10 The branch number B of a linear mapping theta is given by
B(theta) = min(wh(a),wh(theta(a))), for a<>0
-- Daemen, J. 1995. Cipher and Hash Function Design, Strategies Based on Linear and Differential Cryptanalysis. Thesis. Section 6.8.1.
Branch number specifically applies only to a linear mixing. Actually, even that is not quite right: the real problem is keying, not nonlinearity (although in practice, keying may imply nonlinearity). To the extent that we can experimentally traverse the input block, a branch number certainly can be developed for a nonlinear mixer.
But while any particular mixer can have a branch number, a keyed mixer will have a branch number for every possible key. Moreover, we would expect the minimum over all those nonlinear mixings to be very low, just like the minimum strength of any cipher over all possible keys (the opponent trying just one key) is also very low. Yet we do not attempt to characterize ciphers by their minimum strength over all possible keys.
No keyed structure can be properly characterized by the extrema over all keys. When we have random variables such as keying, we should be thinking of the distribution of values, and the probability of encountering extreme values. And that is not branch number.
More insight is available in the description of the SQUARE cipher:
"It is intuitively clear that both linear and differential trails would beneft from a multiplication polynomial that could limit the number of nonzero terms in input and output difference (and selection) polynomials. This is exactly what we want to avoid by choosing a polynomial with a high diffusion power, expressed by the so-called branch number.
Let wh(a) denote the Hamming weight of a vector, i.e., the number of nonzero components in that vector." [Normally, Hamming weight applies to bits, but here it is being used for bytes./tfr] "Applied to a state a, a difference pattern a' or a selection pattern u, this corresponds to the number of non-zero bytes. In [2] the branch number B of an invertible linear mapping was introduced as
B(theta) = for a<>0, min wh(a)+ wh(a)This implies that the sum of the Hamming weights of a pair of input and output difference patterns (or selection patterns) to theta is at least B. It can easily be shown that B is a lower bound for the number of active S-boxes in two consecutive rounds of a linear or differential trail.""In [15] it was shown how a linear mapping over GF(2m)n with optimal B
(B = n + 1) can be constructed from a maximum distance separable code."
-- Daemen, J., L. Knudsen and V. Rijmen." 1997. "The Block Cipher {SQUARE}." Fast Software Encryption, Lecture Notes in Computer ScienceVol. 1267:149-165. Section 4.
In the wide trail strategy, branch number applies to a particular unkeyed and linear diffusion mechanism. In the SQUARE design, branch number also applies to a particular unkeyed and linear polynomial multiplication. So branch number might also describe the simple linear form of Balanced Block Mixing used in Mixing Ciphers. But linear BBM's apparently do not have an optimal branch number (over all possible input data changes), although in most cases they do have a good branch, and are dynamically scalable to both tiny and huge blocks on a block-by-block basis.
Instead of linear diffusion, it should be "intuitively obvious" that nonlinear diffusion would be a better choice for a cipher, if such could be obtained with good quality at reasonable cost. Nonlinear Balanced Block Mixing occurs when the butterfly functions are keyed. Keying is easily accomplished by constructing appropriate orthogonal Latin squares using the fast checkerboard construction. But "branch number" does not apply to these keyed nonlinear constructions.
The "optimal" branch value for the MDS codes in the SQUARE
design is given as
From the Handbook of Applied Cryptography:
"1.23 Definition. An encryption scheme is said to be breakable if a third party, without prior knowledge of the key pair (e,d), can systematically recover plaintext from corresponding ciphertext in some appropriate time frame." [p.14]
"Breaking an information security service (which often involves more than simply encryption) implies defeating the objective of the intended service." [p.15]
The term "break" seems to be a term of art in academic cryptanalysis, where it apparently means a successful attack which takes less effort than brute force (or the cipher design strength, if that is less), even if the effort required is impractical, and even if the attack is easily prevented at the cipher system level. This meaning of the term "break" can be seriously misleading because, in English, "break" means "to render unusable" or "to destroy," and not just "to make a little more dubious."
The academic meaning of "break" is also controversial, as it can be used as a slander to demean both cipher and designer without a clear analysis of whether the attack really succeeds. And even if the attack does succeed, the question is whether it actually reveals data or key material, thus making the cipher dangerous for use in practice.
Everyone understands that a cipher is "broken" when the information in a message can be extracted without the key, or when the key itself can be recovered, with less effort than the design strength. And a break is particularly significant when the work involved need not be repeated on every message. But when the amount of work involved is impractical, the situation is best described as a theoretical or academic break. The concept of an "academic break" is especially an issue for ciphers with a very large keyspace, in which case it is perfectly possible for a cipher with an academic break to be more secure than ciphers with lesser goals which have no "break." It is also at least conceivable that an attack can be surprising and insightful and, thus, "successful" even if it takes more effort than the design strength, which would be no form of "break" at all.
In my view, a documented flaw in a cipher, such as some statistic
which
distinguishes a practical cipher from
some
model, but without an
attack which recovers data or key, at most
should be described as a "theoretical" or "certificational"
weakness.
Unfortunately, even a problem which has no impact on security
is often promoted (improperly, in my view) to the term
academic break or even "break"
itself.
Even when the key length of a cipher is sufficient to prevent brute force attack, that key will be far too small to produce every possible plaintext from a given ciphertext (see Shannon's Perfect Secrecy). Combined with the fact that language is redundant, this means that very few of the decipherings will be words in proper form. So most wrong keys could be identified immediately.
On the other hand, recognizing plaintext may not be easy.
If the plaintext itself
Brute force is the obvious way to attack a cipher, and the way
most ciphers can be attacked, so ciphers are designed to have a
large enough
keyspace to make this much too expensive
to succeed in practice.
Normally, the design
strength of a cipher is based on the
cost of a brute-force attack.
In most FFT diagrams, the input elements are shown in a vertical
column at the left, and the result elements in a vertical column on
the right.
Lines represent signal flow from left to right.
There are two computations, and each requires input from each of
the two selected elements.
In an "in place" FFT, the results conveniently go back into the
same positions as the input elements.
So we have two horizontal lines between the same elements, and
two diagonal lines going to each "other" element, which cross.
This is the "hourglass" shape or "butterfly wings" on edge.
A source of stable power is the most important requirement for any electronic device. In particular, digital logic functions can only be trusted to produce correct results if their power is kept within specified limits. It is up to the designer to provide sufficient correct power and guarantee that it remain within limits despite whatever else is going on.
Most digital logic families use "totem pole" outputs, which means they have a transistor from Vcc or Vdd (power) to the output pin, and another transistor from the output pin to Vss (ground). Normally, only one transistor is ON, but as the output signal passes from one state to another, transiently, both transistors can be ON, leading to short, high-current pulses on both the Vcc and ground rails. These current pulses are essentially RF energy, and can and do produce ringing on power lines and a general increase in system noise. The pulses are also strong enough to potentially change both the Vcc and ground voltage levels in the power distribution system near the device, which can affect nearby logic and operation. Typically this occurs at some random moment when the worst conditions coincide to cause a logic fault. To avoid that, we want to bypass the current pulse away from the power system in general, so other devices are not affected.
For many years, a typical rule of thumb was to use a 0.1uF ceramic disc for each supply at each bipolar chip, plus a 1uF tantalum for every 8 chips. That may still be a good formula for slower analog chips and older digital logic like LSTTL. But as chip speed has increased, bypassing has become more complex.
Ideally, a bypass capacitor will be connected from every supply pin to the ground pin right at each chip. Ideally, there will be no lead left on either end of the capacitor: not 1/4 inch, not 1/8 inch, which is one reason why surface-mount capacitors are desirable. Ideally, any necessary lead will be wide, flat copper. But the ideal system is a goal, not reality.
One of the effects of higher system speeds is that normal system operation now covers the resonant frequency of the bypass capacitors. Unfortunately, this resonance is not a fixed constant, even for a particular type of part. Bypass resonance is instead a circuit condition, involving the reactance of the closest bypass capacitor, plus the inductance in power connections, and reactance in other bypass capacitors. Although it is virtually impossible to remove inductance from PC-board traces, it is possible to use whole copper layers as "power planes" for power distribution.
Resonance means that an impulse causes "ringing," in which energy is propagated back and forth between inductance and capacitance until it finally dissipates in circuit resistance or is radiated away, but the resulting signal from many devices may appear as increased system noise.
Resonance would actually seem to be the ideal bypass situation, in that a resonant bypass presents the minimum impedance to ground. But it does that only at one frequency; lower and higher frequencies are less rejected. It seems quite impractical to tune for resonance with the "random" pulses occurring in complex logic. And, above resonance, inductance dominates and then higher-frequency noise and pulses are more able to affect the rest of the system.
Another approach has been to use various bypass capacitors, typically 0.01uF and 0.1uF in parallel, "sprinkled around" the PC layout. The idea was that self-resonance in any one bypass capacitor would be hidden by the other capacitor of different value and, thus, different resonant frequency. Alone, either a 0.01uF or a 0.1uF cap may do an effective job. However, recent modeling indicated, and experimentation has confirmed, that using both together can be substantially worse than using either value alone.
The inherent limitation in bypassing is that the normal bypass process is not "lossy" or dissipative. Pulse energy can be stored in the inductance of short leads or PC-board traces, and then "ring" in resonance with the usual ceramic bypass capacitors. Having many bypass caps often leads to complex RF filter-like structures which just pass the ringing energy around. An alternative is the wide use of tantalum bypass capacitors, since tantalum becomes increasingly lossy at higher frequencies and will dissipate pulse energy.
Several approaches seem reasonable:
If we know the capacitance C in Farads and the frequency f in Hertz, the capacitive reactance XC in Ohms is:
XC = 1 / (2 Pi f C) Pi = 3.14159...Capacitors in parallel are additive. Two capacitors in series have a total capacitance which is the product of the capacitances divided by their sum.
A capacitor is typically two conductive "plates" or metal foils separated by a thin insulator or dielectric, such as air, paper, or ceramic. An electron charge on one plate attracts the opposite charge on the other plate, thus "storing" charge. A capacitor can be used to collect a small current over long time, and then release a high current in a short pulse, as used in a camera strobe or "flash."
