The Frobenius FFT

Joris van der Hoeven (1) , Robin Larrieu (1)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

Let F_q be the finite field with q elements and let ω be a primitive n-th root of unity in an extension field F_(q^d) of F_q. Given a polynomial P ∈ F_q[x] of degree less than n, we will show that its discrete Fourier transform (P(1),P(ω),…,P(ω^(n-1))) ∈ F_(q^d)^n can be computed essentially d times faster than the discrete Fourier transform of a polynomial Q ∈ F_(q^d)[x] of degree less than n, in many cases. This result is achieved by exploiting the symmetries provided by the Frobenius automorphism of F_(q^d) over F_q.

Mots clés: FFT; finite field; complexity bound; Frobenius automor-phisms

Domaines: Calcul formel [cs.SC]; Complexité [cs.CC]; Algèbre commutative [math.AC]

Joris van der Hoeven, Robin Larrieu. The Frobenius FFT. 2017. ⟨hal-01453269v2⟩