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62 CHAPTER 4. THE EUCLIDEAN ALGORITHM
Before summarizing, let’s do a couple more examples.
Examples:
(1) Does 7 have a multiplicative inverse modulo 26 If so, what is it
This is asking us to find gcd(7, 26). If the gcd is 1, then we may solve
7x ≡ 1 (mod 26).
q
r
c
26
0
3
7
1
1
5
−3
2
2
4
2
1
−11
0
Since gcd(7, 26) = 1, 7 and 26 are relatively prime and 7 does have a
multiplicative inverse modulo 26. It is −11 or its positive version 15
(since −11 ≡ 15 (mod 26)). Thus 7 may be used as the multiplicative
enciphering key in a decimation or linear cipher, and 15 will be (part of)
the deciphering key.
(2) Does 8 have a multiplicative inverse modulo 26 If so, what is it
q
r
c
26
0
3
8
1
4
2
−3
0
Since gcd(8, 26) = 2 is not 1, 8 and 26 are not relatively prime, and so
there is no multiplicative inverse for 8. Hence 8 may not be used as the
multiplicative enciphering key in a Decimation or Linear Cipher.
(3) Find gcd(49, 13) and, if possible, solve 13x ≡ 1 (mod 49)
q
r
c
49
0
3
13
1
1
10
−3
3
3
4
1
1
−12
0
gcd(49, 13) = 1 so they are relatively prime and since −12 ≡ 37 (mod 49),
the inverse of 13 modulo 49 is 37.
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
Euclid’s Extended Algorithm is quite simple. Nonetheless it is very powerful.
Let’s end this section by summarizing it and some of its consequences.
Because we will mostly be using it for computing a deciphering key from a
known enciphering key k and modulus n, we state it in terms of k and n.