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Week #5 – 23/25/27 September 2002 Prof. Marie desJardins
CMSC 203 / 0201 Fall 2002 Week #5 – 23/25/27 September 2002 Prof. Marie desJardins
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TOPICS Integers and algorithms Applications of number theory Matrices
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MON 9/23 INTEGERS AND ALGORITHMS (2.4)
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CONCEPTS / VOCABULARY Euclidean algorithm
Base b expansions of integers (especially binary, hexadecimal) Binary addition, binary multiplication, bit shifting
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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 Prove that if a = bq + r, where a, b, q, and r are integers, then gcd(a,b) = gcd(b,r). Use Lemma to prove that the Euclidean algorithm finds the gcd of its two arguments.
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WED 9/25 APPLICATIONS OF NUMBER THEORY (2.5 & 2.2 revisited)
** Homework #3 due today! ** ** (Ungraded) quiz today! **
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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
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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
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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.)
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FRI 9/27 MATRICES (2.6)
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CONCEPTS / VOCABULARY mxn matrices, rows, columns, equality
Matrix arithmetic, products Identity matrix Transpose At, symmetric matrices Zero-one matrix, join (), meet (), Boolean product
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Examples Example 2.1.1. Let A = 1 1 1 3 [ 2 0 4 6 ] 1 1 3 7
(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?
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Examples II Example 2.6.5: How many additions of integers and multiplications of integers are used by Algorithm to multiply two nxn matrices with integer entries? Example : Let A be an invertible matrix. Show that (An)-1 = (A-1)n whenever n is a positive integer.
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