Many crypto textbooks assume that their readers are pure maths graduates, so let me start off with non-mathematical definitions. Cryptography refers to the science and art of designing ciphers; cryptanalysis to the science and art of breaking them; while cryptology, often shortened to just crypto, is the study of both. The input to an encryption process is commonly called the plaintext or cleartext, and the output the ciphertext. Thereafter, things get somewhat more complicated. There are a number of basic building blocks, such as block ciphers, stream ciphers, and hash functions. Block ciphers may either have one key for both encryption and decryption, in which case they're called shared-key (also secret-key or symmetric), or have separate keys for encryption and decryption, in which case they're called public-key or asymmetric. A digital signature scheme is a special type of asymmetric crypto primitive.
I will first give some historical examples to illustrate the basic concepts. I'll then fine-tune definitions by introducing the security models that cryptologists use, including perfect secrecy, concrete security, indistinguishability and the random oracle model. Finally, I'll show how the more important cryptographic algorithms actually work, and how they can be used to protect data. En route, I'll give examples of how people broke weak ciphers, and weak constructions using strong ciphers.
5.2 Historical background
Suetonius tells us that Julius Caesar enciphered his dispatches by writing ‘D’ for ‘A’, ‘E’ for ‘B’ and so on [1847]. When Augustus Caesar ascended the throne, he changed the imperial cipher system so that ‘C’ was now written for ‘A’, ‘D’ for ‘B’ etcetera. In modern terminology, we would say that he changed the key from ‘D’ to ‘C’. Remarkably, a similar code was used by Bernardo Provenzano, allegedly the capo di tutti capi of the Sicilian mafia, who wrote ‘4’ for ‘a’, ‘5’ for ‘b’ and so on. This led directly to his capture by the Italian police in 2006 after they intercepted and deciphered some of his messages [1538].
The Arabs generalised this idea to the monoalphabetic substitution, in which a keyword is used to permute the cipher alphabet. We will write the plaintext in lower case letters, and the ciphertext in upper case, as shown in Figure 5.1:
abcdefghijklmnopqrstuvwxyz SECURITYABDFGHJKLMNOPQVWXZ
Figure 5.1: Monoalphabetic substitution cipher
OYAN RWSGKFR AN AH RHTFANY MSOYRM OYSH SMSEAC NCMAKO; but it's a pencil and paper puzzle to break ciphers of this kind. The trick is that some letters, and combinations of letters, are much more common than others; in English the most common letters are e,t,a,i,o,n,s,h,r,d,l,u in that order. Artificial intelligence researchers have experimented with programs to solve monoalphabetic substitutions. Using letter and digram (letter pair) frequencies alone, they typically need about 600 letters of ciphertext; smarter strategies such as guessing probable words can cut this to about 150 letters; and state-of-the-art systems that use neural networks and approach the competence of human analysts are also tested on deciphering ancient scripts such as Ugaritic and Linear B [1196].
There are basically two ways to make a stronger cipher – the stream cipher and the block cipher. In the former, you make the encryption rule depend on a plaintext symbol's position in the stream of plaintext symbols, while in the latter you encrypt several plaintext symbols at once in a block.
5.2.1 An early stream cipher – the Vigenère
This early stream cipher is commonly ascribed to the Frenchman Blaise de Vigenère, a diplomat who served King Charles IX. It works by adding a key repeatedly into the plaintext using the convention that ‘A’ = 0, ‘B’ = 1, …, ‘Z’ = 25, and addition is carried out modulo 26 – that is, if the result is greater than 25, we subtract as many multiples of 26 as are needed to bring it into the range [0, …, 25], that is, [A, …, Z]. Mathematicians write this as
So, for example, when we add P (15) to U (20) we get 35, which we reduce to 9 by subtracting 26. 9 corresponds to J, so the encryption of P under the key U (and of U under the key P) is J, or more simply U + P = J. In this notation, Julius Caesar's system used a fixed key
| Plain |
tobeornottobethatisthequestion
|
| Key |
runrunrunrunrunrunrunrunrunrun
|
| Cipher |
KIOVIEEIGKIOVNURNVJNUVKHVMGZIA
|
Figure 5.2: Vigenère (polyalphabetic substitution cipher)
A number of people appear to have worked out how to solve polyalphabetic ciphers, from the womaniser Giacomo Casanova to the computing pioneer Charles Babbage. But the first published solution was in 1863 by Friedrich Kasiski, a Prussian infantry officer [1023]. He noticed that given a long enough piece of ciphertext, repeated patterns will appear at multiples of the keyword length.
In Figure 5.2, for example, we see ‘KIOV’ repeated after nine letters, and ‘NU’ after six. Since three divides both six and nine, we might guess a keyword of three letters. Then ciphertext letters one, four, seven and so on were all enciphered under the same keyletter; so we can use frequency analysis techniques to guess the most likely values of this letter, and then repeat the process for the remaining letters of the key.
5.2.2 The one-time pad
One way to make a stream cipher of this type proof against attacks is for the key sequence to be as long as the plaintext, and to never repeat. This is known as the one-time pad and was proposed by Gilbert Vernam during World War I [1003]; given any ciphertext, and any plaintext of the same length, there's a key that decrypts the ciphertext to the plaintext. So regardless of the amount of computation opponents can do, they're none the wiser, as given any ciphertext, all possible plaintexts of that length are equally likely. This system therefore has perfect secrecy.
Here's an example. Suppose you had intercepted a message from a wartime German agent which you knew started with ‘Heil Hitler’, and the first ten letters of ciphertext were DGTYI BWPJA. So the first ten letters of the one-time pad were wclnb tdefj, as shown in Figure 5.3:
| Plain |
heilhitler
|