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.