1. UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Fall, 2001 Lecture 7 Tuesday, 11/6/01 Number-Theoretic Algorithms Chapter 33

2. 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.)

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

4. Motivation: RSA

5. RSA Encryption source: 91.503 textbook Cormen et al.

6. RSA Digital Signature source: 91.503 textbook Cormen et al.

7. RSA Cryptosystem source: 91.503 textbook Cormen et al.

8. Basic Concepts

9. Division & Remainders source: 91.503 textbook Cormen et al.

10. Equivalence Class Modulo n source: 91.503 textbook Cormen et al.

11. Common Divisors source: 91.503 textbook Cormen et al.

12. Greatest Common Divisor source: 91.503 textbook Cormen et al.

13. Greatest Common Divisor source: 91.503 textbook Cormen et al.

14. Relatively Prime Integers source: 91.503 textbook Cormen et al.

15. Relatively Prime Integers source: 91.503 textbook Cormen et al.

16. Greatest Common Divisor source: 91.503 textbook Cormen et al.

17. Euclid’s GCD Algorithm

18. source: 91.503 textbook Cormen et al.

19. Extended Euclid source: 91.503 textbook Cormen et al.

20. Chinese Remainder Theorem

21. Modular Arithmetic source: 91.503 textbook Cormen et al.

22. Finite Groups source: 91.503 textbook Cormen et al.

23. Finite Groups source: 91.503 textbook Cormen et al.

24. Finite Groups source: 91.503 textbook Cormen et al.

25. Euler’s Phi Function source: 91.503 textbook Cormen et al.

26. Lagrange’s Theorem source: 91.503 textbook Cormen et al.

27. Finite Groups source: 91.503 textbook Cormen et al.

28. Solving Modular Linear Eq source: 91.503 textbook Cormen et al.

29. Solving Modular Linear Eq source: 91.503 textbook Cormen et al.

30. Solving Modular Linear Eq source: 91.503 textbook Cormen et al.

31. Chinese Remainder Theorem source: 91.503 textbook Cormen et al.

32. Chinese Remainder Theorem source: 91.503 textbook Cormen et al.

33. Powers of an Element

34. Theorems of Euler & Fermat source: 91.503 textbook Cormen et al.

35. Modular Exponentiation source: 91.503 textbook Cormen et al.

36. RSA Details

37. RSA Correctness source: 91.503 textbook Cormen et al.