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)

Reading

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)

Existence proof
Constructive proof using the Extended Euclidean algorithm

Modular algorithm for integer multiplication

Reduce inputs mod sufficiently many primes (wordsize)
Multiply mod the primes
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

assign2.mw

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