1 / 13

CMSC 203 / 0201 Fall 2002

Explore concepts such as Euclidean algorithm, base expansions, linear congruence, matrix arithmetic, and more.

egrado
Download Presentation

CMSC 203 / 0201 Fall 2002

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CMSC 203 / 0201Fall 2002 Week #5 – 23/25/27 September 2002 Prof. Marie desJardins

  2. TOPICS • Integers and algorithms • Applications of number theory • Matrices

  3. MON 9/23INTEGERS AND ALGORITHMS (2.4)

  4. CONCEPTS / VOCABULARY • Euclidean algorithm • Base b expansions of integers (especially binary, hexadecimal) • Binary addition, binary multiplication, bit shifting

  5. 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.

  6. WED 9/25APPLICATIONS OF NUMBER THEORY (2.5 & 2.2 revisited) ** Homework #3 due today! ** ** (Ungraded) quiz today! **

  7. 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

  8. 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

  9. 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.)

  10. FRI 9/27MATRICES (2.6)

  11. CONCEPTS / VOCABULARY • mxn matrices, rows, columns, equality • Matrix arithmetic, products • Identity matrix • Transpose At, symmetric matrices • Zero-one matrix, join (), meet (), Boolean product

  12. 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?

  13. 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.

More Related