Frobenius Additive Fast Fourier Transform

Abstract

In ISSAC 2017, van der Hoeven and Larrieu showed that evaluating a polynomial P ∈ F q [x] of degree < n at all n-th roots of unity in F q d can essentially be computed d-time faster than evaluating Q ∈ F q d [x] at all these roots, assuming F q d contains a primitive n-th root of unity [vdHL17a]. Termed the Frobenius FFT, this discovery has a profound impact on polynomial multiplication, especially for multiplying binary polynomials, which finds ample application in coding theory and cryptography. In this paper, we show that the theory of Frobenius FFT beautifully generalizes to a class of additive FFT developed by Cantor and Gao-Mateer [Can89, GM10]. Furthermore, we demonstrate the power of Frobenius additive FFT for q = 2: to multiply two binary polynomials whose product is of degree < 256, the new technique requires only 29,005 bit operations, while the best result previously reported was 33,397. To the best of our knowledge, this is the first time that FFT-based multiplication outperforms Karatsuba and the like at such a low degree in terms of bit-operation count. CCS CONCEPTS• Mathematics of computing → Computations in finite fields; KEYWORDS addtitive FFT, Frobenius FFT, polynomial multiplication

Fast Hermite interpolation and evaluation over finite fields of characteristic two

“…To avoid confusion, we refer to such algorithms as multiplicative FFTs hereafter. Additive FFTs have been investigated as an alternative to multiplicative FFTs for use in fast multiplication algorithms for binary polynomials [28,6,23,8,9,18], and have also found applications in coding theory and cryptography [4,3,10,1].…”