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arXiv:1703.02515

A Discrete Fourier Transform on Lattices with Quantum Applications

Lior Eldar, Peter W. Shor

Published Mar 7, 2017• 1 citation
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Abstract

In this work, we introduce a definition of the Discrete Fourier Transform (DFT) on Euclidean lattices in $\R^n$, that generalizes the $n$-th fold DFT of the integer lattice $\Z^n$ to arbitrary lattices. This definition is not applicable for every lattice, but can be defined on lattices known as Systematic Normal Form (SysNF) introduced in \cite{ES16}. Systematic Normal Form lattices are sets of integer vectors that satisfy a single homogeneous modular equation, which itself satisfies a certain number-theoretic property. Such lattices form a dense set in the space of $n$-dimensional lattices, and can be used to approximate efficiently any lattice. This implies that for every lattice $L$ a DFT can be computed efficiently on a lattice near $L$. Our proof of the statement above uses arguments from quantum computing, and as an application of our definition we show a quantum algorithm for sampling from discrete distributions on lattices, that extends our ability to sample efficiently from the discrete Gaussian distribution \cite{GPV08} to any distribution that is sufficiently "smooth". We conjecture that studying the eigenvectors of the newly-defined lattice DFT may provide new insights into the structure of lattices, especially regarding hard computational problems, like the shortest vector problem.

DOI
10.48550/arxiv.1703.02515

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The Systematic Normal Form (SysNF) is a canonical form of lattices introduced in [Eldar,Shor '16], in which the basis entries satisfy a certain co-primality condition. Using a "smooth" analysis of lattices by SysNF lattices we design a quantum algorithm that can efficiently solve the following variant of the bounded-distance-decoding problem: given a lattice L, a vector v, and numbers b = λ_1(L)/n^{17}, a = λ_1(L)/n^{13} decide if v's distance from L is in the range [a/2, a] or at most b, where λ_1(L) is the length of L's shortest non-zero vector. Improving these parameters to a = b = λ_1(L)/\sqrt{n} would invalidate one of the security assumptions of the Learning-with-Errors (LWE) cryptosystem against quantum attacks.
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We isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over Zp{\mathbb Z}_pZp​ can be efficiently approximated by transforming over Zq{\mathbb Z}_qZq​ for any q in a large range. Our result places no restrictions on the superposition to be transformed, generalizing the result implicit in Shor which applies only to periodic superpositions. In addition, our proof easily generalizes to multi-dimensional transforms for any constant number of dimensions.