For some reason, good cryptography is just much harder than it looks. This field seems to have a continuous flow of experts from other fields who offer cryptographic variations of ideas which are common in their other field. Now, there is nothing wrong with new ideas. But there are in fact many extremely intelligent and extremely well-educated people with wide-ranging scientific interests who are active in this field. It is very common to find that so-called "new" ideas have been previously addressed under another name or as a general concept. Try to get some background before you get in too deep.
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It is possible to transform or encipher a message or plaintext into "an intermediate form" or ciphertext in which the information is present but hidden. Then we can release the transformed message (the ciphertext) without exposing the information it represents.
By using different transformations, we can create many different ciphertexts for the exact same message. So if we select a particular transformation "at random," we can hope that anyone wishing to expose the message ("break" the cipher) can do no better than simply trying all available transformations (or on average, half) one-by-one. This is a brute force attack.
The difference between intermediate forms is the interpretation of the ciphertext data. Different ciphers and different keys will produce different interpretations (different plaintexts) for the exact same ciphertext. The uncertainty of how to interpret any particular ciphertext is how information is "hidden."
Naturally, the intended recipient needs to know how to transform or decipher the intermediate form back into the original message, and this is the key distribution problem.
By itself, ciphertext is literally meaningless, in the sense of having no one clear interpretation. In so-called perfect ciphers, any ciphertext (of appropriate size) can be interpreted as any message, just by selecting an appropriate key. In fact, any number of different messages can produce exactly the same ciphertext, by using the appropriate keys. In other ciphers, this may not always be possible, but it must always be considered. To attack and break a cipher, it is necessary to somehow confirm that the message we generate from ciphertext is the exact particular message which was sent.
Most of us have encountered a simple form of ciphering in grade school, and it usually goes something like this:
On a piece of lined paper, write the alphabet in order, one character per line:
A B C ...Then, on each line, we write another character to the right. In this second column, we also want to use each alphabetic character exactly once, but we want to place them in some different order.
A F B W C A ...
When we have done this, we can take any message and encipher it letter-by-letter.
To encipher a letter, we find that letter in the left column, then use the associated letter from the right column and write that down. Each letter in the right column thus becomes a substitute for the associated letter in the left column.
Deciphering is similar, except that we find the ciphertext letter in the right column, then use the associated plaintext letter from the left column. This is a little harder, because the letters in the right column are not in order. But if we wanted to, we could make a list where the ciphertext letters were in order; this would be the inverse of the enciphering transformation. And if we have both lists, enciphering and deciphering are both easy.
The grade school cipher is a simple substitution cipher, a streaming or repeated letter-by-letter application of the same transformation. That "transformation" is the particular arrangement of letters in the second column, a permutation of the alphabet. There can be many such arrangements. But in this case the key is that particular arrangement. We can copy it and give it to someone and then send secret messages to them. But if that sheet is acquired -- or even copied -- by someone else, the enciphered messages would be exposed. This means that we have to keep the transformation secret.
Now suppose we have a full notebook of lined pages, each of which contains a different arrangement in the second column. Suppose each page is numbered. Now we just pick a number and encipher our message using that particular page. That number thus becomes our key, which is now a sort of numeric shorthand for the full transformation. So even if the notebook is exposed, someone who wishes to expose our message must try about half of the transformations in the book before finding the right one. Since exposing the notebook does not immediately expose our messages, maybe we can leave the notebook unprotected. We also can use the same notebook for messages to different people, and each of them can use the exact same notebook for their own messages to each other. Different people can use the same notebook and yet still cipher messages which are difficult to expose without knowing the right key.
Note that there is some potential for confusion in first calling the transformation a key, and then calling the number which selects that transformation also a key. But both of these act to select a particular ciphertext construction from among many, and they are only two of the various kinds of "key" in cryptography.
The simple substitution used in our grade school cipher is very weak, because it "leaks" information: The more often a particular plaintext letter is used, the more often the associated ciphertext letter appears. And since language uses some letters more than others, simply by counting the number of times each ciphertext letter occurs we can make a good guess about which plaintext letter it represents. Then we can try our guess and see if it produces something we can understand. It usually does not take too long before we can break the cipher, even without having the key. In fact, we develop the ultimate key (the enciphering transformation) to break the cipher.
A "real" cipher will have a far more complex transformation. For example, the usual 64-bit block cipher will encipher 8 plaintext letters at the same time, and a change in any one of those letters will change all 8 letters of the resulting ciphertext. This is still simple substitution, but with a huge alphabet. Instead of using 26 letters, a 64-bit block cipher views each of 264 different block values as a separate letter, which is something like 18,000,000,000,000,000,000 "letters."
