1. CS/COE 1501 Recitation RSA Encryption/Decryption Extended Euclidean Algorithm Digital Signatures

2. Say Alice wants to send a message to Bob 1.Looks up Bob’s public key 2.Convert the message into an integer: m 3.Compute the ciphertext c as: c = m e (mod n) 4.Send c to Bob RSA Encryption

3. Bob can simply: 1.Compute m as: 2.m = c d (mod n) 3.Convert m into Alice’s message RSA Decryption

4. What are public/private keys? How messages encrypted? How are messages decrypted? How are keys generated? Why is it secure? RSA Cryptosystem

5. What are public/private keys? Public Key = (e, n) Private Key = (d, n) How messages encrypted? c = m e (mod n) How are messages decrypted? m = c d (mod n) How are keys generated? RSA Cryptosystem

6. How are keys generated?

7. 1. Choose two prime number p and q p=3, q=11 2. Compute n = p * q n=3*11=33 3. Compute φ(n) φ(n) = φ(p) * φ(q) = (p - 1) * (q - 1) φ(n)=(3-1)*(11-1)=20 4. Choose e such that 1 < e < φ(n), GCD(e, φ(n)) = 1 i.e., e and φ(n) are co-prime We can choose e=3, verify that 1<3< φ(n) =20, 3 and 20 are co- prime An Example

9. Encryption: c = m e (mod n) Decryption: m = c d (mod n) Public Key=(e,n)=(3,33) Private Key=(d,n)=(7,33) Alice said “hello” 7, 4, 11, 11, 14 Encrypt msg: 7 3 mod 33, 4 3 mod 33, 11 3 mod 33, 11 3 mod 33, 14 3 mod 33 Encrypted msg: 13, 31, 11, 11, 5 An Example

10. Encryption: c = m e (mod n) Decryption: m = c d (mod n) Public Key=(e,n)=(3,33) Private Key=(d,n)=(7,33) Bob receive 13, 31, 11, 11, 5 Decrypt msg: 13 7 mod 33, 31 7 mod 33, 11 7 mod 33, 11 7 mod 33, 5 7 mod 33 Decrypt msg: 7, 4, 11, 11, 14 -> ‘hello’ An Example

11. Challenges

12. d = e -1 mod(φ(n)) Means that d = 1/e mod(φ(n)) Means that e * d = 1 (mod φ(n)) Now, this can be equivalently stated as e * d = z * φ(n) + 1 For some z Can further restate this as: e * d - z * φ(n) = 1 Or similarly: 1 = φ(n) * (-z) + e * d How can we solve this? Hint: recall that we know GCD(φ(n), e) = 1 Determine d

13. GCD(a, b) = i = as + bt Let: a = φ(n) b = e s = -z t = d i = 1 GCD(φ(n), e) = 1 = φ(n) * (-z) + e * d We can compute d in linear time! Determine d

14. Extended Euclidean Algorithm

15. We know GCD(a,b)=GCD(b, a%b) Suppose we are computing the GCD(a,b) a*x + b*y = gcd Suppose we already know the GCD(b, a %b), and we find x 1 and y 1 b*x 1 +(a%b)*y 1 =gcd Associate the two formular a%b=a-(a/b)*b gcd= b*x 1 +(a-(a/b)*b)*y 1 = b*x 1 +a*y 1 -(a/b)*b*y 1 = a*y 1 +b*(x 1 -a/b*y 1 ) x=y 1 y=x 1 -a/b*y 1 Extended Euclidean Algorithm

16. public static int[] ExtendedEuclid(int a, int b) { int[] ans = new int[3]; int q; if (b == 0) { /* If b = 0, then we're done... */ ans[0] = a; ans[1] = 1; ans[2] = 0; } else { /* Otherwise, make a recursive function call */ q = a/b; ans = ExtendedEuclid (b, a % b); int temp = ans[1] - ans[2]*q; ans[1] = ans[2]; ans[2] = temp; } return ans; } Extended Euclidean Algorithm

17. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978 2 3 4 5 6

18. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781 2 3 4 5 6

19. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 2 3 4 5 6

20. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 2 3 4 5 6

21. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821 3 4 5 6

22. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 278213 3 4 5 6

23. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3 4 5 6

24. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 32115 4 5 6

25. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 321151 4 5 6

26. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 5 6

27. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 6 5 6

28. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 62 5 6

29. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 623 5 6

30. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 623 563 6

31. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 623 5632 6

32. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 623 56320 6

33. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 623 56320 630

34. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 623 56320 630NaN

35. Extended Euclidean Algorithm

36. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 623 56320 630NaN

37. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 623 56320 630NaN 310

38. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 19978121 27821315 3211516 4 623 56320 630NaN 310

39. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 6233 563203 630NaN 310

40. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 6233 563203 630NaN 310

41. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 6233 5632030 630NaN 310

42. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 6233 5632030 630NaN 310

43. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 6233 56320301 630NaN 310

44. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 6233 56320301 630NaN 310

45. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 6233 56320301 630NaN 310

46. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 62331 56320301 630NaN 310

47. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 62331-2 56320301 630NaN 310

48. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 62331-2 56320301 630NaN 310

49. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163 4 62331-2 56320301 630NaN 310

50. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163-2 41562331-2 56320301 630NaN 310

51. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163-23 41562331-2 56320301 630NaN 310

52. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163-23 41562331-2 56320301 630NaN 310

53. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 278213153 32115163-23 41562331-2 56320301 630NaN 310

54. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 2782131533 32115163-23 41562331-2 56320301 630NaN 310

55. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213 2782131533-11 32115163-23 41562331-2 56320301 630NaN 310

56. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213-11 2782131533-11 32115163-23 41562331-2 56320301 630NaN 310

57. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213-1114 2782131533-11 32115163-23 41562331-2 56320301 630NaN 310

58. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213-1114 2782131533-11 32115163-23 41562331-2 56320301 630NaN 310

59. Find the Bézout numbers and GCD of 99 and 78 Rowaba/ba%bdst 199781213-1114 2782131533-11 32115163-23 41562331-2 56320301 630NaN 310

60. Exercise

61. Hash Functions

62. For Crypto Hash Functions, Output Should Appear Random

63. Digital Signatures – Public Key Cryptography

64. Creating a Digital Signature

65. Digital Signatures Often Use Commutative Operations

67. Plaintext sent by sender

68. Digital Signatures Often Use Commutative Operations Plaintext sent by sender Cryptotext sent by sender using sender’s private key

69. Digital Signatures Often Use Commutative Operations Plaintext sent by sender Cryptotext sent by sender using sender’s private key Sender’s public key

70. Digital Signatures Often Use Commutative Operations Plaintext sent by sender Cryptotext sent by sender using sender’s private key Sender’s public key =

71. Digital Signatures Often Use Commutative Operations Plaintext sent by sender Cryptotext sent by sender using sender’s private key Sender’s public key = Plaintext recovered matches

72. Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original: Digital Signatures and Hashes

73. Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original: Digital Signatures and Hashes Received: HASH ALGORITHM

74. Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original: Digital Signatures and Hashes Received: HASH ALGORITHM

75. Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original: Digital Signatures and Hashes Received: HASH ALGORITHM Compute

76. Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original: Digital Signatures and Hashes Received: HASH ALGORITHM Compute =

77. Because Public-Key crypto can be computationally expensive, often the crypto operations are performed on the securely hashed version of the message rather than the original: Digital Signatures and Hashes Received: HASH ALGORITHM Compute = Match. Signature Verified.

78. Adam J. Lee’s slides from CS 1653 http://www.csee.umbc.edu/~chang/cs203.s09/exteuclid.shtml Acknowledgements