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9.2. HILL CIPHERS 173
We will be using a special type of 2 × 2 matrices known as involutions.
They are special in that enciphering and deciphering is done using the same
matrix. 6 We need to take care in setting up these matrices.
Method for using Hill Ciphers:
1. Pick any number a from 0 to 25.
2. Pick a number b that is relatively prime to 26. Find its multiplicative
inverse β from Figure 3.2.
3. Compute c = (1 − a 2 )β%26 and d = (−a)%26.
4. ( The enciphering ) ( and deciphering ) matrix is
a b a b
=
c d (1 − a 2 .
) · β −a
Examples: Suppose we choose a = 7 and b = 9. From Figure 3.2 b’s inverse
is 3 = β. Then (1 − a 2 ) · β = (1 − 7 2 ) × 3 = −146, ( so c = ) (−146)%26 = 12.
7 9
Since −7 ≡ 19 (mod 26), d = 19. So the matrix is .
12 19
(1) To( encipher ) ( ) matrix ( ) we first convert the plaintext into numbers: matrix
13 20 9
=
, and then compute
1 18 24
So matrix → VSHZSR.
(2) Decipher TIDDYVKM. 7
( ) ( ) ( ) ( 7 9 13 22 V
× ≡ = ,
12 19 1 19 S)
( ) ( ) ( ) (
7 9 20 8 H
× ≡ = ,
12 19 18 26 Z)
( ) ( ( ) ( 7 9 9 19 S
× ≡ =
12 19 24)
18 R)
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
Hill ciphers may be done with matrices of any square size, like 3 × 3 or 5 × 5,
and the entries in the array can be chosen with more latitude than our examples
suggest. See Exercise 9.11 for consideration of the latter.
6 For the use of non-involutions in Hill ciphers see Exercise 9.11.
7 multiply.