UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Fall, 2001 Lecture 7 Tuesday, 11/6/01 Number-Theoretic Algorithms Chapter.

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Presentation transcript:

UMass Lowell Computer Science Analysis of Algorithms Prof. Karen Daniels Fall, 2001 Lecture 7 Tuesday, 11/6/01 Number-Theoretic Algorithms Chapter 33

Chapter Dependencies Ch 33 Number-Theoretic Algorithms RSA Math: Number Theory You’re responsible for material in this chapter that we discussed in lecture. (Note that this does not include sections 33.8 or 33.9.)

Overview ä Motivation: RSA ä Basics ä Euclid’s GCD Algorithm ä Chinese Remainder Theorem ä Powers of an Element ä RSA Details

Motivation: RSA

RSA Encryption source: textbook Cormen et al.

RSA Digital Signature source: textbook Cormen et al.

RSA Cryptosystem source: textbook Cormen et al.

Basic Concepts

Division & Remainders source: textbook Cormen et al.

Equivalence Class Modulo n source: textbook Cormen et al.

Common Divisors source: textbook Cormen et al.

Greatest Common Divisor source: textbook Cormen et al.

Greatest Common Divisor source: textbook Cormen et al.

Relatively Prime Integers source: textbook Cormen et al.

Relatively Prime Integers source: textbook Cormen et al.

Greatest Common Divisor source: textbook Cormen et al.

Euclid’s GCD Algorithm

source: textbook Cormen et al.

Extended Euclid source: textbook Cormen et al.

Chinese Remainder Theorem

Modular Arithmetic source: textbook Cormen et al.

Finite Groups source: textbook Cormen et al.

Finite Groups source: textbook Cormen et al.

Finite Groups source: textbook Cormen et al.

Euler’s Phi Function source: textbook Cormen et al.

Lagrange’s Theorem source: textbook Cormen et al.

Finite Groups source: textbook Cormen et al.

Solving Modular Linear Eq source: textbook Cormen et al.

Solving Modular Linear Eq source: textbook Cormen et al.

Solving Modular Linear Eq source: textbook Cormen et al.

Chinese Remainder Theorem source: textbook Cormen et al.

Chinese Remainder Theorem source: textbook Cormen et al.

Powers of an Element

Theorems of Euler & Fermat source: textbook Cormen et al.

Modular Exponentiation source: textbook Cormen et al.

RSA Details

RSA Correctness source: textbook Cormen et al.