Suppose we have 256 pages of transformations in the notebook; this means there are exactly 256 different keys we can select from. If we write the number 256 in binary we get "100000000"; here the leftmost "1" represents 1 count of 28, and we call this an "8 bit" number. Or we can compute the base 2 logarithm by first taking the natural log of 256 (about 5.545) and dividing that by the natural log of 2 (about 0.693); this result is also 8. So we say that having 256 key possibilities is an "8 bit" keyspace. If we choose one of the 256 key values at random, and use that transformation to encipher a message, someone wishing to break our cipher should have to try about 128 decipherings before happening upon the correct one. The effort involved in trying, on average, 128 decipherings (a brute force attack) before finding the right one, is the design strength of the cipher.
If our notebook had 65,536 pages or keys (instead of just 256), we would have a "16 bit" keyspace. Notice that this number of key possibilities is 256 times that of an "8 bit" keyspace, while the key itself has only 8 bits more than the "8 bit" cipher. The strength of the "16 bit" cipher is the effort involved in trying, on average, 32,768 decipherings before finding the right one.
The idea is the same as a modern cipher: We have a machine which can produce a huge number of different transformations between plaintext and ciphertext, and we select one of those transformations with a key value. Since there are many, many possible keys, it is difficult to expose a message, even though the machine itself is not secret. And many people can use the exact same machine for their own secrets, without revealing those secrets to everyone who has such a machine.
One of the consequences of having a digital electronic machine
for ciphering, is that it operates very, very fast. This means that
someone can try a lot more possibilities than they could with a
notebook of paper pages. For example, a "40 bit" keyspace represents
about 1012 keys, which sounds like a lot. Unfortunately,
special-purpose hardware could try this many decipherings in under 5
seconds, which is not much strength.
A "56 bit" keyspace represents about
Under the theory that if a little is good, a lot is better, some people suggest using huge keys of 56,000 bits, or 1,000,000 bits, or even more. We can build such devices, and they can operate quickly. We can even afford the storage for big keys. What we do not have is a reason for such keys: a 128 bit key is "strong enough" to defeat even unimaginably extensive brute force attacks. While a designer might use a larger key for convenience, even immense keys cannot provide more strength than "strong enough." And while different attacks may show that the cipher actually has less strength, a huge keyspace is not going to solve those problems.
Some forms of cipher need relatively large key values simply to have a sufficiently large keyspace. Most number-theory based public key ciphers are in this class. Basically, these systems require key values in a very special form, so that most key values are unacceptable and unused. This means that the actual keyspace is much smaller than the size of the key would indicate. For this reason, public key systems need keys in the 1,000 bit range, while delivering strength comparable to 128 bit secret key ciphers.
Suppose we want to hide a name: We might think to innovate a different rule for each letter. We might say: "First we have 'T', but 't' is the 3rd letter in 'bottle' so we write '3.'" We can continue this way, and such a cipher could be very difficult to break. So why is this sort of thing not done? There are several reasons:
Modern ciphering is about constructions which attempt to solve these problems. A modern cipher has a large keyspace, which might well be controlled by a hashing computation on a language phrase we can remember. A modern cipher system can handle a wide range of message sizes, with exactly the same key, and normally provides a way to securely re-use keys. And the key can be much, much smaller than a long message.
Moreover, in a modern cipher, we expect the key to not be exposed, even if The Opponent has both the plaintext and the associated ciphertext for many messages (a known-plaintext attack). In fact, we normally assume that The Opponent knows the full construction of the cipher, and has lots of known plaintext, and still cannot find the key. Such designs are not trivial.
Sometimes a novice gives us 40 or 50 random-looking characters and says, "Bet you can't break this!" But that is not very realistic.
In actual use, we normally assume that a cipher will be widely distributed, and thus somewhat available. So we assume The Opponent will somehow acquire either the cipher machine or its complete design. We also assume a cipher will be widely used, so a lot of ciphered material will be around somewhere. We assume The Opponent will somehow acquire some amount of plaintext and the associated ciphertext. And even in this situation, we still expect the cipher to hide the key and other messages.
Potentially, cryptography can hide information while it is in transit or storage. In general, cryptography can:
Cryptography hides words: At most, it can only hide talking about contraband or illegal actions. But in a country with "freedom of speech," we normally expect crimes to be more than just "talk."
Cryptography can kill in the sense that boots can kill; that is, as a part of some other process, but that does not make cryptography like a rifle or a tank. Cryptography is defensive, and can protect ordinary commerce and ordinary people. Cryptography may be to our private information as our home is to our private property, and our home is our "castle."
