A lower bound for polynomial multiplication

Probabilistic algorithms are presented for testing the result of the product of two n-bit integers in O(n) bit operations and for testing the result of the product of two polynomials of degree n over any integral domain in 4n + o(n) algebraic operations with the error probability o(l/n1-e) for any e > 0. The last algorithm does not depend on the constants of the underlying domain.

A note on probabilistically verifying integer and polynomial products

An algorithm for polynomial multiplication that does not depend on the ring constants

An algorithm is presented that allows to multiply two univariate polynomials of degree no more than n with coefficients from an arbitrary (possibly non-commutative) ring in O(n log(n) log(log n)) additions and subtractions and O(n log(n)) multiplications. The arithmetic depth of the algorithm is O(log(n)). This algorithm is a modification of the Schonhage-Strassen procedure to arbitrary radix fast Fourier transforms, and division by the radix is circumvented. By a Kronecker homomorphism the method can be extended to multivariate polynomials. * This material is based upon work supported by the National Science Foundation under Grant Nos. DCR-85-04391 and CCR-87-0563 and by an IBM Faculty Development Award (second author).

Fast Multiplication of Polynomials over Arbitrary Rings

Let R be a ring, and let f(/spl alpha/), g(/spl alph/) /spl epsi/ R[/spl alpha/] be univariate polynomials over R of degree n We Present an algorithm for computing the coefficients of the product f(/spl alpha/)G (/spl alpha/) by O (nlgn) multiplications This algorithm is based on an algorithm for multiplying polynomials over the ring of integers, and does not depend on R Also we prove that multiplying the third degree polynomials over the ripg of integers requires at least nine multiplications This bound is tight

M. Kaminski

Papers

Multiplication Of Polynomials Over The Ring Of Integers