On fast multiplication of polynomials over arbitrary algebras

TLDR: This paper generalizes the well-known Sch6nhage-Strassen algorithm for multiplying large integers to an algorithm for dividing polynomials with coefficients from an arbitrary, not necessarily commutative, not always associative, algebra d, and obtains a method not requiring division that is valid for any algebra.

Abstract: In this paper we generalize the well-known Sch6nhage-Strassen algorithm for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra d . Our main result is an algorithm to multiply polynomials of degree < n in O (n log n) algebra multiplications and O (n log n loglog n) algebra additions/subtractions (we count a subtraction as an addition). The constant implied by the \"0 \"does not depend upon the algebra ~4. The parallel complexity of our algorithm, i.e., the depth of the corresponding arithmetic circuit, is O (log n). When division by 2 is possible, then the Sch6nhage-Strassen [-13] integer multiplication algorithm can be easily reformulated as a polynomial multiplication procedure (c.f. [11]). Sch6nhage [12] investigated the polynomial multiplication problem for arbitrary fields of characteristic 2, in which the standard U-point Discrete Fast Fourier Transform algorithm (DFT) cannot be used because it requires division by 2. The fields over which the DFT is used do not necessarily contain the primitive roots of unity necessary for the computation of the Discrete Fast Fourier Transform and, to use it, such roots must be adjoined to the ground field. It is this which increases the complexity from O (n log n) to O(n log n loglog n). Sch6nhage's algorithm for fields of characteristic 2 uses a 3k-point Fourier transform. When division by 3 is possible, he obtains again an algorithm of complexity O(n log n loglog n). His approach does not appear to generalize to s k transforms, even when s = 5. Here, we exhibit an alternate method that works for order s k for any integer s > 2. By applying this method for two relatively prime values of s, we obtain a method not requiring division. As a result our method is valid for any algebra