The simple physical model of a component which is a simple capacitance and nothing else works well at low frequencies and moderate impedances. But at RF frequencies and modern digital rates, there is no "pure" capacitance. Instead, each capacitance has a series inductance that often does affect the larger circuit. See bypass.
Also see
inductor and
resistor.
The earliest definition of "cascade cipher" I know (1983) does not mention key independence:
"A Cascade Cipher (CC) is defined as a concatenation of block cipher systems, thereafter referred to as its stages; the plaintext of the CC plays the role of the plaintext of the first stage, the ciphertext of the i-th stage is the plaintext of the (i+1)-st stage and the ciphertext of the last stage is the ciphertext of the CC.A modern academic definition is:"We assume that the plaintext and ciphertext of each stage consists of m bits, the key of each stage consists of k bits and there are t stages in the cascade."
[Note the lack of the term "independent."/tfr]
-- Even, S. and O. Goldreich. 1983. "On the power of cascade ciphers." Advances in Cryptology: Proceedings of Crypto '83.43-50.
"[The] Product of several ciphers is also a product cipher, such a design is sometimes called a cascade cipher."[Note the lack of anything like "with independent keys."/tfr]
-- Biryukov, Alex. (Faculty of Mathematics and Computer Science, The Weizmann Institute of Science.) 2000. Methods of Cryptanalysis. "Lecture 1. Introduction to Cryptanalysis."
Similarly, the term
"product encipherment" is defined in
Shannon 1949 (and is quoted here under
Algebra of Secrecy Systems)
as the use of one cipher, then another with independent keys.
Thus, the independent key terminology was defined in cryptography
over half a century ago, and probably 34 years before
"cascade ciphering" was defined for the same idea without the
key independence requirement.
Both terms are commonly and legitimately confused in use.
Anyone using the terms "cascade ciphering" or "product ciphering"
would be well advised to explicitly state what the term is supposed
to mean, or to not complain when someone takes it to mean something
else.
Particularly inappropriate as a description of
multiple encryption, because a
physical "chain" is is only as strong as the weakest link,
while a sequence of ciphers is as strong as the strongest link.
Chance
Chaos
In physics, the state of an analog physical system cannot be fully measured, which always leaves some remaining uncertainty to be magnified on subsequent steps. And, in many cases, a physical system may be slightly affected by thermal noise and thus continue to accumulate new information into its state.
In a
computer, the state of the
digital
system is explicit and
complete, and there is no uncertainty. No noise is accumulated.
All operations are completely
deterministic. This means that, in a
computer, even a "chaotic" computation is completely predictable
and repeatable.
One way to construct a larger square is to take some Latin square and replace each of the symbols or elements with a full Latin square. By giving the replacement squares different symbol sets, we can arrange for symbols to be unique in each row and column, and so produce a Latin square of larger size.
If we consider squares with numeric symbols, we can give each replacement square an offset value, which is itself determined by a Latin square. We can obtain offset values by multiplying the elements of a square by its order:
0 1 2 3 0 4 8 12 1 2 3 0 * 4 = 4 8 12 0 2 3 0 1 8 12 0 4 3 0 1 2 12 0 4 8To simplify the example, we can use the same original square for all of the replacement squares:
0+ 0 1 2 3 4+ 0 1 2 3 8+ 0 1 2 3 12+ 0 1 2 3
1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0
2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1
3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2
4+ 0 1 2 3 8+ 0 1 2 3 12+ 0 1 2 3 0+ 0 1 2 3
1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0
2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1
3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2
8+ 0 1 2 3 12+ 0 1 2 3 0+ 0 1 2 3 4+ 0 1 2 3
1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0
2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1
3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2
12+ 0 1 2 3 0+ 0 1 2 3 4+ 0 1 2 3 8+ 0 1 2 3
1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0
2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1
3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2
which produces the order-16 square:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 0 5 6 7 4 9 10 11 8 13 14 15 12 2 3 0 1 6 7 4 5 10 11 8 9 14 15 12 13 3 0 1 2 7 4 5 6 11 8 9 10 15 12 13 14 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 5 6 7 4 9 10 11 8 13 14 15 12 1 2 3 0 6 7 4 5 10 11 8 9 14 15 12 13 2 3 0 1 7 4 5 6 11 8 9 10 15 12 13 14 3 0 1 2 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 9 10 11 8 13 14 15 12 1 2 3 0 5 6 7 4 10 11 8 9 14 15 12 13 2 3 0 1 6 7 4 5 11 8 9 10 15 12 13 14 3 0 1 2 7 4 5 6 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 13 14 15 12 1 2 3 0 5 6 7 4 9 10 11 8 14 15 12 13 2 3 0 1 6 7 4 5 10 11 8 9 15 12 13 14 3 0 1 2 7 4 5 6 11 8 9 10
Clearly, this Latin square exhibits massive structure at all
levels, but this is just a simple example.
In practice we would create and use a different
There are 576 Latin squares of order 4, any one of which can be
used as any of the 16 replacement squares.
The offset square is another
The construction is also applicable to orthogonal Latin squares. See my articles:
Improved checksums (e.g., Fletcher's checksums) include both data values and data positions and may perform within a factor of 2 of CRC. One advantage of a true summation checksum is a minimal computation overhead in software (in hardware, a CRC is almost always smaller and faster). Another advantage is that when header values are changed in transit, a summation checksum is easily updated, whereas a CRC update is more complex and many implementations will simply re-scan the full data to get the new CRC.
The term "checksum" is sometimes applied to any form of error
detection, including more sophisticated codes like CRC.
In the usual case, "many" random samples are counted by category or separated into value-range "bins." The reference distribution gives us the the number of values to expect in each bin. Then we compute a X2 test statistic related to the difference between the distributions:
X2 = SUM( SQR(Observed[i] - Expected[i]) / Expected[i] )
("SQR" is the squaring function, and we require that each expectation not be zero.) Then we use a tabulation of chi-square statistic values to look up the probability that a particular X2 value or lower (in the c.d.f.) would occur by random sampling if both distributions were the same. The statistic also depends upon the "degrees of freedom," which is almost always one less than the final number of bins. See the chi-square section of the "Normal, Chi-Square and Kolmogorov-Smirnov Statistics Functions in JavaScript" page (locally, or @: http://www.ciphersbyritter.com/JAVASCRP/NORMCHIK.HTM#ChiSquare).
The c.d.f. percentage for a particular chi-square value is the area of the statistic distribution to the left of the statistic value; this is the probability of obtaining that statistic value or less by random selection when testing two distributions which are exactly the same. Repeated trials which randomly sample two identical distributions should produce about the same number of X2 values in each quarter of the distribution (0% to 25%, 25% to 50%, 50% to 75%, and 75% to 100%). So if we repeatedly find only very high percentage values, we can assume that we are probing different distributions. And even a single very high percentage value would be a matter of some interest.
Any statistic probability can be expressed either as the proportion of the area to the left of the statistic value (this is the "cumulative distribution function" or c.d.f.), or as the area to the right of the value (this is the "upper tail"). Using the upper tail representation for the X2 distribution can make sense because the usual chi-squared test is a "one tail" test where the decision is always made on the upper tail. But the "upper tail" has an opposite "sense" to the c.d.f., where higher statistic values always produce higher percentage values. Personally, I find it helpful to describe all statistics by their c.d.f., thus avoiding the use of a wrong "polarity" when interpreting any particular statistic. While it is easy enough to convert from the c.d.f. to the complement or vise versa (just subtract from 1.0), we can base our arguments on either form, since the statistical implications are the same.
It is often unnecessary to use a statistical test if we just want to know whether a function is producing something like the expected distribution: We can look at the binned values and generally get a good idea about whether the distributions change in similar ways at similar places. A good rule-of-thumb is to expect chi-square totals similar to the number of bins, but distinctly different distributions often produce huge totals far beyond the values in any table, and computing an exact probability for such cases is simply irrelevant. On the other hand, it can be very useful to perform 20 to 40 independent experiments to look for a reasonable statistic distribution, rather than simply making a "yes / no" decision on the basis of what might turn out to be a rather unusual result.
Since we are accumulating discrete bin-counts, any fractional expectation will always differ from any actual count. For example, suppose we expect an even distribution, but have many bins and so only accumulate enough samples to observe about 1 count for every 2 bins. In this situation, the absolute best sample we could hope to see would be something like (0,1,0,1,0,1,...), which would represent an even, balanced distribution over the range. But even in this best possible case we would still be off by half a count in each and every bin, so the chi-square result would not properly characterize this best possible sequence. Accordingly, we need to accumulate enough samples so that the quantization which occurs in binning does not appreciably affect the accuracy of the result. Normally I try to expect at least 10 counts in each bin.
But when we have a reference distribution that trails off toward
zero, inevitably there will be some bins with few counts.
Taking more samples will just expand the range of bins, some of which
will be lightly filled in any case. We can avoid quantization error
by summing both the observations and expectations from multiple bins,
until we get a reasonable expectation value (again, I like to see 10
counts or more).
This allows the "tails" of the distribution to be more properly
(and legitimately) characterized.
(The technique of merging adjacent bins is sometimes called
"collapsing.")
Also see:
cryptography,
block cipher,
stream cipher,
substitution,
permutation,
A good cipher can transform secret information into a multitude of different intermediate forms, each of which represents the original information. Any of these intermediate forms or ciphertexts can be produced by ciphering the information under some key value. The intent is that the original information only be exposed by one of the many possible keyed interpretations of that ciphertext. Yet the correct interpretation is available merely by deciphering under the appropriate key.
A cipher appears to reduce the protection of secret information to enciphering under some key, and then keeping that key secret. This is a great reduction of effort and potential exposure, and is much like keeping your valuables in your house, and then locking the door when you leave. But there are also similar limitations and potential problems.
