Refined Analysis of the Asymptotic Complexity of the Number Field Sieve

TL;DR

This paper refines the asymptotic complexity analysis of the Number Field Sieve by precisely estimating the slow convergence of the heuristic function involved, questioning the practical relevance of existing complexity estimates.

Contribution

It provides exact asymptotic estimates for the parameters of NFS, revealing the slow convergence of the heuristic function and challenging previous complexity assumptions.

Findings

The heuristic function $\xi(N)$ converges very slowly, approximately as $4 ext{log} ext{log} ext{log} hinspace N / (3 ext{log} ext{log} hinspace N)$.

Practical estimates of NFS complexity are unreliable due to the slow convergence of $\xi(N)$.

Asymptotic series expansion of $\xi(N)$ indicates convergence only for extremely large N, beyond practical ranges.

Abstract

The classical heuristic complexity of the Number Field Sieve (NFS) is the solution of an optimization problem that involves an unknown function, usually noted $o(1)$ and called $\xi(N)$ throughout this paper, which tends to zero as the entry $N$ grows. The aim of this paper is to find optimal asymptotic choices of the parameters of NFS as $N$ grows, in order to minimize its heuristic asymptotic computational cost. This amounts to minimizing a function of the parameters of NFS bound together by a non-linear constraint. We provide precise asymptotic estimates of the minimizers of this optimization problem, which yield refined formulas for the asymptotic complexity of NFS. One of the main outcomes of this analysis is that $\xi(N)$ has a very slow rate of convergence: We prove that it is equivalent to $4{\log}{\log}{\log}\,N/(3{\log}{\log}\,N)$. Moreover, $\xi(N)$ has an unpredictable behavior for practical estimates of the complexity. Indeed, we provide an asymptotic series expansion of $\xi$ and numerical experiments indicate that this series starts converging only for $N>\exp(\exp(25))$, far beyond the practical range of NFS. This raises doubts on the relevance of NFS running time estimates that are based on setting $\xi=0$ in the asymptotic formula.