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4.4. DECIPHERING DECIMATION AND LINEAR CIPHERS 63
Theorem 2 (The Extended Euclidean Algorithm) Given two integers n
and k with n > k, construct the table
q
r
c
n
0
⋆
k
1
· · ·
· · ·
· · ·
∗
q
by, at each step, from
form
a b
a b r
α β
α β α − qβ
quotient and r the remainder are computed from a ÷ b. Then
⋆
g
d
0
where q the
1. This process eventually terminates by producing a 0 in the “remainder”
line.
2. The last non-zero entry in the remainder line is gcd(n, k).
3. At each step, β × k ≡ b (mod n). In particular, if d is the entry in the
coefficient line directly below the gcd g = gcd(n, k), then a × d ≡ gcd(n, k)
(mod n).
4. When n and k are relatively prime, d is the inverse of i modulo n.
We haven’t formally proven all of this important theorem, although all of
the necessary ingredient have been stated. See any text on number theory, such
as [KRosen], for the missing parts. Nonetheless, the theorem does explain the
oddities in division we noticed at the end of Section 3.2.
Examples: Examples of Division in Modular Arithmetic.
(1) 3x ≡ 9 (mod 7) has the usual solution x = 3.
Since gcd(7, 3) = 1 divides 9 there is exactly gcd = 1 solution.
(2) 3x ≡ 9 (mod 12) has the usual solution x = 3, but also x = 7 and x = 11.
Since gcd(12, 3) = 3 divides 9 there are gcd = 3 solutions.
(3) 3x ≡ 8 (mod 7) has the unusual solution x = 5.
Since gcd(7, 3) = 1 divides 8 there is gcd = 1 solution.
(4) 3x ≡ 8 (mod 12) has no solutions.
Since gcd(12, 3) = 3 doesn’t divide 8 there are no solutions.
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
4.4 Deciphering Decimation and Linear Ciphers
We now finally understand what needs to be “proper” about the multiplying
keynumbers in Decimation and Linear Ciphers. When we work modulo 26,