Polynomial Multiplication over Finite Fields in Time \( O(n \log n \)

Abstract

Assuming a widely believed hypothesis concerning the least prime in an arithmetic progression, we show that polynomials of degree less than \( n \) over a finite field \( \mathbb {F}_q \) with \( q \) elements can be multiplied in time \( O (n \log q \log (n \log q)) \) , uniformly in \( q \) . Under the same hypothesis, we show how to multiply two \( n \) -bit integers in time \( O (n \log n) \) ; this algorithm is somewhat simpler than the unconditional algorithm from the companion paper [ 22 ]. Our results hold in the Turing machine model with a finite number of tapes.

Polynomial modular product verification and its implications

“…It has been showed by Harvey and van der Hoeven in [13] that one can reach a bit complexity of (n log q log(n log q)4 log * (n) ) for polynomial multiplication over q [X ] for any prime field q . We shall mention that very recently, such complexity have been further improved to (n log q log(n log q)) bit operations [15] under some mild hypothesis. For polynomials with integer coefficients bounded by an integer C, the complexity falls down to multiplying two integers of bit length (n(log n + log C)) which gives (n(log 2 n + log C log n + log C log log C) = ˜ (n log C) 1 when we assume that n-bits integer multiplication complexity is I(n) = (n log n) [14].…”

“…Let F = X 14 + 2X 7 + 2, G = 3X 13 + 5X 8 + 3 and H = X 14 − 2X 7 + 2. Then F G = 3X 27 + 5X 22 + 6X 20 + 10X 15 + 3X 14 + 6X 13 + 10X 8 + 6X 7 + 6 has nine terms, while F H = X 28 + 4 has only two.…”

“…We denote by M q (n) the bit complexity of the multiplication of two degree-n dense polynomials over a finite field q . The best known bounds on M q (n) are (n log q log(n log q) 4 log * (n log q) ) [13] unconditionally and (n log q log(n log q)) assuming the existence of some Linnik constant [15]. To simplify the notation, we assume the existence of this Linnik constant.…”

“…Yet the efficiency of Algorithm 6 depends heavily on the integral domain . Indeed the complexity of polynomial multiplication in [X ] can be faster than (n log n log log n) operations in [13,15]. Furthermore, if is small, one operation in ext corresponds to a non-constant number of operations in .…”

“…Polynomial multiplication is the most noticeable problem that attracted a lot of attention since many decades, culminating nowadays with quasi-optimal algorithms [2,15]. Although such algorithms are really efficient in theory and in practice, there are not yet optimal and they often rely on complex approaches that can be error prone.…”