1. UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Spring, 2009 Tuesday, 28 April Number-Theoretic Algorithms Chapter 31

2. Chapter Dependencies Ch 31 Number-Theoretic Algorithms RSA Math: Number Theory You’re responsible for material in this chapter that we discuss in lecture. (Note that this does not include sections 31.8 or 31.9.)

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. 31.5

6. RSA Digital Signature source: 91.503 textbook Cormen et al. 31.6 assume Alice also sends her name so Bob knows whose public key to use ?

7. RSA Cryptosystem (31.19) (31.26) (31.35) (31.36) encode decode source: 91.503 textbook Cormen et al. to be explained later…. need efficient ways to compute P(M), S(C)

8. RSA Dependence ä Correctness:  Euler’s  Function ä Fermat’s Theorem ä Chinese Remainder Theorem ä Efficiency: ä Modular Exponentiation ä Primality Testing ä Security: ä Difficulty of Factoring Large Integers see chart of result dependencies on next slide (courtesy of Mark Micire)… Need to show:

10. Notes on Primality Testing ä Efficient primality testing has been goal for > 2,000 years. ä Early attempts required exponential time. ä Miller-Rabin (Section 31.8) primality test is a randomized polynomial-time algorithm (1980’s). ä Agrawal, Kayal, Saxena provided a deterministic polynomial-time algorithm (2002).

11. Basic Concepts * Indicates that result is on chart of result dependencies

12. Division & Remainders source: 91.503 textbook Cormen et al. 31.1 (3.8) *

13. Equivalence Class Modulo n source: 91.503 textbook Cormen et al. (31.1) (31.2)

14. Common Divisors source: 91.503 textbook Cormen et al. (31.3) (31.4) (31.5) * *

15. Greatest Common Divisor source: 91.503 textbook Cormen et al. (31.6) (31.7) (31.8) (31.9) (31.10) 31.2 (3.8) (31.4) * *

16. Greatest Common Divisor source: 91.503 textbook Cormen et al. 31.3 (31.4) 31.2 31.4 *

17. Relatively Prime Integers source: 91.503 textbook Cormen et al. 31.6 31.2 31.2 *

18. Relatively Prime Integers source: 91.503 textbook Cormen et al. 31.7 31.6 31.1-6 *

19. Greatest Common Divisor source: 91.503 textbook Cormen et al. 31.9 (31.5) (3.8) (31.4) (31.3) (31.4) (31.3) (31.5) (31.14) (31.15) (31.14) (31.15) *

20. Euclid’s GCD Algorithm

21. source: 91.503 textbook Cormen et al. * Also see Java code on course web site

22. Extended Euclid source: 91.503 textbook Cormen et al. (31.16) 31.1 * *

23. Chinese Remainder Theorem

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

25. Finite Groups source: 91.503 textbook Cormen et al. size of this group is 6 size of this group is 8 31.2 Additive group mod 6 Multiplicative group mod 15 elements relatively prime to n

26. Finite Groups source: 91.503 textbook Cormen et al. 31.12

27. Finite Groups source: 91.503 textbook Cormen et al. 31.13 31.6 31.12 31.26

28. Euler’s Phi Function source: 91.503 textbook Cormen et al. (31.19) *

29. Lagrange’s Theorem source: 91.503 textbook Cormen et al. 31.15 *

30. Finite Groups 31.17 * source: 91.503 textbook Cormen et al. 31.18 31.19** additive subgroup generated by a where k

31. Solving Modular Linear Eq source: 91.503 textbook Cormen et al. 31.20 (31.4) *

32. Solving Modular Linear Eq source: 91.503 textbook Cormen et al. 31.22 31.18 31.18 31.22 31.24 * *

33. Solving Modular Linear Eq source: 91.503 textbook Cormen et al. 31.26 * *

34. Chinese Remainder Theorem source: 91.503 textbook Cormen et al. (31.23) 31.27 (31.23) (31.24) (31.25) (31.26) *

35. Chinese Remainder Theorem source: 91.503 textbook Cormen et al. 31.29 *

36. Powers of an Element

37. Theorems of Euler & Fermat source: 91.503 textbook Cormen et al. 31.30 31.31 * * 31.20

38. Modular Exponentiation source: 91.503 textbook Cormen et al. * Also see Java code on course web site

39. RSA Details

40. RSA Encryption source: 91.503 textbook Cormen et al. 31.5

41. RSA Digital Signature source: 91.503 textbook Cormen et al. 31.6 assume Alice also sends her name so Bob knows whose public key to use ?

42. RSA Cryptosystem (31.19) (31.26) (31.35) (31.36) encode decode source: 91.503 textbook Cormen et al. need efficient ways to compute P(M), S(C)

43. RSA Correctness source: 91.503 textbook Cormen et al. (31.35)(31.36) 31.31 31.29 by Thm 31.31 (Fermat)