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Cf. ) Cf. cc] The underlying idea can be derived by closely looking at the convolution of real sequences by the radix-2 FHT. g. in the MFA-based convolution for real input. ’ We want to do FFTs in Z/mZ (the ring of integers modulo some integer m) instead of C, the (field of the) complex numbers. These FFTs are called numbertheoretic transforms (NTTs), mod m FFTs or (if m is a prime) prime modulus transforms. There is a restriction for the choice of m: For a length n FFT we need a primitive n-th root of unity.

Roots of unity of an order different from R are available only for the divisors di of R: rR/di is a di -th root of unity because (rR/di )di = rR = 1. 1) The operations needed in FFTs are addition, subtraction and multiplication. Division is not needed, except for division by n for the final normalization after transform and backtransform. Division by n is multiplication by the inverse of n. 2) Cf. [1], [3], [14] or [2] and books on number theory. 1 Prime modulus: Z/pZ = Fp If the modulus is a prime p then Z/pZ is the field Fp : All elements except 0 have inverses and ‘division is possible’ in Z/pZ.

1n coprime to m ⇐⇒ gcd(n, m) = 1 63 CHAPTER 4. 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.

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