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On fast multiplication of polynomials over arbitrary algebras. (English) Zbl 0766.68055
We generalize the well-known Schönhage-Strassen algorithm [A. Schönhage and V. Strassen, Computing 7, 281–292 (1972; Zbl 0223.68007)] for multiplying large integers to an algorithm for multiplying polynomials with coefficients from an arbitrary, not necessarily commutative, not necessarily associative, algebra $${\mathcal A}$$. Our main result is an algorithm to multiply polynomials of degree $$<n$$ in $$O(n\log n)$$ algebra multiplications and $$O(n\log n\log\log n)$$ algebra additions/subtractions (we count a subtraction as an addition). The constant implied by the “$$O$$” does not depend upon the algebra $${\mathcal A}$$. The parallel complexity of our algorithm, i.e., the depth of the corresponding arithmetic circuit, is $$O(\log n)$$.
Reviewer: D.G.Cantor

##### MSC:
 68W30 Symbolic computation and algebraic computation 68W10 Parallel algorithms in computer science 13P05 Polynomials, factorization in commutative rings
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##### References:
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