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Number Theory Algorithms and Cryptography Algorithms

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  1. Number Theory Algorithms and Cryptography Algorithms Analysis of Algorithms Prepared by John Reif, Ph.D.

  2. Number Theory Algorithms • GCD • Multiplicative Inverse • Fermat & Euler’s Theorems • Public Key Cryptographic Systems • Primality Testing

  3. Number Theory Algorithms (cont’d) • Main Reading Selections: • CLR, Chapter 33

  4. Euclid’s Algorithm • Greatest Common Divisor • Euclid’s Algorithm

  5. Euclid’s Algorithm (cont’d) • Inductive proof of correctness:

  6. Euclid’s Algorithm (cont’d) • Time Analysis of Euclid’s Algorithm for n bit numbers u,v

  7. Euclid’s Algorithm (cont’d) • Fibonacci worst case:

  8. Euclid’s Algorithm (cont’d) • Improved Algorithm

  9. Extended GCD Algorithm

  10. Extended GCD Algorithm (cont’d) • Theorem • Proof

  11. Extended GCD Algorithm (cont’d) • Corollary If gcd(x,y) = 1 then x' is the modular inverse of x modulo y • Proof

  12. Modular Laws • Gives Algorithm for • Modular Laws

  13. Modular Laws (cont’d)

  14. Modular Laws (cont’d)

  15. Fermat’s Little Theorem • If n prime then an = a mod n • Proof by Euler

  16. Fermat’s Little Theorem (cont’d)

  17. Euler’s Theorem • Φ(n) = number of integers in {1,…, n-1} relatively prime to n • Euler’s Theorem • Proof

  18. Euler’s Theorem (cont’d) • Lemma • Proof

  19. Euler’s Theorem (cont’d) • By Law A and Lemma • By Law B

  20. Taking Powers mod n by “Repeated Squaring” • Problem: Compute ae mod b

  21. Taking Powers mod n by “Repeated Squaring” (cont’d) • Time Cost

  22. Rivest, Sharmir, Adelman (RSA) Encryption Algorithm • M = integer message e = “encryption integer” for user A • Cryptogram

  23. Rivest, Sharmir, Adelman (RSA) Encryption Algorithm (cont’d) • Method

  24. Rivest, Sharmir, Adelman (RSA) Encryption Algorithm (cont’d) • Theorem

  25. Rivest, Sharmir, Adelman (RSA) Encryption Algorithm (cont’d) • Proof

  26. Rivest, Sharmir, Adelman (RSA) Encryption Algorithm (cont’d) • By Euler’s Theorem

  27. Security of RSA Cryptosystem • Theorem If can compute d in polynomial time, then can factor n in polynomial time • Proof e· d-1 is a multiple of φ(n) But Miller has shown can factor n from any multiple of φ(n)

  28. Security of RSA Cryptosystem (cont’d)

  29. Rabin’s Public Key Crypto System • Use private large primes p, q public key n=q p message M cryptogram M2 mod n • Theorem If cryptosystem can be broken, then can factor key n

  30. Rabin’s Public Key Crypto System (cont’d) • Proof • In either case, two independent solutions for M give factorization of n, i.e., a factor of n is gcd (n, γ -β).

  31. Rabin’s Public Key Crypto System (cont’d) • Rabin’s Algorithm for factoring n, given a way to break his cryptosystem.

  32. Quadratic Residues

  33. Jacobi Function

  34. Jacobi Function (cont’d) • Gauss’s Quadratic Reciprocity Law • Rivest Algorithm

  35. Jacobi Function (cont’d) • Theorem (Fermat)

  36. Theorem: Primes are in NP • Proof

  37. Theorem & Primes NP (cont’d) • Note

  38. Primality Testing • Testing • Goal of Randomized Primality Testing

  39. Primality Testing (cont’d) • Solovey & Strassen Primality Test quadratic reciprocal law

  40. Definitions

  41. Theorem of Solovey & Strassen • Theorem • Proof

  42. Theorem of Solovey & Strassen (cont’d)

  43. Theorem of Solovey & Strassen (cont’d) • Then by Chinese Remainder Theorem, • Since a is relatively prime to n,

  44. Theorem of Solovey & Strassen (cont’d)

  45. Theorem of Solovey & Strassen (cont’d)

  46. Theorem of Solovey & Strassen (cont’d)

  47. Miller • Miller’s Primality Test

  48. Miller (cont’d) • Theorem (Miller) Assuming the extended RH, if n is composite, then Wn(a) holds for some a ∈ {1,2,…, c log 2 n} • Miller’s Test assumes extended RH (not proved)

  49. Miller – Rabin Randomized Primality Test • Theorem

  50. Number Theory Algorithms and Cryptography Algorithms Analysis of Algorithms Prepared by John Reif, Ph.D.