New Lattice Based Cryptographic Constructions

TL;DR

This paper introduces new lattice-based cryptographic constructions using Fourier analysis, resulting in a stronger public key cryptosystem and novel hash functions with improved security guarantees.

Contribution

It presents the first alternative to Ajtai and Dwork's cryptosystem, with improved security, and introduces a new analysis method for hash functions not relying on iterative steps.

Findings

A new public key cryptosystem with security guaranteed at $O(n^{1.5})$

A family of collision resistant hash functions with enhanced security

A theorem on indistinguishable distributions with potential applications in quantum computing

Abstract

We introduce the use of Fourier analysis on lattices as an integral part of a lattice based construction. The tools we develop provide an elegant description of certain Gaussian distributions around lattice points. Our results include two cryptographic constructions which are based on the worst-case hardness of the unique shortest vector problem. The main result is a new public key cryptosystem whose security guarantee is considerably stronger than previous results ($O(n^{1.5})$ instead of $O(n^7)$). This provides the first alternative to Ajtai and Dwork's original 1996 cryptosystem. Our second result is a family of collision resistant hash functions which, apart from improving the security in terms of the unique shortest vector problem, is also the first example of an analysis which is not based on Ajtai's iterative step. Surprisingly, both results are derived from one theorem which presents two indistinguishable distributions on the segment $[0,1)$. It seems that this theorem can have further applications and as an example we mention how it can be used to solve an open problem related to quantum computation.