Potentially, cryptography can hide secrets, either from others, or during communication. There are many good and non-criminal reasons to have secrets: Certainly, those engaged in commercial research and development (R&D) have "secrets" they must keep. Professors and writers may want to keep their work private, until an appropriate time. Negotiations for new jobs are generally secret, and romance often is as well, or at least we might prefer that detailed discussions not be exposed.
One possible application for cryptography is to secure on-line communications between work and home, perhaps leading to a society-wide reduction in driving, something we could all appreciate.
Cryptography can only hide information after it is encrypted and while it remains encrypted. But secret information generally does not start out encrypted, so there is normally an original period during which the secret is not protected. And secret information generally is not used in encrypted form, so it is again outside the cryptographic envelope every time the secret is used.
Secrets are often related to public information, and subsequent activities based on the secret can indicate what that secret is.
And while cryptography can hide words, it cannot hide:
And cryptography simply cannot protect against:
It is a joke to imagine that cryptography alone could protect most information against Government investigation. Cryptography is only a small part of the protection needed for "absolute" secrecy.
Usually, we arrange to select among a huge number of possible intermediate forms by using some sort of "pass phrase" or key. Normally, this is some moderately-long language phrase which we can remember, instead of something we have to write down (which someone else could then find).
Those who have one of the original keys can expose the information hidden in the message. This reduces the problem of protecting information to:
This is similar to locking our possessions in our house and keeping the keys in our pocket.
The physical key model reminds us of various things that can go wrong with keys:
Even absolutely perfect keys cannot solve all problems, nor can they guarantee privacy. Indeed, when cryptography is used for communications, generally at least two people know what is being communicated. So either party could reveal a secret:
When it is substantially less costly to acquire the secret by means other then a technical attack on the cipher, cryptography has pretty much succeeded in doing what it can do.
It is fairly easy to design a complex cipher program to produce a single complex, intermediate form. In this case, the program itself becomes the "key."
But this means that the deciphering program must be kept available to access protected information. So if someone steals your laptop, they probably will also get the deciphering program, which -- if it does not use keys -- will immediately expose all of your carefully protected data. This is why cryptography generally depends upon at least one remembered key, and why we need ciphers which can produce a multitude of different ciphertexts.
Cryptography deliberately creates the situation of "a needle in a haystack." That is, of all possible keys, only one should recover the correct message, and that one key is hidden among all possible keys. Of course, The Opponent might get lucky, but probably will have to perform about half of the possible decipherings to find the message.
To keep messages secret, it is important that a cipher be able to produce a multitude of different intermediate forms or ciphertexts. Clearly, no cipher can possibly be stronger than requiring The Opponent to check every possible deciphering. If such a brute force search is practical, the cipher is weak. The number of possible ciphertexts is the "design strength" of a cipher.
Each different ciphertext requires a different key. So the number of different ciphertexts which we can produce is limited to the number of different keys we can use. We describe the keyspace by the length in bits of the binary value required to represent the number of possible ciphertexts or keys.
It is not particularly difficult to design ciphers which may have a design strength of hundreds or thousands of bits, and these can operate just as fast as our current ciphers. However, the U.S. Government generally does not allow the export of data ciphers with a keyspace larger than about 40 bits, which is a very searchable value.
Recently, a 56-bit keyspace was searched (with special hardware) and the correct key found in about 56 hours. Note that a 56-bit key represents 216 times as many transformations as a 40-bit key. So, all things being equal, similar equipment might find a 40-bit key in about 3 seconds. But at the same rate, an 80-bit key (which is presumably 224 times as strong as a 56-bit key) would take over 100,000 years.
Keyspace alone only sets an upper limit to cipher strength; a cipher can be much weaker than it appears. An in-depth understanding or analysis of the design may lead to "shortcuts" in the solution. Perhaps a few tests can be designed, each of which eliminates vast numbers of keys, thus in the end leaving a searchable keyspace; this is cryptanalysis.
We understand strength as the ability to resist cryptanalysis. But this makes "strength" a negative quality (the lack of any practical attack), which we cannot measure. We can infer the "strength" of a cipher from the best known attack. We can only hope that The Opponent does not know of something much better.
Every user of cryptography should understand that all known ciphers (including the one time pad) are at least potentially vulnerable to some unknown technical attack. And if such a break does occur, there is absolutely no reason that we would find out about it. However, a direct technical attack may be one of the least likely avenues of exposure.