With a good cipher, the resulting ciphertext can be stored or transmitted otherwise exposed without also exposing the secret information hidden inside. This means that ciphertext can be stored in, or transmitted through, systems which have no secrecy protection. For transmitted information, this also means that the cipher itself must be distributed in multiple places, so in general the cipher cannot be assumed to be secret. With a good cipher, only the deciphering key need be kept secret. (See: Kerckhoffs' requirements, but also security through obscurity.)
Note that a cipher does not, in general, hide the length of a plaintext message, nor the fact that the message exists, nor when it was sent, nor, usually, the addressing to whom and from whom. Thus, even the theoretical one time pad (often said to be "proven unbreakable") does expose some information about the plaintext message. If message length is a significant risk, random amounts of padding can be added to confuse that, although padding can of course only increase message size, and is an overhead to the desired communications or storage. This typically would be handled at a level outside the cipher design proper, see cipher system.
It is important to understand that ciphers are unlike any other modern product design, in that we cannot know when a cipher "works." For example:
In CBC mode the ciphertext value of the preceding block is exclusive-OR combined with the plaintext value for the current block. This randomization has the effect of distributing the resulting block values evenly among all possible block values, and so tends to prevent codebook attacks. But ciphering the first block generally requires an IV or initial value to start the process. And the IV necessarily expands the ciphertext by the size of the IV.
[There are various possibilities other than CBC for avoiding plaintext block statistics in ciphers. One alternative is to pre-cipher, presumably with a different cipher and key, thus producing randomized plaintext blocks (see multiple encryption). Another alternative is to use a block at least 64 bytes wide, which, if it contains language text, can be expected to contain sufficient unknowable randomness to avoid codebook attacks (see huge block cipher advantages).]
Note that the exposed nature of the CBC randomizer (the previous block ciphertext) does not hide plaintext or plaintext statistics. When simple deciphering exposes plaintext, the vast majority of possible plaintexts can be rejected automatically, based on their lack of bit-level and character and word structure. Normal CBC does not improve this situation much at all.
In CBC mode, each randomizing value is the ciphertext from each previous block. Clearly, all the ciphertext is exposed to the opponent, so there would seem to be little benefit associated with hiding the IV, which, after all, is just the first of these randomizing values. Clearly, in the usual case, if the opponent makes changes to a ciphertext block in transit, that will hopelessly garble two blocks (or perhaps just one) of the recovered plaintext. As a result, it is very unlikely that an opponent could make systematic changes in the plaintext simply by changing the ciphertext.
But the IV is a special case: if the IV is not enciphered,
and if the opponents can intercept and change the IV in transit,
they can change the first-block plaintext
Despite howls of protest to the contrary, it is easy to see that the CBC first-block problem is a confidentiality problem, not an authentication problem. To see this, we simply note that all that is necessary to avoid the problem is to keep the IV secret. When the IV is protected, the opponent cannot know which changes to make to reach a desired plaintext. And, since the problem can be fixed without any authentication at all, it is clear that the problem was not a lack of authentication in the first place. Instead, the problem was caused by exposing the IV, and solving that is the appropriate province of the CBC and block level, instead of a MAC at the cipher system and message level.
To fix the CBC first-block problem it is not necessary to check the plaintext for changes by using a MAC. Nor is a MAC necessarily the only way to authenticate a message. But if we are going to use a MAC anyway, that is one way to solve the problem. That works because a MAC can detect the systematic changes which a lack of confidentiality may have allowed to occur. But if a MAC is not otherwise desired, introducing a MAC to solve the CBC first-block problem is probably overkill, because only the block-wide IV needs to be protected, and not the entire message.
The reason we might not want to use a MAC is that a MAC carries some inherent negative consequences. One of those is a processing latency, in that we cannot validate the recovered plaintext until we get to the end and check the digest. Latency can be a serious problem with streaming data like audio and video, and with interactive protocols. But even with an email message we have to buffer the whole message as decrypted and wait for the incoming data to finish before we can do anything with it (or we can make encryption hard and decryption easy, but one side will be a problem). Or we can set up some sort of packet structure with localized integrity checks and ciphertext expansion in each packet. But that seems like a lot of trouble when an alternative is just to encipher the IV.
Even when a MAC is used at a higher level anyway, it may be important for Software Engineering and modular code construction to handle at the CBC level as many of the problems which CBC creates as possible. This avoids forcing the problem on, and depending upon a correct response from, some unknown programmer at the higher level, who may have other things on the mind. Handling security problems where they occur and not passing them on to a higher layer is an appropriate strategy for security programming.
As the problems compound themselves, it seems legitimate to point
out that the CBC first-block problem is a CBC-level security issue
caused by CBC and by transporting the IV in the open.
The CBC first-block problem is easily prevented simply by
transporting the IV securely, by encrypting the IV before including
it with the ciphertext.
Also see "The IV in Block Cipher CBC Mode" conversation
(locally, or @:
http://www.ciphersbyritter.com/NEWS6/CBCIV.HTM).
Also see
traffic analysis,
Software Engineering,
Structured Programming, and
comments in the "Cipher Review Service" document
(locally, or @:
http://www.ciphersbyritter.com/CIPHREVU.HTM).
For the analysis of cipher operation it is useful to collect ciphers into groups based on their functioning (or intended functioning). The goal is to group ciphers which are essentially similar, so that as we gain an understanding of one cipher, we can apply that understanding to others in the same group. We thus classify not by the components which make up the cipher, but instead on the "black-box" operation of the cipher itself.
We seek to hide distinctions of size, because operation is independent of size, and because size effects are usually straightforward. We thus classify conventional block ciphers as keyed simple substitution, just like newspaper amusement ciphers, despite their obvious differences in strength and construction. This allows us to compare the results from an ideal tiny cipher to those from a large cipher construction; the grouping thus can provide benchmark characteristics for measuring large cipher constructions.
We could of course treat each cipher as an entity unto itself, or relate ciphers by their dates of discovery, the tree of developments which produced them, or by known strength. But each of these criteria is more or less limited to telling us "this cipher is what it is." We already know that. What we want to know is what other ciphers function in a similar way, and then whatever is known about those ciphers. In this way, every cipher need not be an island unto itself, but instead can be judged and compared in a related community of similar techniques.
Our primary distinction is between ciphers which handle all the data at once (block ciphers), and those which handle some, then some more, then some more (stream ciphers). We thus see the usual repeated use of a block cipher as a stream meta-cipher which has the block cipher as a component. It is also possible for a stream cipher to be re-keyed or re-originate frequently, and so appear to operate on "blocks." Such a cipher, however, would not have the overall diffusion we normally associate with a block cipher, and so might usefully be regarded as a stream meta-cipher with a stream cipher component.
The goal is not to give each cipher a label, but instead to seek insight. Each cipher in a particular general class carries with it the consequences of that class. And because these groupings ignore size, we are free to generalize from the small to the large and so predict effects which may be unnoticed in full-size ciphers.
In general, absent special coding for transmission (such as converting full binary into base-64 for email) ciphertext should be "random-like." Accordingly, we can run all sorts of tests to try to find any sort of structure or correlation in the ciphertext, or between plaintext, key, and ciphertext. The many available statistical randomness tests should provide ample opportunity for virtually unlimited testing.
The usual or conventional block cipher is intended to emulate a huge, keyed, substitution table. Mathematically, such a function is a bijection, and the symbols in the table are a permutation. These structures might be measured, at least in theory. But very few conventional block ciphers are scalable to tiny size, and the vast size of a real block cipher allows only statistical sampling.
One obvious issue in block cipher construction is diffusion. If the resulting emulated table really is a permutation, if we change the input value in any way, we expect the number of bits which change in the output to occur in a binomial distribution. In addition, we expect each output bit to have a 50 percent probability of changing. We can measure these things.
Typically, we pick some random input value and cipher to get the result; then we change some bit of the input and get the new result and note which and how many bits changed. One advantage of the binomial distribution is that, as block size increases, the distribution becomes increaingly narrow (for any reasonable probability). Thus, we can hope to peer into tremendously small probabilities, which may be about as much error as we can expect to find.
We also can develop a mean value for each output bit, or analyze a particular bit more closely, looking for correlations between input and output, or between key and output, or between the key and some aspect of the transformation between input and output. We might look at correlations between each key bit and each output bit, or between any combination of key bits versus any combination of output bits and so on. With increasingly large experiments, we can perform increasingly fine statistical analyses.
An issue of at least potential concern is that conventional block cipher designs do not implement a completely keyed transformation , but instead implement only a tiny, tiny fraction of all possible tables of the block size. This opens the possibility of weakness in some form of correlation resulting from a tiny subset of implemented permutations. The issue then becomes one of trying to measure possible structural correlations between the set of implemented permutations and the key, including individual bits, or even arbitrary functions of arbitrary multiple bits. At real cipher size, such measurements will be difficult. Or perhaps knowledge of some subset of the transformation could lead to filling out the rest of the transformation; at real cipher size, this may be very difficult to see.
Cipher designs which are scalable can be tested at real size when that is useful, or as tiny "toy" versions, when that is useful. Naturally, the tiny versions are not intended to be as strong as the real-size versions, nor even to be a useful cipher at that size. One purpose is to support exhaustive correlation testing to reveal structural problems which should be easier to discern in the smaller construction. The goal would be to find fault at the tiny size, and then use that to develop insight leading to a scalable attack. That same insight also should help improve the cipher design.
One advantage of scalability is to support attacks on the same cipher at different sizes. Once we find an attack on a toy-size version, we can measure how hard that approach really is by actually doing it. Then we can scale up the cipher slightly and measure how much the difficulty has increased. That can provide true evidence which can be used to extrapolate the strength of the real-size cipher, under the given attack. I see this as vastly more believable information than we have for current ciphers.
Another thing we might do is to measure Boolean function nonlinearity values. This measure at least has the advantage of directly addressing one form of strength: the linear predictability of each key-selected permutation.
Yet another thing we might investigate is the number of keys that are actually different. That is, do any keys produce the same emulated table, and if not, how close are those tables? Can we find any two keys that produce the same ciphertext from the same plaintext? (See population estimation and multiple encryption.)
