Modular Arithmetic

Modular Arithmetic Shirley Moore CS4390/5390 Fall 2013 http://svmoore.pbworks.com/ September 5, 2013

Agenda • Intro to Matlab by Rogelio Long (cont.) (15 min) • Discuss homework from last class (15 min) • Modular arithmetic (20min) • Fermat’s Little Theorem (20 min) • Wrap-up and preparation for next class (5 min)

Modular Arithmetic • The number X (mod Y) is the remainder when X is divided by Y. • For example: 7 (mod 3) is 1 because 7 = 2 * 3 + 1. That is, when you divide 7 by 3, you get a remainder of 1. • The "modulo Y" terminology can also be used in the following way: Z = X (mod Y), meaning that Z and X have the same remainder when divided by Y. For example: 7 = 25 (mod 3)because 7 = 2 * 3 + 1 and 25 = 8 * 3 + 1

Modular Addition • Find the units digit of the sum 2403 + 791 + 688 + 4339 i.e., 2403 + 791 + 688 + 4339 (mod 10) • In general, if a, b, c, and d are integers and m is a positive integer such that a = c (mod m) and b = d (mod m) then a + b = c + d (mod m) Proof:

Modular Multiplication • When you take products of many numbers and you want to find their remainder modulo n, you never need to worry about numbers bigger than the square of n. • Pick any two numbers x and y, and look at their remainders (mod 7): a = x (mod 7) b= y (mod 7) • Compare the remainder modulo 7 of the products xy and ab: xy (mod 7) with ab (mod 7) • For example, try x = 26, y = 80

Modular Multiplication of Many Numbers • If we want to multiply many numbers modulon, we can first reduce all numbers to their remainders. Then, we can take any pair of them, multiply and reduce again. • For example, suppose we want to find X = 36 * 53 * 91 * 17 * 22 (mod 29) • What is the largest number we have to multiply?

Modular Exponentiation • Suppose we would like to calculate 1143 (mod 13). • The straightforward method would be to multiply 11 by 11, then to multiply the result by 11, and so forth. This would require 42 multiplications. • We can save a lot of multiplications if we do the following: • First write 43 as a sum of powers of 2: 43 = 32 + 8 + 2 + 1 • That means that 1143 = 1132 * 118 * 112 * 11 . • How many multiplications are required, and what is the largest number we have to multiply?

Fermat’s Little Theorem • First stated by Pierre de Fermat in 1640 • First published proof by Leonhard Euler in 1736 • Highly useful for simplifying the computation of exponents in modular arithmetic • Corollary by Euler serves as the basis for RSA encryption • Theorem: If p is a prime number and p does not divide a, then ap-1 = 1 (mod p) • Example: p = 5 • Proof: See http://www.youtube.com/watch?v=w0ZQvZLx2KA • Use FLT to find 3100,000 (mod 53)

Use FLT to prove a number is composite without factoring it • To prove n is composite, find some a such that a is not a multiple of n and an-1 ≠ 1 (mod n). • Is 91 a prime number? Try a = 2. • 75 = 1 (mod 6), so is 6 prime? • True or False: If bn-1 = 1 (mod n) for all b such that b is not a multiple of n, then n is prime.

Modular Arithmetic in Matlab • http://www.mathworks.com/help/symbolic/mupad_ug/modular-arithmetic.html • mod • mods • powermod

Preparation for Next Class • Workon Homework 1 (turn in for grade, due September 12). Ask questions next class.

Modular Arithmetic