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      For example, here we have two possible cipher alphabets, and we could encrypt a message by alternating between them. To encrypt the message hello, we would encrypt the first letter according to the first cipher alphabet, so that h becomes A, but we would encrypt the second letter according to the second cipher alphabet, so that e becomes F. To encrypt the third letter we return to the first cipher alphabet, and to encrypt the fourth letter we return to the second alphabet. This means that the first l is enciphered as P, but the second l is enciphered as A. The final letter, o, is enciphered according to the first cipher alphabet and becomes D. The complete ciphertext reads AFPAD. The crucial advantage of Alberti’s system is that the same letter in the plaintext does not necessarily appear as the same letter in the ciphertext, so the repeated l in hello is enciphered differently in each case. Similarly, the repeated A in the ciphertext represents a different plaintext letter in each case, first h and then l.

      Blaise de Vigenère, a French diplomat born in 1523, became acquainted with the writings of Alberti when, at the age of twenty-six, he was sent to Rome on a two-year mission. To start with, his interest in cryptography was purely practical and was linked to his work. Then, at the age of thirty-nine, Vigenère decided that he had accumulated enough money to be able to abandon his career and concentrate on a life of study. It was only then that he examined Albertu’s idea and turned it into a coherent and powerful new cipher, now known as the Vigenère cipher. The strength of the Vigenère cipher lies in its use of not one or two but twenty-six distinct cipher alphabets to encrypt a message. The first step in encipherment is to draw up a so-called Vigenère square, as shown in Table 3, a plaintext alphabet followed by twenty-six cipher alphabets, each shifted by one letter with respect to the previous alphabet. Hence, row 1 represents a cipher alphabet with a Caesar shift of 1, which means that it could be used to implement a Caesar shift cipher in which every letter of the plaintext is replaced by the letter one place further on in the alphabet. Similarly, row 2 represents a cipher alphabet with a Caesar shift of 2, and so on. The top row of the square, in lowercase, represents the plaintext letters. You could encipher each plaintext letter according to any one of the twenty-six cipher alphabets. For example, if cipher alphabet number 2 is used, then the letter a is enciphered as C, but if cipher alphabet number 12 is used, then a is enciphered as M.

      Figure 10 Blaise de Vigenère.

      If the sender were to use just one of the cipher alphabets to encipher an entire message, this would effectively be a simple Caesar cipher, which would be a very weak form of encryption, easily deciphered by an enemy interceptor. However, in the Vigenère cipher a different row of the Vigenère square (a different cipher alphabet) is used to encrypt different letters of the message. In other words, the sender might encrypt the first letter according to row 5, the second according to row 14, the third according to row 21, and so on.

      To unscramble the message, the intended receiver needs to know which row of the Vigenère square has been used to encipher each letter, so there must be an agreed system of switching between rows. This is achieved by using a keyword. To illustrate how a keyword is used with the Vigenère square to encrypt a sample message, let us encipher divert troops to east ridge, using the keyword WHITE. First of all, the keyword is spelled out above the message and repeated over and over again, so that each letter in the message is associated with a letter from the keyword, as shown on page 56. The ciphertext is then generated as follows. To encrypt the first letter, d, begin by identifying the key letter above it, W, which in turn defines a particular row in the Vigenère square. The row beginning with W, row 22, is the cipher alphabet that will be used to find the substitute letter for the plaintext d. We look to see where the column headed by d intersects the row beginning with W, which turns out to be at the letter Z.

Table 3 A Vigenère square.

      Consequently, the letter d in the plaintext is represented by Z in the ciphertext.

      To encipher the second letter of the message, i, the process is repeated. The key letter above i is H, so it is encrypted via a different row in the Vigenère square: the H row (row 7), which is a new cipher alphabet. To encrypt i, we look to see where the column headed by i intersects the row beginning with H, which turns out to be at the letter P. Consequently, the letter i in the plaintext is represented by P in the ciphertext. Each letter of the keyword indicates a particular cipher alphabet within the Vigenère square, and because the keyword contains five letters, the sender encrypts the message by cycling through five rows of the Vigenère square. The fifth letter of the message is enciphered according to the fifth letter of the keyword, E, but to encipher the sixth letter of the message we have to return to the first letter of the keyword. A longer keyword, or perhaps a keyphrase, would bring more rows into the encryption process and increase the complexity of the cipher. Table 4 shows a Vigenère square, highlighting the five rows (i.e., the five cipher alphabets) defined by the keyword WHITE.

      The great advantage of the Vigenère cipher is that it is invulnerable to the frequency analysis described in Chapter 1. For example, a cryptanalyst applying frequency analysis to a piece of ciphertext would usually begin by identifying the most common letter in the ciphertext, which in the case above is Z, and then assume that this represents the most common letter in English, e. In fact, the letter Z represents three different letters, d, r and s, but not e. This is clearly a problem for the cryptanalyst. The fact that a letter that appears several times in the ciphertext can represent a different plaintext letter on each occasion generates tremendous ambiguity. Equally confusing is the fact that a letter that appears several times in the plaintext can be represented by different letters in the ciphertext. For example, the letter o is repeated in troops, but it is substituted by two different letters – the oo is enciphered as HS.

Table 4 A Vigenère square with the rows defined by the keyword WHITE highlighted. Encryption is achieved by switching among the five highlighted cipher alphabets, defined by W, H, I, T and E.

      As well as being invulnerable to frequency analysis, the Vigenère cipher has an enormous number of keys. The sender and receiver can agree on any word in the dictionary or any combination of words, or even fabricate words. A cryptanalyst would be unable to crack the message by searching all possible keys because the number of options is simply too great.

      The traditional forms of substitution cipher, those that existed before

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