Module Lattice Security (Part II): Module Lattice Reduction via Optimal Sign Selection

TL;DR

This paper extends lattice reduction algorithms to module lattices, achieving near-ideal Hermite factors and introducing optimized sign selection and precision control techniques.

Contribution

It generalizes the CDPR lattice reduction to module lattices, introduces CRT-scaled rounding for precision, and formulates sign selection as a MILP problem.

Findings

Achieves Hermite factor $ ilde{O}( oot{n} ext{})$ matching ideal cases.

Introduces CRT-scaled rounding to reduce Gram-Schmidt error.

Determines optimal sign selection discrepancy as a universal constant.

Abstract

We extend the CDPR lattice reduction algorithm from ideal to module lattices, leveraging the trace orthogonality of the power basis to decompose the module into rank-1 submodules and applying CDPR independently to each. This base module reduction achieves a Hermite factor $\exp(\tilde{O}(\sqrt{n}))$ matching the ideal case, with a module reduction factor $O(1)$ independent of the rank, under a balance hypothesis automatically satisfied for MLWE-distributed bases. To control precision, we introduce CRT-scaled rounding at totally split primes, reducing the Gram-Schmidt rounding error and yielding a bounded-precision implementation. We further reformulate the CDPR sign-selection subproblem as a mixed-integer linear program, determining the optimal balanced discrepancy to be a universal constant $\delta^*\approx 0.4407$. All results build on the class number one condition $h_k^+=1$ established in Part I of this series.