# Lecture: Modular Arithmetic

This lecture introduces 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. An algorithm to reconstruct a rational number from its modular image is presented. This algorithm also utilizes the extended Euclidean algorithm and the related concept of continued fraction approximation. Finally, a modular algorithm for integer multiplication is discussed.

### Background Material

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

• Lecture notes on the Euclidean algorithm.
• Also study Maple's Power, mod, mods, numtheory[phi] 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
• Fermat's theorem and Euler's identity
• 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
• Reconstruction of rational numbers
1. Continued fractions and the Euclidean algorithm
2. Convergents and the Extended Euclidean algorithm
3. Approximation by continued fractions
4. RATCONVERT algorithm

### Maple worksheets and programs and other resources

• mod.mw - Maple worksheet on modular arithmetic, fast powering, the CRT, and modular algorithms
• mod.ppt (mod.pdf)- slides on modular arithmetic
• ratconvert.mw - Maple worksheet illustrating an algorithm to construct a rational number from its modular image.
• Paul S. Wang, M. J. T. Guy, and J. H. Davenport, P-adic Reconstruction of Rational Numbers, ACM SIGSAM Bulletin, Volume 16 , Issue 2, 1982, pp. 2-3.
Created: Oct. 5, 2020 by jjohnson AT cs DOT drexel DOT edu