Addition

1. Addition How fast can you add A+B

2. Addition How fast can you add A+B • 1 0 1 0 1 1 1 0 0 1 • 1 0 0 1 0 0 1 1 1

3. Addition How fast can you add A+B • 1 0 1 0 1 1 1 0 0 1 • 1 0 0 1 0 0 1 1 1 • 0

4. Addition How fast can you add A+B • 1 0 1 0 1 1 1 0 0 1 • 1 0 0 1 0 0 1 1 1 • 0 0

5. Addition How fast can you add A+B • 1 0 1 0 1 1 1 0 0 1 • 1 0 0 1 0 0 1 1 1 • 1 0 0

6. Addition How fast can you add A+B • 1 0 1 0 1 1 1 0 0 1 • 1 0 0 1 0 0 1 1 1 1 1 1 1 0 1 1 1 0 0 n-bit numbers  time = O(n)

7. Multiplication How fast can you multiply A*B • 1 0 1 0 1 1 1 0 0 1 • *1 0 1 1

8. Multiplication How fast can you multiply A*B • 1 0 1 0 1 1 1 0 0 1 • *1 0 1 1 • 1 0 1 0 1 1 1 0 0 1 • 1 0 1 0 1 1 1 0 0 11 0 1 0 1 1 1 0 0 1

9. Multiplication How fast can you multiply A*B • 1 0 1 0 1 1 1 0 0 1 • *1 0 1 1 • 1 0 1 0 1 1 1 0 0 1 • 1 0 1 0 1 1 1 0 0 11 0 1 0 1 1 1 0 0 1 n-bit numbers  time = O(n2)

10. Karatsuba-Offman a=2n/2 a1 + a0 b=2n/2 b1 + b0 ab=(2n/2a1+a0)(2n/2b1+b0) = 2n a1 b1 + 2n/2 (a1 b0 + a0 b1) + a0 b0

11. Karatsuba-Offman a=2n/2 a1 + a0 b=2n/2 b1 + b0 Multiply(a,b,n) if n=1 return a*b else R1 Multiply(a1,b1,n/2) R2 Multiply(a0,b1,n/2) R3 Multiply(a1,b0,n/2) R4 Multiply(a0,b0,n/2) return 2n R1+ 2n/2 (R2+R3) + R4

12. Karatsuba-Offman Multiply(a,b,n) if n=1 return a*b else R1 Multiply(a1,b1,n/2) R2 Multiply(a0,b1,n/2) R3 Multiply(a1,b0,n/2) R4 Multiply(a0,b0,n/2) return 2n R1+ 2n/2 (R2+R3) + R4 Recurrence?

13. Karatsuba-Offman Multiply(a,b,n) if n=1 return a*b else R1 Multiply(a1,b1,n/2) R2 Multiply(a0,b1,n/2) R3 Multiply(a1,b0,n/2) R4 Multiply(a0,b0,n/2) return 2n R1+ 2n/2 (R2+R3) + R4 Recurrence? T(n) = 4T(n/2) + O(n)

14. Karatsuba-Offman T(n) = 4T(n/2) + O(n) T(n)=O(n2)

15. Karatsuba-Offman ab=(2n/2a1+a0)(2n/2b1+b0) = 2na1 b1 + 2n/2 (a1 b0 + a0 b1) + a0 b0 Can compute in less than 4 multiplications?

16. Karatsuba-Offman ab=(2n/2a1+a0)(2n/2b1+b0) = 2na1 b1 + 2n/2 (a1 b0 + a0 b1) + a0 b0 Can compute using 3 multiplications: (a0+a1)(b0+b1) = a0b0 + (a1 b0 + a0 b1) + a1 b1

17. Karatsuba-Offman Multiply(a,b,n) if n=1 return a*b else R1 Multiply(a1,b1,n/2) R2 Multiply(a0,b0,n/2) R3 Multiply(a1+a0,b1+b0,n/2+1) R4 R3 – R2 – R1 return 2n R1+ 2n/2 R3 + R2 Recurrence?

18. Karatsuba-Offman Multiply(a,b,n) if n=1 return a*b else R1 Multiply(a1,b1,n/2) R2 Multiply(a0,b0,n/2) R3 Multiply(a1+a0,b1+b0,n/2+1) R4 R3 – R2 – R1 return 2n R1+ 2n/2 R3 + R2 Recurrence? T(n) = 3T(n/2) + O(n)

19. Karatsuba-Offman T(n) = 3T(n/2) + O(n) T(n)=O(nC) C=log2 3  1.58

20. Integer Division r=a mod b a,b  q,r a = q*b + r 0  r < b Can be done in O(n2) time.

21. d divides a DEFINITION: d divides a (denoted d | a) if there exists b such that b*d = a 3|6 3|0 0|3 0|0

22. d divides a DEFINITION: d divides a (denoted d | a) if there exists b such that b*d = a 3|6 yes, b=2 3|0 yes, b=0 0|3 no 0|0 yes, b=?

