Cryptology - Unofficial St. Mary's College of California Web Site

64 CHAPTER 4. THE EUCLIDEAN ALGORITHM
in order for the multiplicative keynumber number k to have a multiplicative
inverse, its greatest common divisor with 26 must be 1. And if so, the Extended
Euclidean Algorithm computes the multiplicative inverse of k. This allows us
to finalize our description of the multiplicative ciphers.
Computing Keys for Decimation and Linear Ciphers:
1) Pick an integer k.
2) Use the Euclidean Algorithm to check that gcd(k, N) = 1.
3) If not, pick a new k.
4) If so, use the Euclidean Algorithm to find the inverse d. (If d is
negative, use N + d instead.)
Examples:
(1) Decipher OXXMT MEB. It was enciphered with key 19m + 12 modulo 26.
A Linear Cipher first multiplies and then divides. To undo this we must go
backwards: first subtract and then “divide.” Dividing, of course, means
multiplying by the inverse of 19 modulo 26, which from Figure 3.2 is 11.
ciphertext O X X M T M E B
ciphernumbers 15 24 24 13 20 13 5 2
−12 3 12 12 1 8 1 -7 -10
×11 33 132 132 11 88 11 -77 -110
%26 7 4 4 9 20 9 15 14
plaintext a d d i t i o n
The answer is addition.
(2) Decipher FRGPN I. The enciphering key was 14 modulo 27.
First we need to find the inverse of 14 modulo 27:
q
1 1 1
r
c
27
0
14
1
13
−1
The inverse is 2. So we decipher by multiplying by 2 modulo 27.
The answer is linear.
1
2
ciphertext F R G P N I
ciphernumbers 6 18 7 16 14 8
×2 12 36 14 32 28 16
%27 12 9 14 5 1 16
plaintext l i n e a r
(3) May 13 be used as the multiplicative part of a key in a Decimation or
Linear Cipher modulo 33
Yes: gcd(33, 13) = 1. (We didn’t even need Euclid’s Algorithm for this
one!)
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