2.8. Cryptology
131
and since 5−1 ≡ 21 (mod 26), Dd (c) = 21(c − 9) = 21c − 7 ∈ Z/26Z = M. Table 2.2 on page 129 provides the numerical equivalents for each element in M = C. Using the above, we wish to decipher the following message and provide plaintext:
JIIXWD
To do this, we first translate each letter into the numerical equivalent in the alphabet of definition, via Table 2.2 on page 129 as follows. 9
8
8
23
22
3.
Then we apply Dd (m) to each of these numerical equivalents m to get the following. 0
5
5
8
13
4,
whose letter equivalents are affine Monoalphabetic ciphers suffer from the weakness that they can be cryptanalyzed via a frequency count of the letters in the ciphertext. For instance, if a letter occurs most frequently in ciphertext, we might guess the plaintext equivalent to be the letter E since E is the most commonly occurring letter in the English alphabet. If correct, this would lead to other decryptions and the cipher would be broken in this manner. For instance if the second most commonly occurring letter is guessed to be T, the second most commonly occurring letter in English, then we have more decryptions. Table 2.4 provides the letter frequencies for the English alphabet.
a 8.167 j 0.153 s 6.327
Relative Letter Frequencies for English Table 2.4 b c d e f g h 1.492 2.782 4.253 12.702 2.228 2.015 6.094 k l m n o p q 0.772 4.025 2.406 6.749 7.507 1.929 0.095 t u v w x y z 9.056 2.758 0.978 2.360 0.150 1.974 0.074
i 6.966 r 5.987
To prevent cryptanalysis via frequency analysis as described above, we may use ciphers that operate on blocks of plaintext rather than individual letters.