CSE 20: Discrete Mathematics for Computer Science Prof. Shachar Lovett

Today’s Topics: 1. GCD 2. Euclid’s algorithm 3. Extended Euclid’s algorithm 2

1. GCD The poor man version of prime factorization 3

GCD  Greatest common divisor  Given two positive integers a,b, their GCD is the largest integer n such that n|a and n|b 4

GCD  What is GCD(20,30)? A. 5 B. 10 C. 20 D. 30 E. Other 5

GCD  What is GCD( , )? 6

GCD  How can we compute GCD(a,b)?  Simple way: (i) Compute prime factorization of a,b (ii) Take common primes and prime powers  Example: if a= and b= then GCD(a,b)=  However, we believe that computing the prime factorization of large numbers is hard…  Euclid’s algorithm provides a much faster way 7

2. Euclid’s algorithm Fast GCD 8

Euclid’s algorithm

(a,b) (b,a mod b)

Euclid’s algorithm

a b Example run: a=20, b=30

Euclid’s algorithm  The same basic questions 1. Does it always terminate? 2. Does it return the correct answer? 3. How fast is it?

Euclid’s algorithm: termination

The value of a keeps decreasing, which proves termination

Euclid’s algorithm: correctness g=gcd(a,b)

Euclid’s algorithm: correctness

g=gcd(a,b) Proved!

Euclid’s algorithm: speed How many iterations?

Euclid’s algorithm: speed

3. Extended Euclid’s algorithm Using algorithms to do math! 28

Extended Euclid’s algorithm