Fast norm computation in smooth-degree Abelian number fields

Abstract 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 $$\Theta (n\log n)$$ Θ ( n log n ) -bit norm of an element of weight $$n^{1/2+o(1)}$$ n 1 / 2 + o ( 1 ) in a degree- n field; this method then uses $$n(\log n)^{3+o(1)}$$ n ( log n ) 3 + o ( 1 ) bit operations. An $$n(\log n)^{O(1)}$$ 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 $$n^2(\log n)^{2+o(1)}$$ 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 $$n/(\log n)^{1+o(1)}$$ n / ( log n ) 1 + o ( 1 ) times faster, and finish norm computations inside S -unit searches $$n^2/(\log n)^{1+o(1)}$$ n 2 / ( log n ) 1 + o ( 1 ) times faster.