Cryptography, Information Theory, and Error-Correction. Aiden A. Bruen

spaces):

      In this case, we have . If is a negative number, then it is replaced by . So, for example gets replaced by . The reasoning is that since . Alternatively, (see Chapter 19) we have .

The Caesar cipher is a simple example of a type of cipher known as a monoalphabetic cipher. Monoalphabetic ciphers belong to a class of ciphers known as substitution ciphers, in which all of the plain text characters in the message are substituted for another letter. Mathematically speaking, the enciphering process of a monoalphabetic cipher can be represented by the mapping of a plain text character to a cipher text character:

      Similarly, deciphering is represented by the mapping

      where is the cipher key, with . In the case of the Caesar cipher, . If is negative, then, as explained above, .

      To break such a cipher, one can decrypt the message by trying all 26 keys (this is referred to as an exhaustive search). For long messages, the likelihood of a cipher text decrypting to two intelligible messages is small.

      We mention here briefly affine ciphers. They are similar to Caesar ciphers in that they are simple substitution ciphers, but they differ in that enciphering involves not only addition, but also multiplication as well.

2.3 The Scytale Cipher

      The Scytale cipher was introduced around 500 BCE by the Spartans, who used this rather simple but effective method to send crucial planning data between generals and bureaucrats. Both the sender and receiver were in possession of a cylindrical tube of the same diameter. To encode the message, the sender would wrap a thin strip of paper around the tube, with the paper spiraling its way down the length of the tube. The message was then written on the strip, with letters being written one beneath the other until the end of the tube was reached. (The message was then continued by starting a new column of letters, and this process was repeated until the message was finished.) To encode, the sender would simply unwrap the paper, leaving a thin strip of unintelligible letters. To decode, the receiver only had to wrap the paper around their similar tube and read the message off in columns.

      It is often much simpler to duplicate the Scytale process with pencil and paper. Using a preselected number of rows (this number is the cipher key), write the message in columns. Then, “unwrap” the message by writing a string of characters consisting of the concatenated rows. For example, the message “THE ENEMY WILL ATTACK AT DAWN” is encrypted as follows:

Knowledge of the cipher key reduces the decryption process to a trivial matter. Using the fact that the key for this example is six, count up the total number of characters and divide by six. Doing so yields the period of the sequence, which is four. Thus, by taking the first, fifth, ninth, characters, one can reconstruct the columns. Upon completion, the message can be read off, column by column.

      The Scytale concept can be modified to create very complex ciphers. By arranging the plain text in varying matrix patterns and “unwrapping” in different ways, messages can be scrambled very effectively. The Scytale cipher, along with its variants, belongs to a class of ciphers called transposition ciphers, in which all plain text characters are present in the cipher text, but they appear in a substantially different order.

2.4 The Vigenère Cipher

      The next classical cipher of great interest is known as the Vigenère cipher. Although it is a relatively simple concept, it has influenced many ciphers used throughout history, including some that are still in use today. The idea is to modify the notion of Caesar ciphers, which were covered in Section 2.2. Instead of a single cipher key, we make use of an entire keyword, which is a key in the form of a string of letters chosen by the user. For example, suppose we decide to use the word “encode” as our keyword. The enciphering process is carried out as follows. The keyword is repeated as many times as necessary to span the length of the message. Each letter corresponds to a number between 0 and 25 (so , , , ). Then, to encipher, we add the corresponding numbers of the keyword and the message, subtracting 26 if the sum is bigger than 25. Then we switch back to letters to get the cipher text. For example, if the message letter is (