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