Paper 2025/986 Original: eprint.iacr.org/2025/986 ↗

The Rényi Smoothing Parameter and Its Applications in Lattice-Based Cryptography

Abstract

The smoothing parameter is a cornerstone concept in lattice-based cryptography. Traditionally defined using the \( L^{\infty} \) distance, this standard formulation can be overly stringent compared to the \( L^1 \) (or statistical) distance more commonly employed in cryptographic contexts. Recent work has proposed relaxed definitions based on Kullback-Leibler (KL) divergence and \( L^1 \) distance, thereby loosening the constraints required for the distance to vanish. However, the additive nature of the \( L^1 \) distance can be limiting for cryptographic applications where probability preservation is essential. In this paper, we introduce the {Rényi smoothing parameter} of a lattice, based on Rényi divergence, to address this limitation. The advantages of Rényi divergence in cryptographic settings are well known thanks to its multiplicative nature. The Rényi smooting parameter provides a tunable framework that interpolates between the \( L^1 \) and \( L^{\infty} \) distances, offering enhanced flexibility. We present two complementary methods to study the averaging behavior of the Rényi flatness factor: one uses classical tools such as the Minkowski-Hlawka ensemble and Rogers’ formula for computing lattice function moments; the other employs Construction A lattices derived from random codes. Finally, we illustrate how this new perspective yields improvements in lattice-based cryptographic constructions.