Cryptographic design may seem as easy as selecting a cipher from a book of ciphers. But ciphers, per se, are only part of a secure encryption system. It is common for a cipher system to require cryptographic design beyond simply selecting a cipher, and such design is much trickier than it looks.
The use of an unbreakable cipher does not mean that the encryption system will be similarly unbreakable. A prime example of this is the man-in-the-middle attack on public-key ciphers. Public-key ciphers require that one use the correct key for the desired person. The correct key must be known to cryptographic levels of assurance, or this becomes the weak link in the system: Suppose an Opponent can get us to use his key instead of the right one (perhaps by sending a faked message saying "Here is my new key"). If he can do this to both ends, and also intercept all messages between them (which is conceivable, since Internet routing is not secure), The Opponent can sit "in the middle." He can decipher each message (now in one of his keys), then re-encipher that message in the correct user key, and send it along. So the users communicate, and no cipher has been broken, yet The Opponent is still reading the conversation. Such are the consequences of system design error.
Cryptanalysis is hard; it is often tedious, repetitive, and very, very expensive. Success is never assured, and resources are always limited. Consequently, other approaches for obtaining the hidden information (or the key!) can be more effective.
Approaches other than a direct technical attack on ciphertext include getting the information by cunning, outright theft, bribery, or intimidation. The room or computer could be bugged, secretaries subverted, files burglarized, etc. Most information can be obtained in some way other than "breaking" ciphertext.
When the strength of a cipher greatly exceeds the effort required to obtain the same information in another way, the cipher is probably strong enough. And the mere fact that information has escaped does not necessarily mean that a cipher has been broken.
Although, in some cases, cryptanalysis might succeed even if the ciphering process was unknown, we would certainly expect that this would make The Opponents' job much harder. It thus can be argued that the ciphering process should remain secret. Certainly, military cipher systems are not actually published (although it may be assumed internally that the equipment is known to the other side). But in commercial cryptography we normally assume (see Kerckhoff's Requirements) that The Opponents will know every detail of the cipher (although not the key, of course). There are several reasons for this:
There is another level of secrecy here, and that is the trade secrecy involved with particular software designs. Very few large companies are willing to release source code for their products without some serious controls, and those companies may have a point. While the crypto routines themselves presumably might be patented, releasing that code alone probably would not support a thorough security evaluation. Source code might reasonably be made available to customers under a nondisclosure agreement, but this will not satisfy everyone. And while it might seem nice to have all source code available free, this will certainly not support an industry of continued cipher design and development. Unfortunately, there appears to be no good solution to this problem.
Currently, most ciphers are implemented in software; that is, by a program of instructions executed by a general-purpose computer. Normally, software is cheaper, but hardware can run faster, and nobody can change it. Of course, there are levels to hardware, from chips (which thus require significant interface software) to external boxes with communications lines running in and out. But there are several possible problems:
One logical possibility is the development of ciphering processors -- little ciphering computers -- in secure packaging. Limited control over the processor might allow a public-key authenticated software update, while otherwise looking like hardware. But probably most users will not care until some hidden software system is exposed on some computers.
There are a whole range of things which can distinguish one cipher from another. But perhaps the easiest and most useful distinction is that between stream ciphers and block ciphers.
Logically, a block cipher is just simple substitution: A block of plaintext data is collected and then substituted into an arbitrary ciphertext value. So a toy version of a block cipher is just a table look-up, much like the amusement ciphers in newspapers. Of course, a realistic block cipher has a block width which is far too large to hold the transformation in any physical table. Because of the large block size, the invertible transformation must be simulated, in some way dynamically constructed for each block enciphered.
In a block cipher, any possible permutation of "table" values is a potential key. So if we have a 64-bit block, there would theoretically be 264 factorial possible keys, which is a huge, huge value. But the well-known 64-bit block cipher DES has "only" 256 keys, which is as nothing in comparison. In part, this is because any real mechanism can only emulate the theoretical ideal of a huge simple substitution. But mostly, 56-bit keys have in the past been thought to be "large enough." Now we expect at least 128 bits, or perhaps somewhat more.
If a block cipher is a huge simple substitution, a stream cipher can be a small substitution which is in some way altered for each bit or byte enciphered. Clearly, repeatedly using a small unchanging substitution (or even a linear transformation) is not going to be secure in a situation where The Opponent will have a substantial quantity of known plaintext. One way to use a small transformation securely is to use a simple additive combiner to mix data with a really random confusion sequence; done properly, this is an "unbreakable" one-time pad.
