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These definitions of addition and multiplication in
Properties 1, 3-5 say that Properties 1-10 establish that Since additive inverses exist in For example, to compute 11 - 18 in We present the Shift Cipher in Figure 1.2. It is defined over REMARK For the particular key K = 3, the cryptosystem is often called the Caesar Cipher, which was purportedly used by Julius Caesar. We would use the Shift Cipher (with a modulus of 26) to encrypt ordinary English text by setting up a correspondence between alphabetic characters and residues modulo 26 as follows: A ↔ 0, B ↔ 1, . . . , Z ↔ 25. Since we will be using this correspondence in several examples, lets record it for future use: A small example will illustrate. Example 1.1 Suppose the key for a Shift Cipher is K = 11, and the plaintext is wewillmeetatmidnight. We first convert the plaintext to a sequence of integers using the specified correspondence, obtaining the following:
Next, we add 11 to each value, reducing each sum modulo 26:
Finally, we convert the sequence of integers to alphabetic characters, obtaining the ciphertext: HPHTWWXPPELEXTOYTRSE To decrypt the ciphertext, Bob will first convert the ciphertext to a sequence of integers, then subtract 11 from each value (reducing modulo 26), and finally convert the sequence of integers to alphabetic characters. REMARK In the above example we are using upper case letters for ciphertext and lower case letters for plaintext, in order to improve readability. We will do this elsewhere as well. If a cryptosystem is to be of practical use, it should satisfy certain properties. We informally enumerate two of these properties now.
The second property is defining, in a very vague way, the idea of security. The process of attempting to compute the key K, given a string of ciphertext y, is called cryptanalysis. (We will make these concepts more precise as we proceed.) Note that, if Oscar can determine K, then he can decrypt y just as Bob would, using dK. Hence, determining K is at least as difficult as determining the plaintext string x. We observe that the Shift Cipher (modulo 26) is not secure, since it can be cryptanalyzed by the obvious method of exhaustive key search. Since there are only 26 possible keys, it is easy to try every possible decryption rule dK until a meaningful plaintext string is obtained. This is illustrated in the following example.
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