Symmetric-Key Cryptography - Codes: The Guide to Secrecy from Ancient to Modern Times - page 110

the Caesar cipher is a simple example. Before giving the formal definition,
we may think of the following type of cipher as a combination of an additive
modular cipher , such as the Caesar cipher where we add a fixed amount, say,
b , to the plaintext numerical equivalent, m , to get m + b (mod n ), for a fixed
modulus n ; and a multiplicative modular cipher , where we take a plaintext

message unit and multiply it by a fixed amount, say, a , to get ciphertext given
by am (mod n ). Each of these values a,b , may be regarded as keys, so that used
in combination we get that the pairs ( a,b ) make up the keyspace . When we
combine all of this into one cryptosystem, we get the following type of cipher.

∈ N

A"ne Ciphers
An a , ne cryptosystem is a symmetric-key block cipher, defined as follows.
Suppose that both the message space M and ciphertext space are both Z /n Z

for some integer n> 1, and let the keyspace be given by

K = { ( a,b ): a,b ∈ Z /n Z and gcd( a,n )=1 } .

Then for e,d , the enciphering and deciphering transfor-

mations are given, respectively, by the following:

∈ K , and m,c ∈ Z /n Z

a − 1 ( c E e ( m ) ≡ am + b (mod n ), and D d ( c ) ≡ − b )(mod n ) .

The Caesar cipher is the simple application of an a0ne cipher, where n = 26,
a = 1, and b = 3. Here is a slightly more complicated example.

Example 3.1 Let n =26 , and M = C = Z / 26 Z . Define an a , ne cipher via,

≡ ≡

c (mod 26) , and D d ( c )

≡ −

E e ( m ) 5 m +11 21( c 11)(mod 26) ,

since 21 5 − 1 (mod 26) . Nowsupposethatweareawareofthefollowingcipher-

text having been enciphered with the above cryptosystem ( Kerckhoff's Principle
in action, see page 76 ) .

≡

c = (11 , 10 , 10 , 25 , 24 , 5 , 21 , 25 , 8 , 20 , 5 , 18 , 23 , 11 , 18 , 5 , 5 , 11 , 23 , 1) .

Then to decipher, we use D d on each ciphertext message unit. For instance,

D d (11) ≡ 21(11 − 11) ≡ 0(mod 26) , D d (10) ≡ 21(10 − 11) ≡ 5(mod 26) ,

andsoon, ( wherethereadercannowfillintheblanks ) ,toachievetheplaintext,

m =(0 , 5 , 5 , 8 , 13 , 4 , 2 , 8 , 15 , 7 , 4 , 17 , 18 , 0 , 17 , 4 , 4 , 0 , 18 , 24) .

Now, if we want the plaintext in English, we go to Table 1.3 on page 11, to get

ane ciphers are easy .