1. COMP 170 L2 Page 1 L06: The RSA Algorithm l Objective: n Present the RSA Cryptosystem n Prove its correctness n Discuss related issues

2. COMP 170 L2 Page 2 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Calculating exponentiation mod n efficiently l The Chinese Remainder Theorem

3. COMP 170 L2 Exponentiation mod n l Encryption with addition and multiplication mod n n Easy to find the way to decrypt l RSA: use exponentiation mod n

4. COMP 170 L2 Exponentiation mod n

5. COMP 170 L2

7. Corollary of Lemma 2.19

8. COMP 170 L2 Page 8 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

9. COMP 170 L2 Public-Key Cryptography

10. COMP 170 L2 RSA Algorithm l Questions to answer

11. COMP 170 L2 One-Way Function

12. COMP 170 L2 RSA Algorithm l Builds a one-way function using n Exponentiation mod n n Prime numbers n gcd n Multiplicative inverse

13. COMP 170 L2 RSA Algorithm

14. COMP 170 L2 RSA Algorithm

15. COMP 170 L2 RSA Example l Key generation

16. COMP 170 L2 RSA Example l Encryption and decryption  Try: http://cisnet.baruch.cuny.edu/holowczak/classes/9444/rsademo/http://cisnet.baruch.cuny.edu/holowczak/classes/9444/rsademo/

17. COMP 170 L2 Page 17 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

18. COMP 170 L2 A Lemma

19. COMP 170 L2

20. Fermat’s Little Theorem

21. COMP 170 L2

22. l What is a is a multiple of p?

23. COMP 170 L2 l Simplifies computation

24. COMP 170 L2

25. Page 25 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

26. COMP 170 L2 Decipherability

27. COMP 170 L2

29. Decipherability

30. COMP 170 L2

33. Decipherability Proved!

34. COMP 170 L2

35. Page 35 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

36. COMP 170 L2

38. Page 38 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

39. COMP 170 L2 Exponentiation mod n efficiently Page 39

40. COMP 170 L2 Exponentiation mod n efficiently

41. COMP 170 L2 Exponentiation mod n efficiently

42. COMP 170 L2 Exponentiation mod n efficiently Page 42

43. COMP 170 L2 Complexity of Repeated Squaring Page 43

44. COMP 170 L2 Page 44 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

45. COMP 170 L2 The Chinese Remainder Theorem

46. COMP 170 L2 The Chinese Remainder Theorem

47. COMP 170 L2 The Chinese Remainder Theorem

48. COMP 170 L2 The Chinese Remainder Theorem

49. COMP 170 L2 The Chinese Remainder Theorem

50. COMP 170 L2

51. The Chinese Remainder Theorem

52. COMP 170 L2 Past Exam Question

53. COMP 170 L2

54. Past Exam Question l About Chinese remainder theorem (CRT) l Think n 36 = 3 * 13, 5 = 3 * 17; not relatively prime, so cannot use CRT n Brute-force  x = q1 * 36 + 12 => x mod 3 = 0  x = q2 * 51 + 5 => x mod 3 = 2  Cannot have solution. n What is 12 is changed 11?

55. COMP 170 L2 l Think: n 35 = 5 * 7; 69 = 3 * 23 n Relatively prime. Also can apply CRT. Unique solution exists. l How to find the solution?