# Lecture 6: Modular Arithmetic and Fast Powering

This lecture introces the concept of modular arithmetic and presents a fast algorithm to compute the power of a number modulo another number. Constructive algorithms, based on the extended Euclidean algorithm, to compute modular inverses and solve the Chinese Remainder problem is presented.  Finally, a modular algorithm for integer multiplication is discussed.

### Background Material

• Equivalence relations.
• Division with quotient and remainder.
• Euclidean algorithm (and extended version)

• Lecture 5 notes on the Euclidean algorithm.
Also study Maple's Power, mod, mods, and chrem functions.

### Topics

• Equivalence relations
• Equivalence modulo n
• Equivalence classes modulo n
• Modular arithmetic (Z_n)
1. definition of + and *
2. proof that it is well defined
• modular inverses
1. Examples
2. Condition for inverses to exist
3. Computing with the extended Euclidean algorithm
4. Computing with Fermat's theorem
5. The finite field Z_p
• Fast algorithm for powering
1. repeated multiplication
2. binary powering
3. Analysis of binary powering
• Chinese Remainder Theorem (CRT)
1. Existence proof
2. Constructive proof using the Extended Euclidean algorithm
• Modular algorithm for integer multiplication
1. Reduce inputs mod sufficiently many primes (wordsize)
2. Multiply mod the primes
3. Recover integer product using the CRT

### Maple worksheets and programs and other resources

• mod.mws - Maple worksheet on modular arithmetic, fast powering, the CRT, and modular algorithms
• mod.ppt (< ahref="mod.pdf">mod.pdf)- slides on modular arithmetic

### Assignments

Created: Oct. 20, 2005 by jjohnson@cs.drexel.edu