The conventional stream cipher consists of a keyed RNG or confusion generator and some sort of data and confusion combiner, usually exclusive-OR. Since exclusive-OR has absolutely no strength of its own, the strength of the classic stream cipher depends solely on the RNG. Such testing is a common activity in cryptography, using various available statistical randomness tests. (But recall that many strengthless statistical RNG's do well on such tests.) I particularly recommend runs up/down, because we can develop a useful non-flat distribution of results and then compare that to the theoretical expectation. We can do similar things with birthday tests, which are also useful in confirming the coding efficiency or entropy of really random generators.
Modern stream ciphers with
nonlinear combiners (see, for example:
Dynamic Substitution)
seem harder to test.
Presumably we can test the ciphertext for
randomness, as usual, yet that would not
distinguish between the combiner and the RNG.
Possibly we could test the combiner with RNG, and then the RNG
separately, and compare distributions.
However, it is not clear what sort of tests would provide useful
insight to this construction.
Alternate suggestions are welcomed.
Ciphertext contains the same information as the original plaintext, hopefully in a form which cannot be easily understood. Cryptography hides information by transforming a plaintext message into any one of a vast multitude of different ciphertexts, as selected by a key. Ciphertext thus can be seen as a code, in which the exact same ciphertext has a vast number of different plaintext interpretations. As a goal, it should be impractical to know which interpretation represents the original plaintext without knowing the key.
Normally, ciphertext will appear
random; the values in the ciphertext
should occur in a generally
balanced way.
Normally, we do not expect ciphertext to
compress to a smaller size; that
implies efficient
coding (also see
entropy), but only for the
It also may happen that the ciphertext can be
encoded inefficiently (perhaps as
Ciphertext expansion is the general situation: Stream ciphers need a message key, and block ciphers with a small block need some form of plaintext randomization, which generally needs an IV to protect the first block. Only block ciphers with a large size block generally can avoid ciphertext expansion, and then only if each block can be expected to hold sufficient uniqueness or entropy to prevent a codebook attack.
It is certainly true that in most situations of new construction
a few extra bytes are not going to be a problem. However, in some
situations, and especially when a cipher is to be installed into
an existing system, the ability to encipher data without
requiring additional storage can be a big advantage. Ciphering
data without expansion supports the ciphering of data structures
which have been defined and fixed by the rest of the system,
provided only that one can place the cipher at the interface
"between" two parts of the system. This is also especially
efficient, as it avoids the process of acquiring a different,
larger, amount of store for each ciphering. Such an installation
also can apply to the entire system, and not require the
re-engineering of all applications to support cryptography in
each one.
CFB is closely related to OFB, and is intended to provide some of the characteristics of a stream cipher from a block cipher. CFB generally forms an autokey stream cipher. CFB is a way of using a block cipher to form a random number generator. The resulting pseudorandom confusion sequence can be combined with data as in the usual stream cipher.
CFB assumes a shift register of the block cipher block size. An IV or initial value first fills the register, and then is ciphered. Part of the result, often just a single byte, is used to cipher data, and the resulting ciphertext is also shifted into the register. The new register value is ciphered, producing another confusion value for use in stream ciphering.
One disadvantage of this, of course, is the need for a full
block-wide ciphering operation, typically for each data byte
ciphered. The advantage is the ability to cipher individual
characters, instead of requiring accumulation into a block
before processing.
In a sense, the idea of a ciphertext-only attack is inherently incomplete. By themselves, symbols and code values have no meaning. So we can have all the ciphertext we want, but unless we can find some sort of structure or relationship to plaintext, we have nothing at all. The extra information necessary to identify a break could be the bit structure in the ASCII code, the character structure of language, or any other known relation. But the ciphertext is never enough if we know absolutely nothing about the plaintext. It is our knowledge or insight about the plaintext, the statistical structure, or even just the known use of one plaintext concept, that allows us to know when deciphering is correct.
In practice, ciphertext-only attacks typically depend on some
error or weakness in the
encryption design which somehow relates
some aspect of
plaintext in the ciphertext. For example,
codes that always encrypt the same words in
the same way naturally leak information about how often those words
are used, which should be enough to identify the plaintext.
And the more words identified, the easier it is to fill in the gaps
in sentences, and, thus, identify still more words. Modern
ciphers are less likely to fall into that
particular trap, making ciphertext-only attacks generally more
academic than realistic (also see
break).
See the documentation:
In a digital system we create a delay or measure time by simply
counting pulses from a stable
oscillator.
Since counting operations are digital, noise effects are virtually
eliminated, and we can easily create accurate delays which are as
long as the count in any counter we can build.
Code values can easily represent not only symbols or characters, but also words, names, phrases, and entire sentences (also see nomenclator). In contrast, a cipher operates only on individual characters or bits. Classically, the meaning of each code value was collected in a codebook. Codes may be open (public) or secret.
Coding is a very basic part of modern computation and generally implies no secrecy or information hiding. In modern usage, a code is often simply a correspondence between information (such as character symbols) and values (such as the ASCII code or Base-64). Because a code can represent entire phrases with a single number, one early application for a public code was to decrease the cost of telegraph messages.
In general, secret codes are weaker than ciphers, because a typical code will simply substitute or transform each different word or letter into a corresponding value. Thus, the most-used plaintext words or letters also become the most-used code or ciphertext values and the statistical structure of the plaintext remains exposed. Then the opponent easily can find the most-used ciphertext values and realize that they represent the most-used plaintext words. Accordingly, it is common to superencipher a coded message in an attempt to hide the codebook values.
A meaningful code is more than just data, being also the interpretation of that data. The main concept of modern cryptography is the use of a key to select one interpretation from among vast numbers of different interpretations, so that meaning is hidden from those who do not have both the appropriate decryption program and key. Each particular ciphertext is interpreted by the decryption system to produce the desired plaintext. The pairing of value plus interpretation to produce or do something occurs in various places:
In real life, many useful things do require a particular thing
to use them.
For example, gasoline provides energy for cars, but only because
cars have the appropriate engine to perform the desired conversion.
Similarly, bullets require guns, radio broadcasting stations
require radios and so on.
But that probably reaches beyond the idea of a code, which is
basically limited to information- or symbol-oriented transformations.
The usual ciphertext only approach depends upon the plaintext having strong statistical biases which make some values far more probable than others, and also more probable in the context of particular preceding known values. While this is not known plaintext, it is a form of known structure in the plaintext. Such attacks can be defeated if the plaintext data are randomized and thus evenly and independently distributed among the possible values (see balance).
When a codebook attack is possible on a block cipher, the complexity of the attack is controlled by the size of the block (that is, the number of elements in the codebook) and not the strength of the cipher. This means that a codebook attack would be equally effective against either DES or Triple-DES.
One way a block cipher can avoid a codebook attack is by having
a large
block size which will contain an unsearchable
amount of plaintext "uniqueness" or
entropy. Another approach is to randomize the
plaintext block, by using an
operating mode such as
CBC, or
multiple encryption.
Yet another approach is to change the
key frequently, which is one role of the
message key introduced at the
cipher system level.
Codebreaking is what we normally think of when hearing the WWII crypto stories, especially the Battle of Midway, because many secrecy systems of the time were codes. According to the story, the Japanese are preparing an attack on Midway island, and have given Midway the coded designation "AF." American cryptanalysts have exposed the designator "AF," but not what it represents. Assuming the "AF" to be Midway, American codebreakers have Midway falsely report the failure of their fresh-water plant in open traffic. Then, two days later, intercepted Japanese traffic states that "AF" is short of fresh water. Thus, "AF" is confirmed as Midway.
Note that there had to be a way to identify the actual target
(plaintext) with the code value
(ciphertext) before the meaning was
exposed.
Simply having the ciphertext itself, without finding structure in
the ciphertext or some relationship to plaintext, is almost never
enough, see
ciphertext-only attack.
The classic example is of a cult who believed the Earth was going to end at a particular time. Supposedly, many members gave up their houses and jobs and so on, but the Earth did not end. As a consequence, less-involved members generally accepted that their belief was false. But more-involved members instead insisted that the actions of the cult showed their faith, which was then rewarded by the Earth not ending.
Obviously it is difficult to use
logic to address issues of faith, but
science is not a faith and does not require
belief.
Therefore, when we find that current scientific positions are wrong,
they can be changed with only minor discomfort and anguish.
Supposedly. (Also see
mere semantics and
old wives' tale.)
n
( ) = C(n,k) = n! / (k! (n-k)!)
k
Also,
n n n
( ) = ( ) = 1 ( ) = n
0 n 1
See the combinations section of the "Base Conversion, Logs,
Powers, Factorials, Permutations and Combinations in JavaScript" page
(locally, or @:
http://www.ciphersbyritter.com/JAVASCRP/PERMCOMB.HTM#Combinations).
Also see
permutation.
Consider a conventional block cipher: For any given size block, there is some fixed number of possible messages. Since every enciphering must be reversible (deciphering must work), we have a 1:1 mapping between plaintext and ciphertext blocks. The set of all plaintext values and the set of all ciphertext values is the same set; particular values just have different meanings in each set.
Keying gives us no more ciphertext values, it only re-uses the values which are available. Thus, keying a block cipher consists of selecting a particular arrangement or permutation of the possible block values. Permutations are a combinatoric topic. Using combinatorics we can talk about the number of possible permutations or keys in a block cipher, or in cipher components like substitution tables.
Permutations can be thought of as the number of unique
arrangements of a given length on a particular set. Other
combinatoric concepts include
binomials
and
combinations
(the number of unique given-length subsets of a given set).
Reversible combiners are pretty much required to encipher plaintext into ciphertext in a stream cipher. The ciphertext is then deciphered into plaintext using a related inverse or extractor mechanism. The classic examples are the stateless and strengthless linear additive combiners, such as addition, exclusive-OR, etc.
