Symmetric-Key Cryptography - Codes: The Guide to Secrecy from Ancient to Modern Times - page 112

modulo n (see Definition A.20 in Appendix A on page 477). Then for m ∈ M ,

e ∈ K , c ∈ C , the enciphering transformation is given by

E e ( m )= me = c,

and the deciphering transformation is given by

D d ( c )= cd = ce − 1 .

Note that

e is invertible, namely, d = e − 1 exists if and only if gcd(det( e ) ,n )=1 .

(See Appendix A, page 493.)

The above definition tells us that changing one character of plaintext will
usually change r letters of ciphertext. Hence, frequency analysis of ciphertext
is less effective, especially for large r . However, the Hill cipher succumbs to
known plaintext attacks (see Footnote 2.9 on page 84). Now we illustrate it
with a simple example, which is intended for the uninitiated reader. Merely
revisit pages 491-493 in Appendix A to see the simple methods for two-by-two
matrices illustrated there.

Example 3.2 Let n =26 and r =2 ,so

) 2 , A = Z / 26 Z , M = C =( Z / 26 Z

and K consists of invertible 2 × 2 matrices with entries from Z / 26 Z . Thus, if

e , then gcd(det( e ) , 26) = 1 ( see part (b) of Theorem A.25 on page 493) .

For instance, take

∈ K

e = 25
34

for which det( e )= − 7 . Suppose that we want to encipher the plaintext:

message by matrix .

First we get the numerical equivalents from Table 1.3 on page 11:

12 , 4 , 18 , 18 , 0 , 6 , 4 , 1 , 24 , 12 , 0 , 19 , 17 , 8 , 23 . (3.1)

Thus, we may set

m 1 = (12 , 4) , m 2 = (18 , 18) , m 3 =(0 , 6) , m 4 =(4 , 1) , m 5 = (24 , 12) ,

m 6 =(0 , 19) , m 7 = (17 , 8) , and m 8 = (23 , 25) ,
where we have used a “z” with numerical value 25 to make up the last ordered
pair m 8 . Now use the enciphering transformation defined in the Hill Cipher.
( Remember that once we get the entries in the final ciphertext matrix, we have