Understanding Modular Arithmetic and Exponentiation Algorithms

1. This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License. CS 312: Algorithm Analysis Lecture #3: Algorithms for Modular Arithmetic,Modular Exponentiation Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, Sean Warnick

2. Announcements • HW #1 Due Now • Always start of class • Always show work • FERPA protects your student record • Need waiver to return graded work without cover sheet

3. Objectives • Add the Max Rule to your asymptotic analysis toolbox • Review modular arithmetic • Discuss and analyze algorithms for: • modular arithmetic • modular exponentiation

4. Max. rule • Another useful rule for Asymptotic analysis. O( f(n) + g(n) ) = O( max( f(n), g(n) ) ) • Examples:

5. Goal for Ch. 1 • Appreciate the role of theoretical analysis in the security of RSA. • Requires: Solve, analyze, and use (!) two important and related problems: • Factoring: Given a number N, express it as a product of its prime numbers • Primality Testing: Given a number N, determine whether it is prime • Which one is harder?

6. Algorithms for Integer Arithmetic • Computing Device: • Binary operations are constant time • Arithmetic operations on arbitrary length integers may require more time • For an integer , we talk about its representation in bits: • Pad length of to the next power of 2 (using 0s) if necessary.

7. Algorithms for Integer Arithmetic • Addition • Multiplication • Division

8. Algorithms for Integer Arithmetic • Addition: • Multiplication: • Division:

9. Modular Arithmetic

10. Congruency

11. An important distinction • Congruency • Equality, using the modulus operator

12. Properties • Associativity: • Commutativity: • Distributivity:

13. Substitution Rule

14. Useful Consequence xy (x mod z)y (mod z) xy mod z = (x mod z)y mod z • Example:

15. Modular Addition

16. Modular Multiplication

17. Goal: Modular Exponentiation • We need to compute xy mod N for values of x, y, and N that are several hundred bits long. • Can we do so quickly?

18. Sequential Exponentiation function seqexp(x, y) Input: An n-bit integer x and a non-negative integer exponent y (arbitrarily large) Output: xy if y=0: return 1 r = x for i = 1 to y-1 do r = r x return r Describe a simple algorithm for doing exponentiation:

19. Analysis of Sequential Exponentiation function seqexp (x, y) Input: An n-bit integer x and a non-negative integer exponent y (arbitrarily large) Output: xy if y=0: return 1 r = x for i = 1 to y-1 do r = r x return r

20. Modular Exponentiation, Take I function modexp (x, y, N) Input: Two n-bit integers x and N, a non-negative integer exponent y (arbitrarily large) Output: xy mod N if y=0: return 1 r = x mod N for i = 1 to y-1 do r = (r x) mod N return r

21. Modular Exponentiation, Take I function modexp (x, y, N) Input: Two n-bit integers x and N, a non-negative integer exponent y (arbitrarily large) Output: xy mod N if y=0: return 1 r = x mod N for i = 1 to y-1 do r = (r x) mod N return r

22. New Ideas • Represent y (the exponent) in binary • Then break down xy into factors using the non-zero bits of y • Also: compute the factors using repeated squaring • Reduce factors using substitution rule

23. Modular Exponentiation, Take II function modexp(x, y, N) Input: Two n-bit integers x and N, a non-negative integer exponent y (arbitrarily large) Output: xy mod N if y=0: return 1 z = modexp(x, floor(y/2), N) if y is even: return z2 mod N else: return x z2 mod N Recursive call Right shift Multiplication

24. Analysis of Modular Exponentiation • Each multiplication is Q(n2) • Each modular reduction is Q(n2) • There are log(y)=m of them • Thus, modular exponentiation is in Q(n2 log y) = Q(n2 m) function modexp(x, y, N) if y=0: return 1 z = modexp(x, floor(y/2), N) if y is even: return z2 mod N else: return x z2 mod N

25. Modular Exponentiation (II),Iterative Formulation function modexp(x, y, N) Input: Two n-bit integers x and N, a non-negative integer exponent y (arbitrarily large) Output: xy mod N if y = 0: return 1 i = y; r = 1; z = x mod N while i > 0 if i is odd: r = r z mod N z = z2 mod N i = floor(i/2) return r

26. Modular Exponentiation • xy mod N • Key Insights: • Exponent y can be represented in binary • Problem can be factored into one factor per binary digit • Each factor can be reduced mod N (substitution rule)

27. We’re employingsame insights and a little more cleverness than thealgorithm. Example

28. Example #2 Strictly tracing the algorithm. function modexp(x, y, N) Input: Two n-bit integers x and N, an integer exponent y (arbitrarily large) Output: xy mod N if y = 0: return 1 i = y; r = 1; z = x mod N while i > 0 if i is odd: r = r z mod N z = z2 mod N i = floor(i/2) return r

29. Example #2 function modexp(x, y, N) Input: Two n-bit integers x and N, an integer exponent y (arbitrarily large) Output: xy mod N if y = 0: return 1 i = y; r = 1; z = x mod N while i > 0 if i is odd: r = r z mod N z = z2 mod N i = floor(i/2) return r

30. Example Needed: two volunteers: Volunteer A: use our final modexp() to compute it. Volunteer B: compute 320 then reduce mod 10

31. Efficiency • The key point is that xy mod N is easy • modexpis in Q(n2 log y) • In fact, it requires about 1.5 log2 y multiplications for typical y • seqexp required y-1 multiplications • When x, y, and N are 200 digit numbers • Assume 1 multiplication of two 200 digit numbers takes 0.001 seconds • modexp typically takes about 1 second • seqexp would require 10179 times the Age of the Universe! • Only works when y is an integer.

32. Assignment • Read: Section 1.4 • HW #2: • Problem 1.25 using modexp, • Then redo 1.25 but replace 125 with 126 for the exponent • Implement modular exponentiation now as a step toward finishing Project #1

33. Next • Primality Testing