Reversible and nonlinear keyable combiners with state are a result of the apparently revolutionary idea that not all stream cipher security need reside in the keying sequence. Examples include:
Irreversible or non-invertible combiners are often proposed to mix multiple RNG's into a single confusion sequence, also for use in stream cipher designs. But that is harder than it looks. For example, see:
Also see
balanced combiner,
complete, and also
"The Story of Combiner Correlation: A Literature Survey,"
locally or @:
http://www.ciphersbyritter.com/RES/COMBCORR.HTM.
Also see:
associative and
distributive.
Completeness does not require that an input bit change an output bit for every input value (which would not make sense anyway, since every output bit must be changed at some point, and if they all had to change at every point, we would have all the output bits changing, instead of the desired half). The inverse of a complete function is not necessarily also complete.
As originally defined in Kam and Davida:
"For every possible key value, every output bit ci of the SP network depends upon all input bits p1,...,pn and not just a proper subset of the input bits." [p.748]-- Kam, J. and G. Davida. 1979. Structured Design of Substitution-Permutation Encryption Networks. IEEE Transactions on Computers.C-28(10):747-753.
To build an appropriate algebra and make complex numbers a
field,
the rectangular representation is written as (x+iy) [or (x+jy)],
where i [or j] has the value SQRT(-1).
The symbol i is called "imaginary," but we might just consider it a
way for the algebra to relate the values in the ordered pair.
Clearly,
With appropriate rules like:
addition: (a+bi) + (c+di) = (a+c) + (b+d)i
multiplication: (a+bi) * (c+di) = (ac-bd) + (bc+ad)i
c+di ac+bd ad-bc
division: ---- = ----- + (-----)i
a+bi aa+bb aa+bb
we get complex algebra, and can perform most operations and
even evaluate trignometric and other complex functions like we do
with reals.
In cryptography, perhaps the most common use of complex numbers occurs in the FFT, which typically transforms values in rectangular form. Sometimes we want to know the magnitude or length of the implied vector, which we can get by converting the rectangular (x,y) representation into the (mag,ang) representation:
magnitude: mag(z) = SQRT( x*x + y*y ) angle: ang(z) = arctan( y / x) Note: Computer arctan(x) functions are generally unable to place the angle in the proper quadrant, but arctan2(x,y) routines -- with two input parameters -- may be available to do so.
The most successful components are extremely general and can be used in many different ways. Even as a brick is independent of the infinite variety of brick buildings, a flip-flop is independent of the infinite variety of logic machines which use flip-flops.
The source of the ability to design and build a wide variety of different electronic logic machines is the ability to interconnect and use a few very basic but very general parts.
Electronic components include
The use of individual components to produce a working complex system in production requires: first, a comprehensive specification for each part; and next, full testing to guarantee that each part actually meets the specification (see: quality management).
Digital logic is normally specified to operate correctly over a range of supply voltage, temperature, loading, clock rates, and other appropriate parameters. Specified limits (minimum's or maximum's) guarantee that a working part will operate correctly even with the worst case of all parameters simultaneously. This process allows large, complex systems to operate properly in practice, provided the designer makes sure that none of the parameters can exceed their correct range.
Cryptographic system components include:
A logic machine with:
The general model of mechanical computation is the finite state machine, which is absolutely deterministic and, thus, predictable.
Also see:
source code,
object code,
software,
system design,
Software Engineering and
Structured Programming.
As a rule of thumb, a cubic centimeter (cc) of a solid has about 1024 or 1E24 atoms. In a metal, usually each atom contributes one or two electrons, so a metal has about 1024 (1E24) free electrons per cc. This massive number of free electrons has a tiny resistance to current flow of something like 10-6 ohms across a cubic centimeter of copper, or about one microhm per cm3. Apparently the International Annealed Copper Standard (IACS) says that annealed copper with a cross sectional area of a square centimeter should have a resistance of about 1.7241 microhms/cm (at 20 degrees Celsius), which is satisfactorily close.
A cube with one millimeter sides has 1/100 the cross sectional area of a centimeter cube (and is about like AWG 17 wire), and so would have 100x the resistance per cm., but also is only 1/10 the length, for about 17 microhms per millimeter copper cube. A meter of AWG 17 wire would have 1000 millimeter-size cubes at 17 microhms each, so we would expect it to have about 17 milliohms total resistance. As a check, separate wire tables give the resistance of AWG 17 at 5.064 ohms per 1000ft (304.8m), which is 0.017 ohms (17 milliohms) per meter.
RESISTANCE RELATIVE TO COPPER (cm cube = 1.7241 microhms)
Resist Temp Coef Thermal Cond Melts (deg C)
Silver (Ag) 0.95 0.0038 4.19 960.5
Copper (Cu) 1.00 0.00393 3.88 1083
Gold (Au) 1.416 0.0034 2.96 1063
Aluminum (Al) 1.64 0.0039 2.03 660
Bronze (Cu+Sn) 2.1 --- --- 1280
Tungsten (W) 3.25 0.0045 1.6 3370
Zinc (Zn) 3.4 0.0037 1.12 419
Brass (Cu+Zn) 3.9 0.002 1.2 920
Nickel (Ni) 5.05 0.0047 0.6 1455
Iron (Fe) 5.6 ~0.005 0.67 1535
Tin (Sn) 6.7 0.0042 0.64 231.9
Chromium (Cr) 7.6 --- --- 2170
Steel (Fe+C) ~10 --- 0.59 1480
Lead (Pb) 12.78 0.0039 0.344 327
Titanium (Ti) 47.8 --- 0.41 1800
Stainless (-->) 52.8 --- 0.163 1410 (Fe+Cr+Ni+C)
Mercury (Hg) 55.6 0.00089 0.063 -38.87
Nichrome (Ni+Cr) 65 0.00017 0.132 1350
Graphite (C) 590 --- --- 3800
Carbon (C) 2900 -0.0005 --- 3500
In a conspiracy, multiple individuals can each contribute a
minor action to accumulate a large effect.
One obvious approach is to use gossip to give the impression that
all right-thinking people are against some one or some thing.
A conspiracy can be difficult to oppose, because a major effect
can be achieved with minor actions that individually do not call
for a major response.
In number theory we say than integer a (exactly) divides
integer b (denoted
In number theory we say that integer a is congruent to
integer b
modulo m, denoted
Used in the analysis of signal processing to develop the response
of a processing system to a complicated real-valued input signal.
The input signal is first separated into some number of discrete
impulses. Then the system response to an impulse
It is apparently possible to compute the convolution of two
sequences by taking the
FFT of each, multiplying these results
term-by-term, then taking the inverse FFT. While there is an
analogous relationship in the
FWT, in this case the "delays" between the
sequences represent
mod 2 distance differences, which may or may
not be useful.
Copyright protects a particular expression, but not the underlying idea, process or function it may perform, which is the province of patent protection. Copyright protects form, not content: Copyright can protect particular text and diagrams, but not the described concept. In general, copyright comes into existence simply by creating a picture or manuscript or making a selection; theoretically, no notice or registration is required. (See the Library of Congress circular "Copyright Basics": http://www.loc.gov/copyright/circs/circ1.html#cr). However, formal registration is required before a lawsuit can be filed, and registration within 3 months of publication supports recovery of statutory damages and attorney fees; otherwise, apparently only actual damages can be recovered. Similarly, no copyright notice is required, but having one like this:
Copyright 1991 Terry Ritter. All Rights Reserved.may avoid an "innocent infringement" defense. Protection currently lasts 70 years beyond the death of the author, or 95 years from date of publication for works for hire. Copyright is not handled by the PTO but instead by the United States Copyright Office (http://lcweb.loc.gov/copyright/) in the Library of Congress.
One way to evaluate a common correlation of two real-valued
sequences is to multiply them together term-by-term and sum all
results.
If we do this for all possible "delays" between the two sequences,
we get a "vector" or
"The correlation coefficient associated with a pair of Boolean functions f(a) and g(a) is denoted byC(f,g) and is given byC(f,g) = 2 * prob(f(a) = g(a)) - 1 ." -- Daemen, J., R. Govaerts and J. Vanderwalle. 1994. Correlation Matrices. Fast Software Encryption. 276. Springer-Verlag.
There are two classes: A local counterexample refutes a
lemma, but not necessarily the
main conjecture.
A global counterexample refutes the main conjecture.
Note that integer counting produces perhaps the best possible
signal for investigating block cipher deficiencies in the rightmost
bits. Accordingly, incrementing by some large random constant, or
using some sort of
LFSR or other polynomial counter which changes
about half its bits on each step may be more appropriate.
CRC error-checking is widely used in practice to check the data recovered from magnetic storage. When data are written to disk, a CRC of the original data is computed and stored along with the data itself. When data are recovered from disk, a new CRC is computed from the recovered data and that result compared to the recovered CRC. If the CRC's do not match, we have a "CRC error."
Computer disk-read operations always have some chance of a "soft error" which does not re-occur when the same sector is re-read, so the usual hardware response is to try again, some number of times. If that does not solve the problem, the error may be reported to the user and could indicate the start of serious disk problems.
A CRC operation is essentially a remainder over the huge numeric
value which is the data; the mod 2 polynomials make this "division"
both faster and simpler than one might expect.
Related techniques like integer or floating point division can have
similar power, but are unlikely to be as simple.
In general, "division" techniques only miss errors which are some
product of the divisor, and so
The CRC result is an excellent (but linear) hash value corresponding to the data. Compared with other hash alternatives, CRC's are simple and straightforward. They are well-understood. They have a strong and complete basis in mathematics, so there can be no surprises. CRC error-detection is mathematically tractable and provable without recourse to unproven assumptions. And CRC hashes do not need padding. None of this is true for most cryptographic hash constructions.
For error-detection, the CRC register is first initialized to some fixed value known at both ends, nowadays typically "all 1's." Then each data element is processed, each of which changes the CRC value. When all of the data have been processed, the CRC result is sent or stored at the end of the data. Frequently the CRC result first will be complemented, so that a CRC of the data and the complemented result will produce a fixed "magic number." This allows efficient hardware error-checking, even when the hardware does not know how large the data block will be in advance. (Typically, the end of transmission, after the CRC, is indicated by a hardware "done" signal.)
