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plaintext of the same language long enough to fill one sheet or so, and then we count the occurrences of each letter. We call the most frequently occurring letter the ‘first’, the next most occurring letter the ‘second’, the following most occurring letter the ‘third’, and so on, until we account for all the different letters in the plaintext sample.

Then we look at the ciphertext we want to solve and we also classify its symbols. We find the most occurring symbol and change it to the form of the ‘first’ letter of the plaintext sample, the next most common symbol is changed to the form of the ‘second’ letter, and the following most common symbol is changed to the form of the ‘third’ letter, and so on, until we account for all symbols of the cryptogram we want to solve.

Figure 6 The first page of al-Kind

’s manuscript On Deciphering Cryptographic Messages, containing the oldest known description of cryptanalysis by frequency analysis. Ibrahim A. Al-Kadi and Mohammed Mrayati, King Saud University, Riyadh.

Al-Kind

’s explanation is easier to explain in terms of the English alphabet. First of all, it is necessary to study a lengthy piece of normal English text, perhaps several, in order to establish the frequency of each letter of the alphabet. In English, e is the most common letter, followed by t, then a, and so on, as given in Table 1. Next, examine the ciphertext in question, and work out the frequency of each letter. If the most common letter in the ciphertext is, for example, J then it would seem likely that this is a substitute for e. And if the second most common letter in the ciphertext is P, then this is probably a substitute for t, and so on. Al-Kind

’s technique, known as frequency analysis, shows that it is unnecessary to check each of the billions of potential keys. Instead, it is possible to reveal the contents of a scrambled message simply by analysing the frequency of the characters in the ciphertext.

Table 1 This table of relative frequencies is based on passages taken from newspapers and novels, and the total sample was 100,362 alphabetic characters. The table was compiled by H. Beker and F. Piper, and originally published in Cipher Systems: The Protection Of Communication.

However, it is not possible to apply al-Kind

’s recipe for cryptanalysis unconditionally, because the standard list of frequencies in Table 1 is only an average, and it will not correspond exactly to the frequencies of every text. For example, a brief message discussing the effect of the atmosphere on the movement of striped quadrupeds in Africa would not yield to straightforward frequency analysis: ‘From Zanzibar to Zambia and Zaire, ozone zones make zebras run zany zigzags.’ In general, short texts are likely to deviate significantly from the standard frequencies, and if there are less than a hundred letters, then decipherment will be very difficult. On the other hand, longer texts are more likely to follow the standard frequencies, although this is not always the case. In 1969, the French author Georges Perec wrote La Disparition, a 200-page novel that did not use words that contain the letter e. Doubly remarkable is the fact that the English novelist and critic Gilbert Adair succeeded in translating La Disparition into English, while still following Perec’s shunning of the letter e. Entitled A Void, Adair’s translation is surprisingly readable (see Appendix A). If the entire book were encrypted via a monoalphabetic substitution cipher, then a naive attempt to decipher it might be stymied by the complete lack of the most frequently occurring letter in the English alphabet.

Having described the first tool of cryptanalysis, I shall continue by giving an example of how frequency analysis is used to decipher a ciphertext. I have avoided peppering the whole book with examples of cryptanalysis, but with frequency analysis I make an exception. This is partly because frequency analysis is not as difficult as it sounds, and partly because it is the primary cryptanalytic tool. Furthermore, the example that follows provides insight into the modus operandi of the cryptanalyst. Although frequency analysis requires logical thinking, you will see that it also demands guile, intuition, flexibility and guesswork.

Cryptanalysing a Ciphertext

PCQ VMJYPD LBYK LYSO KBXBJXWXV BXV ZCJPO EYPD KBXBJYUXJ LBJOO KCPK. CP LBO LBCMKXPV XPV IYJKL PYDBL, QBOP KBO BXV OPVOV LBO LXRO CI SX’XJMI, KBO JCKO XPV EYKKOV LBO DJCMPV ZOICJO BYS, KXUYPD: ‘DJOXL EYPD, ICJ X LBCMKXPV XPV CPO PYDBLK Y BXNO ZOOP JOACMPLYPD LC UCM LBO IXZROK CI FXKL XDOK XPV LBO RODOPVK CI XPAYOPL EYPDK. SXU Y SXEO KC ZCRV XK LC AJXNO X IXNCMJ CI UCMJ SXGOKLU?’

OFYRCDMO, LXROK IJCS LBO LBCMKXPV XPV CPO PYDBLK

Imagine that we have intercepted this scrambled message. The challenge is to decipher it. We know that the text is in English, and that it has been scrambled according to a monoalphabetic substitution cipher, but we have no idea of the key. Searching all possible keys is impractical, so we must apply frequency analysis. What follows is a step-by-step guide to cryptanalysing the ciphertext, but if you feel confident then you might prefer to ignore this and attempt your own independent cryptanalysis.

The immediate reaction of any cryptanalyst upon seeing such a ciphertext is to analyse the frequency of all the letters, which results in Table 2. Not surprisingly, the letters vary in their frequency. The question is, can we identify what any of them represent, based on their frequencies? The ciphertext is relatively short, so we cannot slavishly apply frequency analysis. It would be naive to assume that the commonest letter in the ciphertext, O, represents the commonest letter in English, e, or that the eighth most frequent letter in the ciphertext, Y, represents the eighth most frequent letter in English, h. An unquestioning application of frequency analysis would lead to gibberish. For example, the first word PCQ would be deciphered as aov.

However, we can begin by focusing attention on the only three letters that appear more than thirty times in the ciphertext, namely O, X and P. It is fairly safe to assume that the commonest letters in the ciphertext probably represent the commonest letters in the English alphabet, but not necessarily in the right order. In other words, we cannot be sure that O = e, X = t, and P = a, but we can make the tentative assumption that:

O = e, t or a, X = e, t or a, P = e, t or a.

Table 2 Frequency analysis of enciphered message.

In order to proceed with confidence, and pin down the identity of the three most common letters, O, X and P, we need a more subtle form of frequency analysis. Instead of simply counting the frequency of the three letters, we can focus on how often they appear next to all the other letters. For example, does the letter O appear before or after several other letters, or does is tend to neighbour just a few special letters? Answering this question will be a good indication of whether O represents a vowel or a consonant. If O represents a vowel it should appear before and after most of the other letters, whereas if it represents a consonant, it will tend to avoid many of the other letters. For example, the letter e can appear before and after virtually every other letter, but the letter t is rarely seen before or after b, d, g, j, k, m, q or v.

The table below takes the three most common letters in the ciphertext, O, X and P, and lists how frequently each appears before or after every letter. For example, O appears before A on 1 occasion, but never appears immediately after it, giving a total of 1 in the first box. The letter O neighbours the majority of letters, and there are only 7 that it avoids completely, represented by the 7 zeros in the O row. The letter X is equally sociable, because it too neighbours most of the letters, and avoids only 8 of them. However, the letter P is much less friendly. It tends to lurk around just a few letters, and avoids 15 of them. This evidence suggests that O and X represent vowels, while P represents a consonant.

Now we

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