The hardness of computing an eigenform

TL;DR

This paper demonstrates that computing Fourier coefficients of Hecke eigenforms at composite indices is as hard as factoring integers, linking a key problem in number theory to computational complexity.

Contribution

It establishes a conditional hardness result connecting the computation of Fourier coefficients of eigenforms to the difficulty of factoring RSA moduli.

Findings

Computing Fourier coefficients at composite indices is as hard as factoring RSA moduli.

Existence of a polynomial-time algorithm for Fourier coefficients implies polynomial-time factoring.

Provides evidence that these problems are computationally equivalent under certain conditions.

Abstract

In this article, we give evidence that computing Fourier coefficients of the Hecke eigenforms for composite indices is no easier than factoring integers. In particular, we show that the existence of a polynomial time algorithm that, given n, computes the n-th Fourier coefficient of a (fixed) Hecke eigenform implies that we can factor most RSA moduli (numbers that are products of two distinct primes) in polynomial time.