Nowadays, CRC's are often computed in software which is generally more efficient with larger data quantities. Thus we see 8-bit, 16-bit or 32-bit data elements being processed. However, CRC's can be computed on individual data bits, and on records of arbitrary bit length, including zero bits, one bit, or any uneven or dynamic number of bits. As a consequence, no padding is ever needed for CRC hashing.
Here is a code snippet for a single-bit left-shift CRC operation:
if (msb(crc) == databit)
crc = crc << 1;
else
crc = (crc << 1) ^ poly;
This fragment needs to execute 8 times to compute the CRC for a
full data byte.
However, a better way to process a byte in software is to
pre-compute a 256-element table representing every possible CRC
change corresponding to a single byte.
The table value is selected by a data byte XORed with the current
top byte of the CRC register (in a left-shift implementation).
In the late 60's and early 70's, the first CRC's were initialized as "all-0's." Then it was noticed that extra or missing 0-bits at the start of the data would not be detected, so it became virtually universal to init the CRC as "all-1's." In this case, extra or missing zeros at the start are detected, and extra or missing ones at the start are detected as well.
It is possible for multiple errors to occur and the CRC result to end up the same as if there were no error. But unless the errors are introduced intentionally, this is very unlikely. Various common errors are detected absolutely, such as:
If we have enough information, it is relatively easy to compute error patterns which will take a CRC value to any desired CRC value. Because of this, data can be changed in ways which will produce the original CRC result. Consequently, no CRC has any appreciable cryptographic strength, but some applications in cryptography need no strength:
On the other hand, a CRC, like most computer hashing operations, is normally used so that we do not have "enough information." When substantially more information is hashed than the CRC can represent, any particular CRC result will be produced by a vast number of different input strings. In this way, even a linear CRC can be considered an irreversible "one way" or "information reducing" transformation. Of course, when a string shorter than the CRC polynomial is hashed, it should not be too difficult to find the one string that could produce any particular CRC result.
The CRC polynomial need not be particularly special. Unlike the generator polynomials used in LFSR's, a CRC poly need not be primitive nor even irreducible. Indeed, the early 16-bit CRC polys were composite with a factor of "11" which is equivalent to the information produced by a parity bit. (Since parity was the main method of error-detection at the time, the "11" factor supported the argument that CRC was better.) However, modern CRC polys generally are primitive, which allows the error detection guarantees to apply over larger amounts of data. It also allows the CRC operation to function as an RNG. But the option exists to use secret random polynomials to detect errors without being as predictable as a standard CRC. Polynomial division does not require mathematical structure (such as an irreducible or primitive), beyond the basic mod 2 operations.
Different CRC implementations can shift left or right, take data lsb or msb first, and be initialized as zeros or ones, each option naturally producing different results. Various CRC standards specify different options. Obviously, both ends must do things the same way, but it is not necessary to conform to a standard to have quality error-detection for a private or new design. Variations in internal handling can make a CRC with one set of options produce the same result as a CRC with other options.
When the logical complement of a CRC result is appended to the
data and processed msb first, the CRC across that data and the result
produces a "magic" value which is a constant for a particular poly and
set of options.
In general, the sequence reverse of a good poly is also a good poly, and
there is some advantage to having CRC polys which are about half 1's.
In some notations we omit the msb which is always 1 (as is the lsb).
For notational convenience, we can write
Name Hex Set Bits
CRC16 8005 16,15,2,0
CCITT 1021 16,12,5,0
CRC24a 800063
CRC24b 800055
CRC24c 861863
CRC32 04c11db7 32,26,23,22,16,
12,11,10,8,7,5,4,2,1,0
SWISS-PROT 64,4,3,1,0
SWISS-PROT Impr 64,63,61,59,58,56,55,52,49,48,
(D. Jones) 47,46,44,41,37,36,34,32,
31,28,26,23,22,19,16,
13,12,10,9,6,4,3,0
Also see: reverse CRC, and
In normal cryptanalysis we start out knowing plaintext, ciphertext, and cipher construction. The only thing left unknown is the key. A practical attack must recover the key. (Or perhaps we just know the ciphertext and the cipher, in which case a real attack would recover plaintext.) Simply finding a distinguisher (showing that the cipher differs from the chosen model) is not, in itself, an attack or break.
Because no theory guarantees strength for any conventional cipher (see, for example, the one time pad and proof), ciphers traditionally have been considered "strong" when they have been used for a long time with "nobody" knowing how to break them easily.
Expecting cipher strength because a cipher is not known to have been broken is the logic fallacy of ad ignorantium: a belief which is claimed to be true because it has not been proven false.
Cryptanalysis seeks to extend this admittedly-flawed process by applying known attack strategies to new ciphers (see heuristic), and by actively seeking new attacks. Unfortunately, real attacks are directed at particular ciphers, and there is no end to different ciphers. Even a successful break is just one more trick from a virtually infinite collection of unknown knowledge.
In cryptanalysis it is normal to assume that at least known-plaintext is available; often, defined-plaintext is assumed. The result is typically some value for the amount of work which will achieve a break (even if that value is impractical); this is the strength of the cipher under a given attack. Different attacks on the same cipher may thus imply different amounts of strength. While cryptanalysis can demonstrate "weakness" for a given level of effort, cryptanalysis cannot prove that there is no simpler attack (see, for example, attack tree and threat model):
Indeed, when ciphers are used for real, the opponents can hardly be expected to advertise a successful break, but will instead work hard to reassure users that their ciphers are still secure. The fact that apparently "nobody" knows how to break a cipher is somewhat less reassuring from this viewpoint. (Also see the discussion: "The Value of Cryptanalysis," locally, or @: http://www.ciphersbyritter.com/NEWS3/MEMO.HTM). For this reason, using a wide variety of different ciphers can make good sense: That reduces the value of the information protected by any particular cipher, which thus reduces the rewards from even a successful attack. Having numerous ciphers also requires the opponents to field far greater resources to identify, analyze, and automate breaking (when possible) of each different cipher. Also see: Shannon's Algebra of Secrecy Systems.
In general, a cipher can be seen as a key-selected set of transformations from plaintext to ciphertext (and vise versa for deciphering). A conventional block cipher takes a block of data or a block value and transforms that into some probably different value. The result is to emulate a huge, keyed substitution table, so, for enciphering, we can write this very simple equation:
E[K][PT] = CT, or E[K,PT] = CT
where PT = plaintext block value, K = Key, and
CT = ciphertext block value.
The brackets "[ ]" mean the operation of indexing: the selection of
a particular position in an array and returning that element value.
Here,
D[K][CT] = PT, or D[K,PT] = CT
where D[K] represents an
inverse or decryption table.
But an attacker does not know and thus must somehow develop the
decryption table.
We assume that an opponent has collected quite a lot of information, including lots of plaintext and the associated ciphertext (a condition we call known plaintext). The opponent also has a copy of the cipher and can easily compute every enciphering or deciphering transformation. What the opponent does not have, and what he is presumably looking for, is the key. The key would expose the myriad of other ciphertext block values for which the opponent has no associated plaintext.
We might imagine the opponent attacking a cipher with a deciphering machine having a huge "channel-selector" dial to select a key value. As one turns the key-selector, each different key produces a different deciphering result on the display. So all the opponent really has to do is to turn the key dial until the plaintext message appears. Given this extraordinarily simple attack (known as brute force), how can any cipher be considered secure?
In a real cipher, we make the key dial very, very, very big! The keyspace of a real cipher is much too big, in fact, for anyone to try each key in a reasonable amount of time, even with massively-parallel custom hardware. That leaves the opponent with a problem: brute force does not work.
Nevertheless, the cipher equation seems exceedingly simple. There is one particular huge emulated table as selected by the key, and the opponent has a sizable set of positions and values from that table. Moreover, all the known and unknown entries are created by exactly the same mechanism and key. So, if the opponent can in some way relate the known entries to the rest of the table, thus predicting unknown entries, the cipher may be broken. Or if the opponent can somehow relate known plaintexts to the key value, thus predicting the key, the key may be exposed. And with the key, ciphertext for which there is no corresponding plaintext can be exposed, thus breaking the cipher. Finding these relationships is where the cleverness of the individual comes in. In a real sense, a cipher is a puzzle, and we currently cannot guarantee that there is no particular "easy" way for a smart team to solve it.
One peculiarity of conventional block ciphers is that they cannot emulate all possible tables, but instead only a tiny, tiny fraction thereof (see block cipher). Even what we consider a huge key simply cannot select from among all possible tables because there are far too many. Now, the "tiny fraction" of tables actually emulated is still too many to traverse (this is a "large enough" keyspace), but, clearly, some special selection is happening which might be exploited. Having even one particular value at one known table position is sufficiently special that we expect that only one key would produce that particular relationship in a conventional cipher. So, in practice, just one known-plaintext pair generally should be sufficient to identify the correct key, if only we could find some way to do it.
In academic cryptanalysis we normally assume that we do not know the key, but do know the cipher and everything about it. We also assume essentially unlimited amounts of known plaintext to use in an attack to find the key. In practice things are considerably different.
In practical cryptanalysis we may not know which cipher has been used. The cipher may not ever have been published, or may have been modified from the base version in various ways. Even a cipher we basically know may have been used in a way which will disguise it from us, for example:
Selecting among different ciphers is part of Shannon's 1949 Algebra of Secrecy Systems. In a modern computer implementation, we could select ciphers dynamically. The number of selectable transformations increases exponentially when several ciphers are used in sequence (multiple encryption). Considering Shannon's academic work in this area, the use of well-known standardized designs is, ironically and rather sadly, current cryptographic orthodoxy (see risk analysis).
The general mathematical model of ciphering is that of a keyed transformation (a mapping or function). Numerically, we can make the general model work for a system of multiple ciphers by allowing some "key" bits to select the cipher, with the rest of the key bits going to key that cipher. But in the adapted model, different parts of the key will have vastly different difficulties for the opponent. Finding the correct key within a cipher may be hard, yet could be much, much easier than finding the exact cipher actually being used. Differences in the difficulty of finding different key bits are simply glossed over in the adapted general model.
