Paper 2026/1208

Preimage sampleable function families without discrete Gaussians

Abstract

We construct preimage sampleable function families on $q$-ary lattices [Gentry–Peikert–Vaikuntanathan, STOC'08] for which preimage sampling reduces to sampling uniform points of unitriangular lattices from simple polytopes: affine linear transforms on the $\ell_1$ and $\ell_\infty$ balls, and a scaled intersection thereof. We build the necessary samplers by adapting an algorithm of [Kannan–Vempala, STOC'97] and improving its analysis. This sampling requires only uniform bits and affine linear transforms on the uniform distribution on $[0,1]$. The collision resistance of these families relies on the short integer solutions problem [Ajtai, STOC'96] in various $\ell_p$ norms. Considering the Lee metric as the $\ell_1$ norm in $q$-ary lattices, we answer an open question to construct such families for the Lee metric [Hörmann–van Woerden, CRYPTO'24]. We also answer an open question of [Plançon--Prest, PKC'21] by sampling from polytopes with inradius smaller by a factor almost square root in the lattice rank. We provide a generic framework for constructing preimage sampleable function families from polytopes with sufficient conditions for realising it. While our parameters are worse than prior discrete Gaussian based constructions, such distributions are challenging from a physical security perspective.