Lecture 11-12 Implementations

1. Lecture 11-12 Implementations

2. The efficiency of a particular cryptographic scheme based on any one of the algebraic structures will depend on a number of factors, such as parameter size, time-memory tradeoffs, processing power available, software and/or hardware optimization, and mathematical algorithms. This lecture is concerned primarily with mathematical algorithms for efficiently carrying out computations in the underlying algebraic structure.

3. The algorithms described in this lecture are those which, for the most part, have received considerable attention in the literature. Although some attempt is made to point out their relative merits, no detailed comparisons are given.

4. Outline • Prime Number Issue • Exponentiation • Exponent Recoding • Multi-Exponentiation • Chinese Remainder Theorem for RSA • Montgomery Reduction Method

5. 1 Prime Number Issue

6. 1.1 Miller-Rabin Test

7. 1.1 Miller-Rabin Test (Continued)

8. 1.1 Miller-Rabin Test (Continued)

9. 1.1 Miller-Rabin Test (Continued)

10. 1.1 Miller-Rabin Test (Continued)

11. 1.1 Miller-Rabin Test (Continued)

12. 1.1 Miller-Rabin Test (Continued)

13. 1.1 Miller-Rabin Test (Continued)

14. 1.2 Prime Number Generation Prime number generation differs from primality testing as before, but may and typically does involve the latter. The former allows the construction of candidates of a fixed form which may lead to more efficient testing than possible for random candidates.

15. 1.2.1 Random Search for Probable Primes

16. 1.2.1 Random Search for Probable Primes (Continued)

17. 1.2.1 Random Search for Probable Primes (Continued)

18. 1.2.1 Random Search for Probable Primes (Continued)

19. 1.2.1 Random Search for Probable Primes (Continued)

20. 1.2.2 Strong Primes

21. 1.2.2 Strong Primes (Continued)

22. 1.2.2 Strong Primes (Continued)

23. 1.2.3 Generating DSA Primes

24. 1.2.3 Generating DSA Primes (Continued)

25. 1.2.3 Generating DSA Primes (Continued)

26. 1.2.3 Generating DSA Primes (Continued)

27. 1.2.3 Generating DSA Primes (Continued)

28. 2 Exponentiation

30. 2.1 Problem Model 2.1.1 Addition Chains

31. 2.1.2 Addition–Subtraction Chains

32. 2.1.3 Addition Sequences and Vector Addition Chains

33. 2.1.3 Addition Sequences and Vector Addition Chains (Continued)

34. 2.1.3 Addition Sequences and Vector Addition Chains (Continued)

35. 2.2 Techniques for General Exponentiation 2.2.1 The Binary Method

36. 2.2.1 The Binary Method (Continued)

37. 2.2.1 The Binary Method (Continued)

38. 2.2.1 The Binary Method (Continued)

39. 2.2.1 The Binary Method (Continued)

40. 2.2.1 The Binary Method (Continued)

41. 2.2.1The Binary Method (Continued)

42. 2.2.2 k-ary Method

43. 2.2.2 k-ary Method (Continued)

44. 2.2.2 k-ary Method (Continued)

45. 2.2.2 k-ary Method (Continued)

46. 2.2.2 k-ary Method (Continued)

47. 2.2.3 Sliding-Window Exponentiation

48. 2.2.3 Sliding-Window Exponentiation (Continued)

49. 2.3 Fixed-Exponent Exponentiation Algorithms There are numerous situations in which a number of exponentiations by a fixed exponent must be performed. Examples include RSA encryption and decryption, and ElGamal decryption.

50. 2.3 Fixed-Exponent Exponentiation Algorithms (Continued)