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For now, we can simply compute it by trial and error: For a given a we simply try all possible values a−1 until we obtain: a · a−1 ≡ 1 mod 26 For instance, if a = 3, then a−1 = 9 since 3 · 9 = 27 ≡ 1 mod 26. Note that a−1 also always fulfills the condition gcd(a−1 , 26) = 1 since the inverse of a−1 always exists. In fact, the inverse of a−1 is a itself. Hence, for the trial-and-error determination of a−1 one only has to check the values given in Eq. 2). 12. Let the key be k = (a, b) = (9, 13), and the plaintext be ATTACK = x1 , x2 , .

For further reading on Vernam’s machine, the book by Kahn [97] is recommended. 1 Random Number Generators As we saw in the previous section, the actual encryption and decryption of stream ciphers is extremely simple. The security of stream ciphers hinges entirely on a “suitable” key stream s0 , s1 , s2 , . .. Since randomness plays a major role, we will first learn about the three types of random number generators (RNG) that are important for us. 2 Random Numbers and an Unbreakable Stream Cipher 35 True Random Number Generators (TRNG) True random number generators (TRNGs) are characterized by the fact that their output cannot be reproduced.

4 Modular Arithmetic and More Historical Ciphers 15 12 ≡ 3 mod 9, 3 is a valid remainder since 9|(12 − 3) 12 ≡ 21 mod 9, 21 is a valid remainder since 9|(21 − 3) 12 ≡ −6 mod 9, −6 is a valid remainder since 9|(−6 − 3) where the “x|y” means “x divides y”. There is a system behind this behavior. The set of numbers {. . , −24, −15, −6, 3, 12, 15, 24, . } form what is called an equivalence class. There are eight other equivalence classes for the modulus 9: {. . , −27, −18, −9, 0, 9, 18, 27, . } {.

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