23. d divides a 3|6 yes, b=2 3|0 yes, b=0 0|3 no 0|0 yes, b=? d | a  a | c  d | c Proof: a = b*d, c=b’*a  c=(b*b’)*d

24. Divisibility poset 0 8 10 9 4 6 3 5 7 2 1

25. GCD GCD (a,b) “largest” d such that d|a, d|b

26. GCD GCD (a,b) “largest” d such that d|a, d|b d|a, d|b (c; c|a,c|b) : c|d GCD(3,6) GCD(0,8) GCD(0,0)

27. GCD GCD (a,b) “largest” d such that d|a, d|b d|a, d|b (c; c|a,c|b) : c|d GCD(3,6) = 3 GCD(0,8) = 8 GCD(0,0) = 0

28. GCD How quickly can we compute GCD (a,b) ?

29. GCD How quickly can we compute GCD (a,b) ? Euclid GCD(a,b) = GCD(b,a mod b)

30. GCD wlog a>b GCD(a,b) if b=0 then return a else return GCD(b,a mod b) Running time?

31. GCD wlog a>b GCD(a,b) if b=0 then return a else return GCD(b,a mod b) Running time? (a,b)(b,a mod b)(a mod b, ?) (a mod b) < a/2

32. GCD (a,b)(b,a mod b)(a mod b, ?) (a mod b) < a/2 2(log2 a)=O(n) iterations each mod O(n2) time  O(n3) time total

33. Modular exponentiation (a,b,m)  ab mod m

34. Modular exponentiation (a,b,m)  ab mod m b = 10101 a mod m a2 mod m a4 mod m a8 mod m a16 mod m ... ab mod m

35. Modular exponentiation (a,b,m)  ab mod m mod-ex(a,b,m) if b=0 then RETURN 1 else if b mod 2 = 0 then RETURN mod-ex(a,b/2,m)2 mod m else RETURN a*mod-ex(a,(b-1)/2,m)2 mod m

36. Algorithms so far a,b,m n-bit integers addition a+b O(n) time multiplication a*b O(n1.58) time division a/b,a mod b O(n2) time gcd(a,b) O(n3) time ab mod m O(n3) time

37. GROUP (G,) is a group if  : GG  G (ab)c = a(bc) exists  G (aG) a = a a  a-1 aa-1=

38. Modular arithmetic modulo m G = {0,...,m-1} = Zm ab = a+b mod m (G,) is a group if  : GG  G (ab)c = a(bc) exists  G (aG) a = a a  a-1 aa-1=

39. Modular arithmetic modulo m G = {0,...,m-1} = Zm ab = a+b mod m (G,) is a group if  : GG  G (ab)c = a(bc) exists  G (aG) a = a a  a-1 aa-1= IS A GROUP

40. Modular arithmetic modulo m G = {0,...,m-1} = Zm ab = a*b mod m (G,) is a group if  : GG  G (ab)c = a(bc) exists  G (aG) a = a a  a-1 aa-1=

41. Modular arithmetic modulo m G = {0,...,m-1} = Zm ab = a*b mod m (G,) is a group if  : GG  G (ab)c = a(bc) exists  G (aG) a = a a  a-1 aa-1= b; ab=1 [mod m]  GCD(a,m)=1

42. Modular arithmetic modulo m G = Z*m ={a | GCD(a,m)=1 } ab = a*b mod m (G,) is a group if  : GG  G (ab)c = a(bc) exists  G (aG) a = a a  a-1 aa-1= IS A GROUP

43. Fermat’s little Theorem p a prime ap-1 = 1 [mod p] {ak | k Z} is a subgroup of Z*p

44. Fermat’s little Theorem a(m)=1 [mod m] (m) = | Z*m | m=p1a1 p2a2 ... pkak (m) = (1-1/p1) ... (1-1/pk) m

45. Fermat’s little Theorem m=p1a1 p2a2 ... pkak (m) = (1-1/p1) ... (1-1/pk) m E.g. if m=pq p,q primes (m)=

46. Fermat’s little Theorem m=p1a1 p2a2 ... pkak (m) = (1-1/p1) ... (1-1/pk) m E.g. if m=pq p,q primes (m)=(p-1)(q-1)

47. Fermat’s little Theorem a(p-1)(q-1) =1 [mod pq] E.g. if m=pq p,q primes (m)=(p-1)(q-1)

48. RSA • choose primes p,q • let n  pq • choose e • compute • d=e-1 [mod (p-1)(q-1)] • 5) announce n,e

49. RSA • choose primes p=13,q=17 • let n  pq • choose e • compute • d=e-1 [mod (p-1)(q-1)] • 5) announce n,e

50. RSA • choose primes p=13,q=17 • let n  pq=221 • choose e • compute • d=e-1 [mod (p-1)(q-1)] • 5) announce n,e