Advanced Algebraic Algorithms on Integers and Polynomials

1. Advanced Algebraic Algorithms on Integers and Polynomials Analysis of Algorithms Prepared by John Reif, Ph.D.

2. Integer and Polynomial Computations • Newton Iteration: application to division • Evaluation and Interpolation via Chinese Remaindering Main Lecture Material: Algorithms on Integers Advanced Lecture Material: Extension of Algorithms to Polynomials

3. Taylor Expansion

4. Taylor Expansion (cont’d) • To find root f(x), use Newton Iteration: • Example: • To find reciprocal of x choose find root

5. Application of Newton Iteration to Reciprocal of an Integer • Input integer x, accuracy bound k • Initialize if x has n bits

6. Application of Newton Iteration to Reciprocal of an Integer (cont’d) • TheoremInteger Reciprocal can be computed to accuracy 2-n in O(log n) integer mults and additions.

7. Steven Cooks’s Improvement (His Harvard Ph.D. Thesis) • Key Trick: Since we need only compute yk up to 2k+1 bit accuracy of error on kth stage • Total Time Cost • Where M(n) is time cost to multiply two n bit integers

8. Other Applications of Newton Iterations on Integers • O(M(n)) time algorithms: • Quotient + divisor of integer division • Square root • Sin, cosine, etc. • Used in practice!

9. Advanced Topic: Extension to Polynomial ReciprocalDefinition of Polynomial Reciprocal • Reciprocal = polynomial r(x) • where and (x) has degree < n-1

10. Algorithm: Reciprocal (P(x)) • Input polynomial degree n-1, n is power of 2

11. Proof of the Reciprocal Algorithm • Theorem: The Algorithm Correctly Computes Reciprocal(P(x)) • Proof by induction

12. Proof Algorithm Computes Reciprocal (P(x)) (cont’d) • By induction hypothesis, if

13. Proof Algorithm Computes Reciprocal (P(x)) (cont’d) • At line [3] we compute

14. Proof Algorithm Computes Reciprocal (P(x)) (cont’d)

15. Modular Arithmetic • Assume relatively prime P0, P1, …, Pk-1 • Let given x, 0 < x < p

16. Applications to Arithmetic • But doesn’t extend to division(overflow problems)

17. Super Moduli Computation • Input p0, p1, …, pk-1, where pi < 2b • Output Super modular Tree:

18. Super Moduli Computation (cont’d) • Time Cost

19. Algorithm Residue Computation • Input • Outputx0, x1, …, xk-1, when xi =x mod Pi i=0, and k-1 • Recursive algorithm [1] compute quotient and remainders:

20. Algorithm Residue Computation (cont’d) [2] recursively compute[3] output for i = 0, 1, …, k-1

21. Time Cost for Residue Computation • Let D(n) = time cost for integer division = O(M(n)) • Total Time for input size n = k· b

22. Proof of Algorithm for Residue Computation • Idea of Proof of algorithm Uses fact: if x = q v + r and v mod Pi = 0, then X mod Pi = r mod Pi

23. Advanced Topic: Residue Computation on Polynomials • Input moduli P0(x), P1(x), …, Pk-1(x) assume each degree < d and relatively prime • Algorithm uses similar Super modular Tree tre , but using polynomials rather than integers

24. Advanced Topic: Residue Computation on Polynomials (cont’d) • Output for i=0, …, k-1 Qi(x) = Q(x) mod Pi(x) Q(x) has degree < kd • Theorem The Residue Computation can be done in time O(M(n) log n) where n = k· d • Proof Idea use same algorithm as in integer case

25. Advanced Topic: Multipoint Evaluation of Polynomials by Residue Computation • Input polynomial f(x) degree n-1 and points x0, x1, …, xn-1 [1] for i=0, …, n-1 let Pi(x) = (x-xi) [2] By Residue Algorithm Computer for i=0, …, n-1 f(xi) = f(x) mod Pi(x) [3] output f(x0), …, f(xn-1) • Time Cost O(M(n) log n), = O(n(log n)2)

26. Polynomial Interpolation • Input evaluation points x0, …, xn-1 values y0, …, yn-1 • Output P(x) where yk = P(xk) for k=0, …, n-1

27. Polynomial Interpolation (cont’d) • Interpolation formula: • Where

28. Proof of Polynomial Interpolation • Proof uses identities:

29. Using Chinese Remaindering for Integer Interpolation • Inputrelatively prime P0, P1, …, Pn-1and yi ∈ {0,…Pi-1} for i=0,…,n-1 • Problem compute y < P0 P1 … Pn-1 s.t. yi = y mod Pi i=0, …, n-1

30. Using Chinese Remaindering for Integer Interpolation • Generalized Interpolation Formula: • Where • proof

31. Advanced Topic: Preconditioned Interpolation Preconditioned Case assumes coefficients {ak | k=0, …, n-1} precomputed • Use Divide & Conquer

32. Preconditioned Interpolation (cont’d)

33. Time Cost for Preconditioned Interpolation • Assuming {a0, …, an-1} precomputed

34. Precomputation of {a0, …, an-1} • Compute • Compute bk where bkPk = P mod (Pk)2by Residue Computation O(M(n) log n) • Compute ak = (bk)-1 mod Pk by Extended GCD algorithm

35. Proof of Precomputation of {a0, …, an-1} • proof

36. Precomputation of {a0, …, an-1} for Polynomial Interpolations • Here Pi = (x-xi) for i=0, …, n-1  reduces to multipoint evaluation of derivative of Q(x) O(M(n) log n) time!

37. Conclusion • Polynomial and Integer Computations use similar divide and conquer techniques to solve: • Multiplication • Division • Interpolation and evaluation • Open Problem Reduce from time O(M(n)log n) to O(M(n))

38. Newton Iteration and Polynomial Computation Analysis of Algorithms Prepared by John Reif, Ph.D.