Multiplication of polynomials modulo xn

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Abstract

Let n, be positive integers with 2n1. Let R be an arbitrary nontrivial ring, not necessarily commutative and not necessarily having a multiplicative identity and R[x] be the polynomial ring over R. In this paper, we give improved upper bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most (n1) modulo xn over R. Moreover, we introduce a new complexity notion, the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most (n1) modulo x over R. This new complexity notion provides improved bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most (n1) modulo xn over R.

Keywords

Multiplication of polynomials
Multiplicative complexity
Multiplication algorithms
Multiplication of power series

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