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UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Spring, 2010

UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Spring, 2010. Tuesday, 27 April Number-Theoretic Algorithms Chapter 31. Chapter Dependencies. Ch 31 Number-Theoretic Algorithms RSA. Math: Number Theory.

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UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Spring, 2010

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  1. UMass Lowell Computer Science 91.503Analysis of AlgorithmsProf. Karen DanielsSpring, 2010 Tuesday, 27 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 • Motivation: RSA • Basics • Euclid’s GCD Algorithm • Chinese Remainder Theorem • Powers of an Element • RSA Details

  4. Motivation: RSA

  5. 31.5 RSA Encryption source: 91.503 textbook Cormen et al.

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

  7. (31.19)* (31.26) RSA Cryptosystem source: 91.503 textbook Cormen et al., 3rd edition to be explained later…. (31.20) (31.36) (31.35) Assume M < n decode encode + EXAMPLE need efficient ways to compute P(M), S(C)

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

  9. EUCLID GCD EXTENDED-EUCLID (Eqn. 31.20) 2002 with thanks to Mark Micire

  10. Notes on Primality Testing • Efficient primality testing has been goal for > 2,000 years. • Early attempts required exponential time. • Miller-Rabin (Section 31.8) primality test is a randomized polynomial-time algorithm (1980’s). • 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 31.1 + EXAMPLE * (3.8) source: 91.503 textbook Cormen et al.

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

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

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

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

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

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

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

  20. Euclid’s GCD Algorithm

  21. Euclid’s GCD Algorithm * + EXAMPLE Also see Java code on course web site source: 91.503 textbook Cormen et al.

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

  23. Chinese Remainder Theorem

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

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

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

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

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

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

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

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

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

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

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

  35. Chinese Remainder Theorem Corollary 31.28. If n1, n2, …, nk are pairwise relatively prime and n = n1n2…nk, then, for any integers a1, a2, …, ak, the set of simultaneous equations for i = 1, 2, …, k, has a unique solution modulo n for the unknown x. * 31.29 source: 91.503 textbook Cormen et al.

  36. NumTheory source: 91.503 textbook Cormen et al. & Prof. Pecelli Example. Given the two equations what is a mod 65? Note that 65 = 5•13. The table of moduliwrt 5 and 13 for all integers in Z65. Table can be generated diagonally.

  37. NumTheory source: 91.503 textbook Cormen et al. & Prof. Pecelli Knowing that find a mod 65. We have a1 = 2, n1 = 5 , m1 = n/n1 = 13, a2 = 3, n2 = 13, m2 = n/n2 = 5. We can compute:

  38. Powers of an Element

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

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

  41. RSA Details

  42. 31.5 RSA Encryption source: 91.503 textbook Cormen et al.

  43. 31.6 RSA Digital Signature ? assume Alice also sends her name so Bob knows whose public key to use source: 91.503 textbook Cormen et al.

  44. (31.19) (31.26) RSA Cryptosystem source: 91.503 textbook Cormen et al., 3rd edition (31.20) (31.36) (31.35) decode encode need efficient ways to compute P(M), S(C)

  45. RSA Correctness (31.37) (31.38) 31.31) p by Thm 31.31 (Fermat) q 31.29 source: 91.503 textbook Cormen et al. 3rd edition

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