Paper 2024/1039

Reduction from Average-Case M-ISIS to Worst-Case CVP Over Perfect Lattices

Abstract

This paper presents a novel reduction from the average-case hardness of the Module Inhomogeneous Short Integer Solution (M-ISIS) problem to the worst-case hardness of the Closest Vector Problem (CVP) by defining and leveraging “perfect” lattices for cryptographic purposes. Perfect lattices, previously only theoretical constructs, are characterized by their highly regular structure, optimal density, and a central void, which we term the “Origin Cell.” The simplest Origin Cell is a hypercube with edge length 1 centered at the origin, guaranteed to be devoid of any valid lattice points. By exploiting the unique properties of the Origin Cell, we recalibrate the parameters of the M-ISIS and CVP problems. Our results demonstrate that solving M-ISIS on average over perfect lattices is at least as hard as solving CVP in the worst case, thereby providing a robust hardness guarantee for M-ISIS. Additionally, perfect lattices facilitate exceptionally compact cryptographic variables, enhancing the efficiency of cryptographic schemes. This significant finding enhances the theoretical foundation of lattice-based cryptographic problems and confirms the potential of perfect lattices in ensuring strong cryptographic security. The Appendix includes SageMath code to demonstrate the reproducibility of the reduction process from M-ISIS to CVP.