Somehow obtaining and breaking every cipher which possibly could have been used is a vastly larger problem than the relatively small increase in keyspace indicated by the number of possible ciphers. For example, if we think we have found the key and want to check it, on a known cipher that has essentially no cost and may take a microsecond. But if we want to check the key on an unknown cipher, we first have to obtain that cipher. That may require the massive ongoing cost of maintaining an intelligence field service to obtain copies of secret ciphers. Once the needed cipher is obtained, finding a practical break may take experts weeks or months, if a break is even found. Taken together, this is a vast increase in difficulty for the opponent per cipher choice compared to the difficulty per key choice within a single cipher.
Just as it may be impossible to try every 128-bit key at even a nanosecond apiece, it also may be impossible to keep up with a far smaller but continuing flow of new secret ciphers which take hundreds of billions of times longer to handle. This advantage seems to be exploited by NSA in keeping cipher designs secret (also see security through obscurity). Given the stark contrast of yet another real example which contradicts the current cryptographic wisdom, crypto academics continue to insist that standardizing and exposing the cipher design makes sense. Surely, exposing a cipher does support gratuitous analysis and help to expose some cipher weakness, but does not, in the end, give us a proven strong cipher. In the end, exposing the cipher may turn out to benefit opponents far more than users.
In practice, an individual attacker mainly must hope that the cipher to be broken is flawed. An attacker can collect ciphertext statistics and hope for some irregularity, some imbalance or statistical bias that will identify the cipher class, or maybe even a well-known design. An attacker can make plaintext assumptions and see if some key will produce those words. But enciphering guessed plaintext seems an unlikely path to success when every possible cipher, and every possible modification of that cipher, is the potential encryption source. All this is a very difficult problem, and far different than the normal academic analysis.
Many academic attacks are essentially theoretical, involving huge amounts of data and computation. But even when a direct technical attack is practical, that may be the most difficult, expensive and time-consuming way to obtain the desired information. Other methods include making a paper copy, stealing a copy, bribery, coercion, and electromagnetic monitoring. No cipher can keep secret something which has been otherwise revealed. Information security thus involves far more than just cryptography, and a cryptographic system is more than just a cipher (see: cipher system). Even finding that information has been revealed does not mean that a cipher has been broken, although good security virtually requires that assumption. (Of course, when we can use only one cipher, we cannot change ciphers anyway.)
Unfortunately, we have no way to know how strong a cipher appears to our opponents. Even though the entire reason for using cryptography is a belief that our cipher has sufficient strength, science provides no basis for such belief. At most, cryptanalysis can give us only an upper limit to the strength of a cipher, which is not particularly helpful, and can only do that when a cipher actually can be broken. But when a cipher is not broken, cryptanalysis has told us nothing about the strength of a cipher, and unbroken ciphers are the only ones we use.
The ultimate goal of cryptanalysis is not to break every possible cipher (that would be the end of an industry and also the end of new PhD's in the field). Instead, the obvious goal is understanding why some ciphers are weak, and why other ciphers seem strong. It is not much of a leap from that to expect cryptanalysts to work with, or at least interact with, cipher designers, with a common goal of producing better ciphers.
Unfortunately, cryptanalysis is ultimately limited by what can
be done: there are no ciphering techniques which guarantee strength,
and there is no test which tells us how weak an arbitrary cipher
really is.
Accordingly, exposing a particular weakness in a particular cipher
may be about as much as cryptanalysis can offer, even if that means
a deafening silence about similar designs, ciphers which
have been repaired, or significant cipher designs which remain
both unbroken and undiscussed.
Apparent agreement among academics does not imply a lack of academic controversy, since many will side with the conventional wisdom, while others step back to consider the arguments. Since reality is not subject to majority rule, even universal academic agreement would not constitute a scientific argument, which instead requires facts and exposed logical reasoning. Controversy may even imply that academic cryptographers are unaware of the issue, or have not really considered it in a deep way. For if clear, understandable and believable explanations already existed, there would be little room for debate. Controversy arises when the given explanations are false, or obscure, or unsatisfactory.
Scientific controversy is less about conflict than exposing Truth. That happens by doing research, then taking a stand and supporting it with facts and scientific argument. Many of these issues should have indisputable answers or expose previously ignored consequences. Wishy-washy statements like "some people think this, some think that," not only fail to inform, but also fail to frame a discussion to expose the real answer.
One aspect of science is the creation of quantitative models which describe or predict reality. Since poor models lead to errors in reasoning, a science reacts to poor predictions by improving the models. In contrast, cryptography reacts by making excuses about why the model really is right after all, or does not apply, or does not matter. Examples include:
In my view, cryptography has presided over a fundamental breakdown in logic, perhaps created by awe of supposedly superior mathematical theory. Upon detailed examination, however, theoretical math often turns out to be inapplicable to the case at hand. Practical results which conflict with theory are ignored or dismissed, even though confronting reality is how science improves models. Demanding belief in conventional cryptographic wisdom requires people to think in ways which accept logical falsehood as truth, and then they apply that lesson. Reasoning errors have become widespread, accepted, and prototypes for future thought. Disputes in cryptography are commonly argued with logic fallacies, and may be "won" with arguments that have no force at all. Since experimental results are rare in cryptography, we cannot afford to lose reason, because that is almost all we have.
A major logical flaw in conventional cryptography is the belief that one good cipher is enough.
But since cipher strength occurs only in the context of our opponents, how could we ever know that we have a "good" cipher, or how "good" it is?
In particular:
So if we cannot measure "good," and cannot prove "good," then exactly how do we know our ciphers are "good?" The answer, of course, is that we do not and can not know any such thing. In fact, nobody on our side can know that our ciphers are "good," no matter how well educated, experienced or smart they may be, because that is determined by our opponents in secret. Anyone who feels otherwise should try to put together what they see as a scientific argument to prove their point.
When conventional cryptography accepts a U.S. Government standard cipher as "good" enough, there is no real need:
Conventional cryptography encourages a belief in known cipher strength, thus ignoring both logic and the lessons of the past. That places us all at risk of cipher failure, which probably would give no indication to the user and so could be happening right now. That we have no indication of any such thing is not particularly comforting, since that is exactly what our opponents would want to portray, even as they expose our information.
When an ordinary person makes a claim, they can be honestly wrong. But when a trained expert in the field makes a claim that we know cannot be supported, and continues to make such claims, we are pretty well forced into seeing that as either professional incompetence or deliberate deceit. Encouraging people to use only one cipher by claiming they need nothing else is exactly what one would expect from an opponent who knows how to break the cipher. Maybe that is just coincidence.
Cryptographic controversies include:
In my view, cryptography often does not understand or attempt to address controversial issues in a scientific way. In areas where cryptography cannot distinguish between truth and falsehood, it cannot advance.
If anyone has any other suggestions for this list, please let
me know.
Uses might include:
Requirements for such an RNG will vary depending upon use, but might include:
We normally assume that the opponents have a substantial amount of known plaintext to use in their work (see cryptanalysis). So the situation for the opponents involves taking what is known and trying to extrapolate or predict what is not known. That is similar to building a scientific model intended to predict larger reality on the basis of many fewer experiments. Since the whole idea is to make prediction difficult for the opponent, unpredictability can be called the essence of cryptography.
Cryptography is a part of cryptology, and is further divided into secret codes versus ciphers. As opposed to steganography, which seeks to hide the existence of a message, cryptography seeks to render a message unintelligible even when the message is completely exposed.
Cryptography includes at least:
In practice, cryptography should be seen as a system or game which includes both users and opponents: True scientific measures of strength do not exist when a cipher has not been broken, so users can only hope for their cipher systems to protect their messages. But opponents may benefit greatly if users can be convinced to adopt ciphers which opponents can break. Opponents are thus strongly motivated to get users to believe in the strength of a few weak ciphers. Because of this, deception, misdirection, propaganda and conspiracy are inherent in the practice of cryptography. (Also see trust and risk analysis.)
And then, of course, we have the natural response to these negative possibilities, including individual paranoia and cynicism. We see the consequences of not being able to test cipher security in the arrogance and aggression of some newbies. Even healthy users can become frustrated and fatalistic when they understand cryptographic reality. Cryptography contains a full sea of unhealthy psychological states.
For some reason (such as the lack of direct academic statements on the issue), some networking people who use and depend upon cryptography every day seem to have a slightly skewed idea about what cryptography can do. While they seem willing to believe that ciphers might be broken, they assume such a thing could only happen at some great effort. Apparently they believe the situation has been somehow assured by academic testing. But that belief is false.
Ciphers are like puzzles, and while some ways to solve the puzzle may be hard, other ways may be easy. Moreover, once an easy way is found, that can be put into a program and copied to every "script kiddie" around. The hope that every attacker would have to invest major effort to find their own fast break is just wishful thinking. And even as their messages are being exposed, the users probably will think everything is fine, just like we think right now. Cipher failure could be happening to us right now, because there will be no indication when failure occurs.
What are the chances of cipher failure? We cannot know! Ciphers are in that way different from nearly every other constructed object. Normally, when we design and build something, we measure it to see that it works, and how well. But with ciphers, we cannot measure how well our ciphers resist the efforts of our opponents. Since we have no way to judge effectiveness, we also cannot judge risk. Thus, we simply have no way to compare whether the cipher design is more likely to be weak than the user, or the environment, or something else. As sad as this situation may seem, it is what we have.
When compared to the alternative of blissful ignorance, it should be a great advantage to know that ciphers cannot be depended upon. First, design steps could be taken to improve things (although that would seem to require a widespread new understanding of the situation that has always existed). Next, we note that ciphers can at most reveal only what they try to protect: When protected information is not disturbing, or dangerous, or complete, or perhaps not even true, exposure becomes much less of an issue.
Modern cryptography generally depends upon translating a message into one of an astronomical number of different intermediate representations, or ciphertexts, as selected by a key. If all possible intermediate representations have similar appearance, it may be necessary to try all possible keys (a brute force attack) to find the key which deciphers the message. By creating mechanisms with an astronomical number of keys, we can make this approach impractical.
