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60 CHAPTER 4. THE EUCLIDEAN ALGORITHM
of the Euclidean Algorithm will do this for us.
For example, suppose we had used k = 9 as our multiplicative key (modulo
26). To decipher we need to use the key c satisfying (9c)%26 = 1, or, equivalently,
9 × c ≡ 1 (mod 26). To find c we extend Euclid’s Algorithm by adding a
new line, a “coefficient line,” that begins with a 0 and a 1 and is continued by
the formula
old c − current q × current c = new c.
Examples:
(1) Find gcd(9, 26) and the inverse of 9 modulo 26.
For this example we will first work one step at a time.
Begin with gcd entries and 0 and 1:
From 26 ÷ 9,
From 9 ÷ 8,
From 8 ÷ 1,
{
q = 2, r = 8, and
0 − (2 × 1) = −2 :
{
q = 1, r = 1, and
1 − (1 × −2) = 3 :
{
q = 8, r = 0, and
2 − (8 × 3) = −26 :
q
r
c
q
r
c
q
r
c
q
r
c
26
0
26
0
26
0
26
0
2
9
1
2
9
1
2
9
1
9
1
8
-2
1
8
−2
1
8
−2
1
3
8
1
3
0
-26
Just as the final (non-zero) entry in the remainder line is the gcd, the entry
in the coefficient line under the gcd gives us the solution to 9 × x ≡ gcd
(mod 26). So 9 × 3 ≡ 1 (mod 26). (The −26 at the end of the coefficient
line is ignored.)