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3.4. DECIPHERING DECIMATION CIPHERS 43
To decipher a decimation cipher with key K we will need to find the inverse
of K modulo 26 (the number that satisfies (K × )%26 = 1). It turns out
that the inverse of 5 when working modulo 26 is 21: (5 × 21)%26 = 1. Since
multiplying by 1 changes nothing, following a multiplication by 5 with another
by 21 brings us back to our original message.
The same is true for several other pairs of numbers. (The numbers that have
multiplicative inverses are “proper” in the sense of the definition of Decimation
ciphers on page 40.) We will learn how to produce these pairs in Chapter 4.
For now, we simply list them.
enciphering key 1 3 5 7 9 11 15 17 19 21 23 25
deciphering key 1 9 21 15 3 19 7 23 11 5 17 25
Figure 3.2: Enciphering/Deciphering pairs modulo 26.
Notice that the deciphering key is seldom the same as the enciphering key. Do
not make the mistake of simply reusing the enciphering key, or using its negative.
A Decimation Cipher will only decipher properly when the correct key is used.
Examples:
(1) Decipher EJWAV DEOY, whose enciphering key was k = 5
From Figure 3.2 the multiplicative inverse of 5 is 21. So if we multiply the
ciphernumbers by 21 and then find their remainders mod 26 we should
have our message back. Let’s see.
ciphertext E J W A V D E O Y
ciphernumbers 5 10 23 1 22 4 5 15 25
×21 105 210 483 21 462 84 21 63 105
%26 1 2 15 21 20 6 1 3 5
plaintext a b o u t f a c e
Multiplying by 21 really did undo the multiplication by 5 and so did
decipher our message.
(2) Decipher MKCCKFI, if the enciphering number was 7.
The key was multiply by 7. However, if we try to divide by 7 to decipher
we have troubles:
ciphertext M K C C K F I
ciphernumbers 13 11 3 3 11 6 9
divide by 7 1.857 1.571 .429 .429 1.571 .857 1.286
plaintext