LWE from Non-commutative Group Rings

TL;DR

This paper explores LWE problems on non-commutative group rings, proposing cryptographic schemes that enhance security by avoiding weaknesses linked to principal ideal lattices, especially using dihedral groups.

Contribution

It introduces LWE on non-commutative group rings, specifically dihedral groups, and constructs secure public key encryption schemes with improved security features.

Findings

LWE on non-commutative group rings can be used for cryptography.

Cryptographic schemes based on dihedral group rings are efficient and more secure.

The approach generalizes LWE beyond cyclic groups to non-commutative settings.

Abstract

The Ring Learning-With-Errors (LWE) problem, whose security is based on hard ideal lattice problems, has proven to be a promising primitive with diverse applications in cryptography. There are however recent discoveries of faster algorithms for the principal ideal SVP problem, and attempts to generalize the attack to non-principal ideals. In this work, we study the LWE problem on group rings, and build cryptographic schemes based on this new primitive. One can regard the LWE on cyclotomic integers as a special case when the underlying group is cyclic, while our proposal utilizes non-commutative groups, which eliminates the weakness associated with the principal ideal lattices. In particular, we show how to build public key encryption schemes from dihedral group rings, which maintains the efficiency of the ring-LWE and improves its security.