The Hidden Lattice Problem

TL;DR

This paper investigates the Hidden Lattice Problem, proposing two algorithms—one adapted from existing methods and a new variant—analyzing their effectiveness and relevance to cryptanalysis and related cryptographic problems.

Contribution

It introduces a new lattice-based algorithm for the Hidden Lattice Problem and compares it with existing methods, providing insights into cryptographic applications.

Findings

The new algorithm offers advantages over previous methods.

Both algorithms are effective for cryptographic problems like Hidden Subset Sum.

The study highlights the problem's relevance to cryptanalysis.

Abstract

We consider the problem of revealing a small hidden lattice from the knowledge of a low-rank sublattice modulo a given sufficiently large integer -- the {\em Hidden Lattice Problem}. A central motivation of study for this problem is the Hidden Subset Sum Problem, whose hardness is essentially determined by that of the hidden lattice problem. We describe and compare two algorithms for the hidden lattice problem: we first adapt the algorithm by Nguyen and Stern for the hidden subset sum problem, based on orthogonal lattices, and propose a new variant, which we explain to be related by duality in lattice theory. Following heuristic, rigorous and practical analyses, we find that our new algorithm brings some advantages as well as a competitive alternative for algorithms for problems with cryptographic interest, such as Approximate Common Divisor Problems, and the Hidden Subset Sum Problem. Finally, we study variations of the problem and highlight its relevance to cryptanalysis.