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

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

Taylor Expansion

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

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

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

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

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!

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

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

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

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

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

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

Modular Arithmetic Assume relatively prime P 0, P 1, …, P k-1 Let given x, 0  x < p

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

Super Moduli Computation Input p 0, p 1, …, p k-1, where p i < 2 b Output Super modular Tree:

Super Moduli Computation (cont’d) Time Cost

Algorithm Residue Computation Input Output x 0, x 1, …, x k-1, when x i =x mod P i i=0, and k-1 Recursive algorithm [1] compute quotient and remainders:

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

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

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

Advanced Topic: Residue Computation on Polynomials Input moduli P 0 (x), P 1 (x), …, P k-1 (x) and each degree  d and relatively prime Algorithm uses similar Super modular Tree tre, but using polynomials rather than integers

Advanced Topic: Residue Computation on Polynomials (cont’d) Output for i=0, …, k-1 Q i (x) = Q(x) mod P i (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

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

Polynomial Interpolation Input evaluation points x 0, …, x n-1 values y 0, …, y n-1 Output P(x) where y k = P(x k ) for k=0, …, n-1

Polynomial Interpolation (cont’d) Interpolation formula: Where

Proof of Polynomial Interpolation Proof uses identities:

Using Chinese Remaindering for Integer Interpolation Input relatively prime P 0, P 1, …, P n-1 and y i  {0,…P i-1 } for i=0,…,n-1 Problem compute y <  P i s.t. y i = y mod P i i=0, …, n-1

Generalized Interpolation Formula: Where proof Using Chinese Remaindering for Integer Interpolation

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

Preconditioned Interpolation (cont’d)

Time Cost for Preconditioned Interpolation Assuming {a 0, …, a n-1 } precomputed

Precomputation of {a 0, …, a n-1 } 1)Compute 2)Compute b k where b k P k = P mod (P k ) 2 by Residue Computation O(M(n) log n) 3)Compute a k = (b k ) -1 mod P k by Extended GCD algorithm

Proof of Precomputation of {a 0, …, a n-1 } proof

Precomputation of {a 0, …, a n-1 } for Polynomial Interpolations Here P i = (x-x i ) for i=0, …, n-1  reduces to multipoint evaluation of derivative of Q(x) O(M(n) log n) time!

Conclusion Polynomial and Integer Computations use similar divide and conquer techniques to solve: 1.Multiplication 2.Division 3.Interpolation and evaluation Open Problem Reduce from time O(M(n)log n) to O(M(n))

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