Keying is the essence of modern cryptography. It is not possible to have a strong cipher without keys, because it is the uncertainty about the key which creates the "needle in a haystack" situation which is conventional strength. (A different approach to strength is to make every message equally possible, see: Ideal Secrecy.)
Nor is it possible to choose a key and then reasonably expect to use that same key forever. In cryptanalysis, it is normal to talk about hundreds of years of computation and vast effort spent attacking a cipher, but similar effort may be applied to obtaining the key. Even one forgetful moment is sufficient to expose a key to such effort. And when there is only one key, exposing that key also exposes all the messages that key has protected in the past, and all messages it will protect in the future. Only the selection and use of a new key terminates insecurity due to key exposure. Only the frequent use of new keys makes it possible to expose a key and not also lose all the information ever protected.
Cryptography is not an engineering science: It is not possible to know when cryptography is "working," nor how close to not-working it may be:
Cryptography may also be seen as a zero-sum game, where a
cryptographer competes against a
cryptanalyst. We might call
this the cryptography war.
Note that the successful cryptanalyst must keep good attacks secret, or the opposing cryptographer will just produce a stronger cipher. This means that the cryptographer is in the odd position of never knowing whether his or her best cipher designs are successful, or which side is winning.
Cryptographers are often scientists who are trained to ignore unsubstantiated claims. But the field of cryptography often turns the scientific method on its head, because almost never is there a complete proof of cryptographic strength in practice. In cryptography, scientists accept the failure to break a cipher as an indication of strength (that is the ad ignorantium fallacy), and then demand substantiation for claims of weakness. But there will be no substantiation when a cipher system is attacked and broken for real, while continued use will endanger all messages so "protected." Evidently, the conventional scientific approach of requiring substantiation for claims is not particularly helpful for users of cryptography.
Since the scientific approach does not provide the assurance of cryptographic strength that users want and need, alternative measures become appropriate:
Sometimes also said to include:
It is especially important to consider the effect the underlying
equipment has on the design.
Even apparently innocuous operating system functions, such
as the multitasking "swap file," can capture supposedly secure
information, and make that available for the asking.
Since ordinary disk operations generally do not even attempt to
overwrite data on disk, but instead simply make that storage
free for use, supposedly deleted data is, again, free for the
asking.
A modern cryptosystem will at least try to address such issues.
Quartz is a piezoelectric material, so a voltage across the terminals forces the quartz wafer to bend slightly, thus storing mechanical energy in physical tension and compression of the solid quartz. The physical mass and elasticity of quartz cause the wafer to mechanically resonate at a natural frequency depending on the size and shape of the quartz blank. The crystal will thus "ring" when the electrical force is released. The ringing will create a small sine wave voltage across electrical contacts touching the crystal, a voltage which can be amplified and fed back into the crystal, to keep the ringing going as oscillation.
Crystals are typically used to make exceptionally stable electronic oscillators (such as the clock oscillators widely used in digital electronics) and the relatively narrow frequency filters often used in radio.
It is normally necessary to physically grind a crystal blank to the desired frequency. While this can be automated, the accuracy of the resulting frequency depends upon the effort spent in exact grinding, so "more accurate" is generally "more expensive."
Frequency stability over temperature depends upon slicing the
original crystal at precisely the right angle.
Temperature-compensated crystal oscillators (TCXO's) improve
temperature stability by using other components which vary with
temperature to correct for crystal changes.
More stability is available in oven-controlled crystal oscillators
(OCXO's), which heat the crystal and so keep it at a precise
temperature despite ambient temperature changes.
Sometimes suggested in cryptography as the basis for a TRNG, typically based on phase noise or frequency variations. But a crystal oscillator is deliberately designed for high frequency stability; it is thus the worst possible type of oscillator from which to obtain and exploit frequency variations. And crystal oscillator phase noise (which we see as edge jitter) is typically tiny and must be detected on a cycle-by-cycle basis, because it does not accumulate. Detecting a variation of, say, a few picoseconds in each 100nSec period of a typical 10 MHz oscillator is not something we do on an ordinary computer.
Another common approach to a crystal oscillator TRNG is to
XOR
many such oscillators, thus getting a complex high-speed waveform.
(The resulting
digital
signal rate increases as the sum of all the oscillators.)
Unfortunately, the high-speed and asynchronous nature of the wave
means that
setup and
hold
times cannot be guaranteed to latch that data for subsequent use.
(Latching is inherent in, say, reading a value from a computer
input port.)
That leads to statistical bias and possible
metastable
operation.
Futher, the construction is essentially
linear
and may power up similarly each time it is turned on.
Current is analogous to the amount of water flow, as opposed
to pressure or
voltage.
A flowing electrical current will create a
magnetic field around the
conductor.
A changing electrical current may create an
electromagnetic field.
In some
RNG constructions, (e.g.,
BB&S and the
Additive RNG)
the system consists of multiple independent cycles, possibly of
differing lengths.
Since having a cycle of a guaranteed length is one of the main
requirements for an RNG, the possibility that a short cycle may
exist and be selected for use can be disturbing.
Sometimes people claim that they have a method to compress a file, and that they can compress it again and again, until it is only a byte long. Unfortunately, it is impossible to compress all possible files down to a single byte each, because a byte can only select 256 different results. And while each byte value might represent a whole file of data, only 256 such files could be selected or indicated.
Normally, compression is measured as the percentage size reduction; 60 percent is a good compression for ordinary text.
In general, compression occurs by representing the most-common data values or sequences as short code values, leaving longer code values for less-common sequences. Understanding which values or sequences are more common is the "model" of the source data. When the model is wrong, and supposedly less-common values actually occur more often, that same compression may actually expand the data.
Data compression is either "lossy," in which some of the information is lost, or "lossless" in which all of the original information can be completely recovered. Lossy data compression can achieve far greater compression, and is often satisfactory for audio or video information (which are both large and may not need exact reproduction). Lossless data compression must be used for binary data such as computer programs, and probably is required for most cryptographic uses.
Compressing plaintext data has the advantage of reducing the size of the plaintext, and, thus, the ciphertext as well. Further, data compression tends to remove known characteristics from the plaintext, leaving a compressed result which is more random. Data compression can simultaneously expand the unicity distance and reduce the amount of ciphertext available which must exceed that distance to support attack. Unfortunately, that advantage may be most useful with fairly short messages. Also see: Ideal Secrecy.
One goal of cryptographic data compression would seem to be minimize the statistical structure of the plaintext. Since such structure is a major part of cryptanalysis, that would seem to be a major advantage. However, we also assume that our opponents are familiar with our cryptosystem, and they can use the same decompression we use. So the opponents get to see the structure of the original plaintext simply by decompressing any trial decryption they have. And if the decompressor cannot handle every possible input value, it could actually assist the opponent by identifying wrong decryptions.
When using data compression with encryption, one pitfall is that many compression schemes add recognizable data to the compressed result. Then, when that compressed result is encrypted, the "recognizable data" represents known plaintext, even when only the ciphertext is available. Having some guaranteed known plaintext for every message could be a very significant advantage for opponents, and unwise cryptosystem design.
It is normally impossible to compress random-like ciphertext. However, some cipher designs do produce ciphertext with a restricted alphabet which can of course be compressed. Also see entropy.
Another possibility is to have a data decompressor that can take any random value to some sort of grammatical source text. That may be what is sometimes referred to as bijective compression. Typically, a random value would decompress into a sort of nonsensical "word salad" source text. However, the statistics of the resulting "word salad" could be very similar to the statistics of a correct message. That could make it difficult to computationally distinguish between the "word salad" and the correct message. If "bijective compression" imposes an attack requirement for human intervention to select the correct choice, that might complicate attacks by many orders of magnitude. The problem, of course, is the need to devise a compression scheme that decompresses random values into something grammatically similar to the expected plaintext. That typically requires a very extensive statistical model, and of course at best only applies to a particular class of plaintext message.
An extension of the "bijective" approach would be to add random
data to compressed text.
Obviously, there would have to be some way to delimit or otherwise
distinguish the plaintext from the added data, but that may be
part of the compression scheme anyway.
More importantly, the random data probably would have to be added
between the compressed text in some sort of keyed way, so that it
could not easily be identified and extracted.
The keying requirement would make this a form of encryption.
The result would be a
homophonic encryption, in that the
original plaintext would have many different compressed
representations, as selected by the added random data.
Having many different but equivalent representations allows the same
message to be sent multiple times, each time producing a different
encrypted result.
But it is also potentially dangerous, in that the compressed message
expands by the amount of the random data, which then may represent
a hidden channel.
Since, for encryption purposes, any random data value is as good
as another, that data could convey information about the key and
the user would never know.
Of course, the same risk occurs in
message keys or, indeed, almost any
nonce.
Most
electronic devices require DC
Contrary to naive expectations, a complex system almost never performs as desired when first realized. Both hardware and software system design environments generally deal with systems which are not working. When a system really works, the design and development process is generally over.
Debugging involves identifying problems, analyzing the source of those problems, then changing the construction to fix the problem. (Hopefully, the fix will not itself create new problems.) This form of interactive analysis can be especially difficult because the realized design may not actually be what is described in the schematics, flow-charts, or other working documents: To some extent the real system is unknown.
The most important part of debugging is to understand in great detail exactly what the system is supposed to do. In hardware debugging, it is common to repeatedly reset the system, and start a known sequence of events which causes a failure. Then, if one really does know the system, one can probe at various points and times and eventually track down the earliest point where the implemented system diverges from the design intent. The thing that causes the divergence is the bug. Actually doing this generally is harder than it sounds.
Software debugging is greatly aided by a design and implementation process that decomposes complex tasks into small, testable procedures or modules, and then actually testing those procedures. Of course, sometimes the larger system fails anyway, in which case the procedure tests were insufficient, but they can be changed and the fixed procedure re-tested. Sometimes the hardest part of the debugging is to find some s