Fast norm computation in smooth-degree Abelian number fields

This paper presents a fast method to compute algebraic norms of integral elements of smooth-degree cyclotomic fields, and, more generally, smooth-degree Galois number fields with commutative Galois groups. The typical scenario arising in S -unit searches (for, e.g., class-group computation) is computing a Θ (nlog n) -bit norm of an element of weight n^1/2+o(1) in a degree- n field; this method then uses n(log n)^3+o(1) bit operations. An n(log n)^O(1) operation count was already known in two easier special cases: norms from power-of-2 cyclotomic fields via towers of power-of-2 cyclotomic subfields, and norms from multiquadratic fields via towers of multiquadratic subfields. This paper handles more general Abelian fields by identifying tower-compatible integral bases supporting fast multiplication; in particular, there is a synergy between tower-compatible Gauss-period integral bases and a fast-multiplication idea from Rader. As a baseline, this paper also analyzes various standard norm-computation techniques that apply to arbitrary number fields, concluding that all of these techniques use at least n^2(log n)^2+o(1) bit operations in the same scenario, even with fast subroutines for continued fractions and for complex FFTs. Compared to this baseline, algorithms dedicated to smooth-degree Abelian fields find each norm n/(log n)^1+o(1) times faster, and finish norm computations inside S -unit searches n^2/(log n)^1+o(1) times faster.