Pseudorandom Knapsacks and the Sample Complexity of LWE Search-to-Decision Reductions

Daniele Micciancio and Petros Mol
University of California, San Diego, USA

Abstract. We study the pseudorandomness of bounded knapsack functions over arbitrary finite abelian groups. Previous works consider only specific families of finite abelian groups and 0-1 coefficients. The main technical contribution of our work is a new, general theorem that provides sufficient conditions under which pseudorandomness of bounded knapsack functions follows directly from their one-wayness. Our results generalize and substantially extend previous work of Impagliazzo and Naor (J. Cryptology 1996).

As an application of the new theorem, we give sample preserving searchto- decision reductions for the Learning With Errors (LWE) problem, introduced by (Regev, STOC 2005) and widely used in lattice-based cryptography. Concretely, we show that, for a wide range of parameters, m LWE samples can be proved indistinguishable from random just under the hypothesis that search LWE is a one-way function for the same number m of samples.

Keywords: Bounded knapsacks, LWE, Fourier analysis, pseudorandomness, sample complexity.