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Si ) the m-dimensional Fourier transform ck (k = (k1 , k2 , k3 , . . , km ), ki ∈ 0, 1, 2, . . k ... x1 =0 x2 =0 where z = e± 2 π i/n , n = S1 S2 . . k where S = (S1 − 1, S2 − 1, . . 79) x=0 The inverse transform is again the one with the minus in the exponent of z. 80) x=0 which shows that the 2-dimensional FT can be accomplished by using 1-dimensional FTs to transform first the rows and then the columns8 . 13 (row column FFT) Compute the two dimensional FT of a[][] using the row column method procedure rowcol_ft(a[][], R, C) { complex a[R][C] // R (length-C) rows, C (length-R) columns for r:=0 to R-1 // FFT rows { fft(a[r][], C, is) } complex t[R] // scratch array for columns for c:=0 to C-1 // FFT columns { 7 Imagine 8 or a R × C matrix of R rows (of length C) and C columns (of length R).

NUMBERTHEORETIC TRANSFORMS (NTTS) 64 Roots of unity are available for the maximal order R = p−1 and its divisors: Therefore the first condition on n for a length-n mod p FFT being possible is that n divides p − 1. This restricts the choice for p to primes of the form p = v n + 1: For length-n = 2k FFTs one will use primes like p = 3 · 5 · 227 + 1 (31 bits), p = 13 · 228 + 1 (32 bits), p = 3 · 29 · 256 + 1 (63 bits) or p = 27 · 259 + 1 (64 bits)2 . The elements of maximal order in Z/pZ are called primitive elements, generators or primitive roots modulo p.

Second remember that the FT is the special case z = e±2 π i/n of the ZT: With the chirp ZT algorithm one also has an (arbitrary length) FFT algorithm The transform takes a few times more than an optimal transform (by direct FFT) would take. The worst case (if only FFTs for n a power of 2 are available) is n = 2p + 1: One must perform 3 FFTs of length 2p+2 ≈ 4 n for the computation of the convolution. So the total work amounts to about 12 times the work a FFT of length n = 2p would cost. It is of course possible to lower this ‘worst case factor’ to 6 by using highly composite L slightly greater than 2 n.

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