Linear Ciphers - Introduction to Cryptography with Java Applets - page 111

E XAMPLE .

Encipher

WAR LOST

using an affine transformation with the ordinary alphabet. Use 7 as the multiplier, and 10
as the shift. Then recover the plaintext. The ordinary alphabet associations are shown in
Table 5.3:

ABCDE F GHI J KL MNOP QRS T UVWXYZ
0123456789 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5

TABLE 5.3

The plaintext message, when the letters are converted to their numerical equivalents,
yields

22 0 17 11 14 18 19

We then compute the following congruences:

C 7 P + 10 7 22 + 10 8 (mod 26)

C 7 P + 10 7 0 + 10 10 (mod 26)

C 7 P + 10 7 17 + 10 25 (mod 26)

C 7 P + 10 7 11 + 10 9 (mod 26)

C 7 P + 10 7 14 + 10 4 (mod 26)

C 7 P + 10 7 18 + 10 6 (mod 26)

C P

7 + 10 7 19 + 10 13 (mod 26)

The results of these calculations produce the ciphertext (in numbers)

8 10 25 9 4 6 13

or, the corresponding letters,

IKZJE GN

To recover the plaintext, we must solve the congruence

C 7 P + 10 (mod 26)

for P. Since 7 is relatively prime to 26, an inverse of it exists modulo 26, and it can be found
solving the congruence

7 x 1 (mod 26)

for x . Quick calculations using the extended Euclidean algorithm yield

x 15 (mod 26).