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CMSC 203 / 0201 Fall 2002 Week #5 – 23/25/27 September 2002 Prof. Marie desJardins September1999 TOPICS Integers and algorithms Applications of number theory Matrices September1999 October 1999 MON 9/23 INTEGERS AND ALGORITHMS (2.4) September1999 CONCEPTS / VOCABULARY Euclidean algorithm Base b expansions of integers (especially binary, hexadecimal) Binary addition, binary multiplication, bit shifting September1999 October 1999 Examples Exercise 2.4.9: Devise a simple method (algorithm) for converting from hexadecimal notation to binary notation. (p. 128) Apply the Euclidean algorithm to find the greatest common divisor of 91 and 287. Lemma 2.4.1. Prove that if a = bq + r, where a, b, q, and r are integers, then gcd(a,b) = gcd(b,r). Use Lemma 2.4.1 to prove that the Euclidean algorithm finds the gcd of its two arguments. September1999 October 1999 WED 9/25 APPLICATIONS OF NUMBER THEORY (2.5 & 2.2 revisited) ** Homework #3 due today! ** ** (Ungraded) quiz today! ** September1999 CONCEPTS / VOCABULARY gcd as linear combination Linear congruence Fermat’s Little Theorem Applications: From Section 2.3: Hashing, pseudorandom numbers, cryptology From Section 2.5: Chinese remainder theorem, computer arithmetic, pseudoprimes / Fermat’s Little Theorem, public key cryptography, RSA encryption/decryption September1999 October 1999 Examples Exercise 2.5.1: Express the gcd of each of the following pairs of integers as a linear combination of these integers: (c) 36, 48 (e) 117, 213 (h) 3454, 4666 September1999 October 1999 Examples II Exercise 2.5.9: Show that if a and m are relatively prime positive integers, then the inverse of a modulo m is unique modulo m. (Hint: Assume that there are two solutions b and c of the congruence ax = 1 mod m. Use Theorem 2 to show that b = c mod m.) September1999 October 1999 FRI 9/27 MATRICES (2.6) September1999 CONCEPTS / VOCABULARY mxn matrices, rows, columns, equality Matrix arithmetic, products Identity matrix Transpose At, symmetric matrices Zero-one matrix, join (), meet (), Boolean product September1999 October 1999 Examples Example 2.1.1. Let A = 1113 [2046] 1137 (a) What size is A? (b) What is the third column of A? (c) What is the second row of A? (d) What is the element of A in the (3,2)th position? (e) What is At? What is AA? What is AAt? September1999 October 1999 Examples II Example 2.6.5: How many additions of integers and multiplications of integers are used by Algorithm 2.6.1 to multiply two nxn matrices with integer entries? Example 2.6.21: Let A be an invertible matrix. Show that (An)-1 = (A-1)n whenever n is a positive integer. September1999 October 1999