Structural properties of complex residue rings applied to number theoretic Fourier transforms

M Vanwormhoudt

doi:10.1109/tassp.1978.1163041

Abstract

The complex residue ring C(m) modulo m is first defined. Because of the existence of a ring isomorphism known as the Chinese remainder theorem (CRT), the study of C(m) can be limited to the cases where m = p e , p being a prime. C(p e ) contains a multiplicative group, the group Q(p e ) of the invertible elements. Q(p e ) is shown to be the product of a group of order p 2e-2 and of a group R(p e ), which is of order (p - 1) 2 when 4 divides (p - 1), and of order (p 2 - 1) when 4 is no divisor of (p - 1). It is shown that there exists an isomorphic mapping of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q(p) \longleftrightarrow R(p^{e})</tex> . Consequently, the study of the orders of the elements of R(p e ) can be reduced to studying those of Q(p). When 4 is not a divisor of (p - 1), Q(p) and R(p e ) are cyclic groups of order (p 2 - 1). When 4 divides (p - 1), the elements of C(p) can be isomorphically mapped on Z(p) × Z(p), Z(p) being the set of real residue classes, mod p. In this case, the order of the elements of Q(p) and of R(p e ) are limited to the divisors of (p - 1). When 4 does not divide (p - 1), all elements of R(p e ) satisfy a set of orthogonality relations. This property also holds true for some of the elements of R(p e ) when 4 divides (p - 1). The foregoing results are applied to number theoretic Fourier transforms in C(m). A necessary and sufficient condition is derived for N to be a possible transform length. It is shown that all reductions mod <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p\min{i}\max{e_{i}}</tex> of the transform factor where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p\min{i}\max{e_{i}}</tex> represent the prime power factors of m, must belong to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(p\min{i}\max{e_{i}}</tex> and be of order N. Where Fermat number transforms (FNT) do not lead to transform lengths that are larger in C(m) than in Z(m), Mersenne number transforms result in a very large increase of the allowable values of N. The paper ends with a discussion on how a search procedure in Q(p) or in Z(p) allows to determine all available transform factors in Q(p e ) for a given N.

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