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40 CHAPTER 3. THE INTRODUCTION OF NUMBERS
Why this happens is a very interesting question, but one that we will leave
unanswered for the moment. What we do need to take away from this example
instead is simply that division is more complicated in modular arithmetic. That
is not to say that it cannot be done, but rather that we will have to be careful
when we do it.
3.3 Decimation Ciphers
Before getting caught up with remainders and equivalence we were trying to
build a cipher built on multiplication rather than addition, but had difficulty
with translating the ciphernumbers back into cipherletters. With our work with
the remainder operator, this is now easy.
To use a Decimation Cipher:
1) Choose a proper keynumber k.
2) Convert the plaintext to plainnumbers.
3) Multiply each number by k to produce ciphernumbers.
4) Find the remainder %n of each ciphernumber.
5) Convert the reduced cipher numbers back into letters.
(We will study the meaning of “proper” in Chapter 4.)
Examples: Enciphering using a Decimation 7 Cipher.
(1) Encipher About Face using k = 5. (k for “key”.)
We follow the steps given in the definition:
plaintext a b o u t f a c e
plain numbers 1 2 15 21 20 6 1 3 5
multiplied numbers 5 10 75 105 100 30 5 15 25
%26 5 10 23 1 22 4 5 15 25
ciphertext E J W A V D E O Y
The ciphertext is EJWAV DEOY.
(2) Encipher Midnight using k = 5.
(3) Encipher Revolution using k = 19. 8
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
Recall that the weakness in the shift ciphers was that adjacent letters in
the plaintext alphabet remained adjacent even after being enciphered. The
7 Why “Decimation” It is not clear. In Roman times, to decimate your troops meant
to have the troops choose one-tenth of their numbers by lots, and then kill those soldiers.
Hopefully this cipher will not be that painful.
8 (2) MSTRS INV, (3) DQBYT IPOYF