The Frobenius FFT
Résumé
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.
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