R S A

1. R S A POON TENG HIN

2. Main topic • RSA • Shamir’s Three-Pass Protocol • Other issues

3. A IQ question:

4. Encryption • The locks in computer network • 1-1 mapping function f so that c = f(m)

5. Decryption • The keys in computer network so that  f-1(c) = f-1(f(m)) = (f-1f)(m) = m

6. RSA We need: • Function • Modulo Operation • Greatest Common Divisor • Multiplicative Inverse • Number theory  • Prime number

7. Multiplicative Inverse • (x × y) mod n = 1. The integer y is called a multiplicative inverse of x, usually denoted x−1 (it is unique if it exists).

8. Prime number People keep finding large prime numbers for computer Security. How the prime number are used?

9. RSA • RSA is an algorithm for public-key cryptography • By Ron Rivest, Adi Shamir, Leonard Adleman

10. Many application • Because of security, high strength • Encryption • Digital signatures • E.g electronic transactions, • software certification.

11. RSA encryption and decryption • Encryption: C = Me mod n • Decryption: M = Cd mod n

12. Let’s try it: • ABCDEFGHIJKLMNOPQRSTUVWXYZ • 1234…………………………………26 • Public key: n = 35, e = 5 C = Me mod n • Private key: d = 5 M = Cd mod n • My word: • “17 21 14 33 8” • “ 6 30 11” • Also, try to give me your words

13. The Security of the RSA • p, q, (n) must be kept secret. • It is believed that determine (n) given n is equivalent to factoring n. • With presently known algorithms, determining d given e and n, appears to be at least as time-consuming as the factoring problem. • So use factoring as the benchmark for security evaluation.

14. ASCII • http://www.cs.drexel.edu/~jpopyack/IntroCS/HW/ASCII.html • A website of ASCII code

15. Term Plaintext: M ( M = {0,1}*) Cipher text: C (C = {0,1}*) It needs two distinct primes p and q Φ(n) = (p-1)(q-1) select an integer e such that gcd(e, Φ(n)) = 1 Where n = pq, n>M Compute the d where ed = 1 (mod Φ(n)) Public key: (e,n) Private key: d

16. n p and q • Randomly choose p and q And n = p X q A sample n from http://www.rsa.com/rsalabs/node.asp?id=2093 RSA-576: 188198812920607963838697239461650439807163563379417382700763356422988859715234665485319060606504743045317388011303396716199692321205734031879550656996221305168759307650257059

17. e • gcd(e, Φ(n)) = 1 and e > 1 • A table to find e and d:

18. Euler’s TotientFunctopmΦ(n) • Φ(n) is the number of positive integers less than n that is relative prime to n • Example Φ(6) : • the GCD(x,6) = 1 when x = 1,5 so Φ(6) = 2

19. Euler’s Totient Function Φ(n) Φ(p) = p-1 for any prime number p Φ(pq) = (p-1)(q-1) for any two distinct primes p and q

20. Fermat’s and Euler’s Theorem Euler’s: For every integer a and n that are relatively prime, aΦ(n)mod n = 1 Fermat’s : If n = p is prime, ap-1 mod p = 1

21. d • ed = 1 (mod Φ(n)) or d = e-1 mod n • Such that ex + Φ(n) y = 1 and d is the value of x • One of the method is Euclidean algorithm http://www.di-mgt.com.au/euclidean.html

22. d example: Fo example Φ(n) =20, e =3 Firstly, gcd(20,3) = 1 if the inverse exists. We use Euclidean algorithm: 20 = 3 x 6 +2 3 = 2 x 1 + 1 1 = 3 – 1X2 = 3 – 1 X (20 – 6 X 3) = -1 X 20 + 7 X 3 (ex + ny = 1) so d = 7

23. Another example ofEuclidean algorithm • 66 = 1 × 35 + 31 gcd(35, 31) • 35 = 1 × 31 + 4 gcd(31, 4) • 31 = 7 × 4 + 3 gcd(4, 3) • 4 = 1 × 3 + 1 gcd(3, 1) • 3 = 3 × 1 + 0 gcd(1, 0) • So, • gcd(66, 35) = gcd(35, 31) = gcd(31, 4) = gcd(4, 3) = gcd(3, 1) = gcd(1, 0) = 1.

24. See it again • Encryption: C = Me mod n • Decryption: M = Cd mod n Needs two distinct primes p and q And Φ(n) = (p-1)(q-1) select an integer e such that gcd(e, Φ(n)) = 1 Where n = pq, n>M Compute the d where ed = 1 (mod Φ(n)) Public key: (e,n) Private key: d

25. RSA calculation • http://www-cs-students.stanford.edu/~tjw/jsbn/rsa2.html • http://www.cs.drexel.edu/~jpopyack/IntroCS/HW/RSAWorksheet.html

26. Correctness of Decryption

27. Correctness of Decryption

28. Correctness of Decryption

29. Answer of IQ question • 1.A lock the box by his lock A • 2.A------------- B (Box with lock A) • 3.B lock the box by his lock B • 4.B---------------A (Box with lock A & B) • 5.A unlock his lock A • 6.A --------------- B (Box with lock B) • 7. B unlock his lock B ~ • ~finish~

30. Shamir’s Three-Pass Protocol • This is the protocol similar to the answer of the IQ question • This is different to RSA In this protocol, we need a prime p which is a public knowledge.

31. A and B • A selects a random number a with gcd(a, p-1) = 1 • B selects a random number b with gcd(b,p-1) = 1 a-1 and b-1 are the inverse of a and b of mod p-1

32. The protocol • A computes k1 = ka mod pand send k1 to B • B computes k2 = k1b mod p and send k2 to A • A computes k3 = k2a-1mod p and send k3 to B • Finally, B computes k = k3b-1 mod p and get k.

33. Homework • Q1.Using slide 13, what is the message under: • “12 21 10 24 20 4 15 14” • “15 14 10” • “”4 24 • “6 4 14 4 24 8 10 9” • Q2. Find d if Φ(n) = 58, e = 27 (use Euclidean algorithm)

34. Others • Others issues I would like to share. • I suggest you may think about them.

35. Comp364 • Computer and Communications Security COMP364 • By Prof. Cunsheng Ding

36. Bridge • People like math will like this game.

37. Classical One-key Cipher or Cryptosystem • Encryption: c = Ek(m), where Ek is usually applied to blocks of the plaintext m. • Decryption: m = Dk(c), where Dk is usually applied to blocks or characters of the ciphertext c.

38. Example: Transposition Ciphers • Example: Let d = 4 and define f by • i : 0 1 2 3 • f(i) : 2 0 3 1 • Then f is a permutation of Z4. • The inverse permutation f−1 is given by • i : 0 1 2 3 • f-1(i) : 1 3 0 2

39. Simple Substitution Ciphers • E.g

40. http://www.blog.republicofmath.com/archives/4120

41. Is that a Paradox?

42. Example: (Condorcet, [1785]1994) A B C 1| plan1 plan3 plan2 2| plan2 plan1 plan3 3| plan3 plan2 plan1 Conclusion: Most people think that: plan1 is better than plan2 plan2 is better than plan3 plan3 is better than plan1

43. END • ByeBye