Logically, a stream cipher can be seen as the general concept of repeatedly using a block transformation to handle more than one block of data. I would say that even the simple repeated use of a block cipher in ECB mode would be "streaming" the cipher. And use in more complex chaining modes like CBC are even more clearly stream meta-ciphers which use block transformations.
One common idea that comes up again and again with novice cryptographers is to take a textual key phrase, and then add (or exclusive-OR) the key with the data, byte-by-byte, starting the key over each time it is exhausted. This is a very simple and weak stream cipher, with a short and repeatedly-used running key and an additive combiner. I suppose that part of the problem in seeing this weakness is in distinguishing between different types of stream cipher "key": In a real stream cipher, even a single bit change in a key phrase would be expected to produce a different running key sequence, a sequence which would not repeat across a message of any practical size. In the weak version, a single bit change in the short running key would affect only one bit each time it was used, and would do so repeatedly, as the keying sequence was re-used over and over again. In any additive stream cipher, the re-use of a keying sequence is absolutely deadly. And a real stream cipher would almost certainly use a random message key as the key which actually protects data.
Public key ciphers are generally block ciphers, with the unusual property that one key is used to encipher, and a different, apparently unrelated key is used to decipher a message. So if we keep one of the keys private, we can release the other key (the "public" key), and anyone can use that to encipher a message to us. Then we use our private key to decipher any such messages. It is interesting that someone who enciphers a message to us cannot decipher their own message even if they want to.
The prototypical public key cipher is RSA, which uses the arithmetic of huge numeric values. These values may contain 1,000 bits or more (over 400 decimal digits), in which each and every bit is significant. The keyspace is much smaller, however, because there are very severe constraints on the keys; not just any random value will do. So a 1,000-bit public key may have a brute-force strength similar to a 128-bit secret key cipher.
Because public key ciphers operate on huge values, they are very slow, and so are normally used just to encipher a random message key. The message key is then used by a conventional secret key cipher which actually enciphers the data.
At first glance, public key ciphers apparently solve the key distribution problem. But in fact they also open up the new possibility of a man-in-the-middle attack. To avoid this, it is necessary to assure that one is using exactly the correct key for the desired user. This requires authentication (validation or certification) via some sort of secure channel, and that can take as much effort as a secure secret key exchange. A man-in-the-middle attack is extremely worrisome, because it does not involve breaking any cipher, which means that all the effort spent in cipher design and analysis and mathematical proofs and public review would be completely irrelevant.
The most important book in cryptography is:
The Codebreakers is the detailed history of cryptography, a book of style and adventure. It is non-mathematical and generally non-technical. But the author does explain why simple ciphers fail to hide information; these are the same problems addressed by increasingly capable cryptosystems. Various accounts show how real cryptography is far more than just schemes for enciphering data. A very good read.
Other important books include
Good books on "The Vietnam War" (and which have nothing to do with cryptography) include:
Normally, cryptanalysis is thought of as the way ciphers are broken. But cryptanalysis is really analysis -- the ways we come to understand a cipher in detail. Since most ciphers have weaknesses, a deep understanding can expose the best attacks for a particular cipher.
Two books often mentioned as introductions to classical cryptanalysis are:
Another problem with monoalphabetic substitution is that the most frequently used letters in the plaintext become the most frequently used letters in the ciphertext, and statistical techniques can be used to help identify which letters are which. Classically, multiple different alphabets (Polyalphabetic Substitution) or multiple ciphertext letters for a single plaintext letter (Homophonic Substitution) were introduced to avoid this. But in a modern computer version, we can continue to permute the single alphabet, as in Dynamic Substitution (see my article). Moreover, if the original "plaintext" is evenly distributed (which can be assured by a previous combining), then statistical techniques are little help.
Thus, it was often the restrictions on the general design -- necessary for "pen and paper" practicality -- which made these classical ciphers easy to attack. And the attacks which work well on specific classical versions may have very little chance on a modern very-general version of the same cipher.
Other books on cryptanalysis:
A perhaps overly famous book for someone programming existing ciphers or selecting protocols is:
Some other books I like include:
Although most authors recommend a background in Number Theory, I recommend some background in Coding Theory:
Those who would design ciphers would do well to follow the few systems whose rise and fall are documented in the open literature. Ciarcia [1] and Pearson [5] are an excellent example of how tricky the field is; first study Ciarcia (a real circuit design), and only then read Pearson (how the design is broken). Geffe [2] and Siegenthaler [8] provide a more technical lesson. Retter [6,7] shows that the MacLaren-Marsaglia randomizer is not cryptographically secure, and Kochanski [3,4] cracks some common PC cipher programs.