Abstract
At the center of many lattice-based constructions is an algorithm that samples a short vector \(\mathbf{s}\), satisfying \([\mathbf{A}|\mathbf{A}\mathbf{R}-\mathbf{H}\mathbf{G}]\mathbf{s}=\mathbf{t}\text { mod }q\) where \(\mathbf{A},\mathbf{A}\mathbf{R}, \mathbf{H}, \mathbf{G}\) are public matrices and \(\mathbf{R}\) is a trapdoor. Although the algorithm crucially relies on the knowledge of the trapdoor \(\mathbf{R}\) to perform this sampling efficiently, the distribution it outputs should be independent of \(\mathbf{R}\) given the public values. We present a new, simple algorithm for performing this task. The main novelty of our sampler is that the distribution of \(\mathbf{s}\) does not need to be Gaussian, whereas all previous works crucially used the properties of the Gaussian distribution to produce such an \(\mathbf{s}\). The advantage of using a non-Gaussian distribution is that we are able to avoid the high-precision arithmetic that is inherent in Gaussian sampling over arbitrary lattices. So while the norm of our output vector \(\mathbf{s}\) is on the order of \(\sqrt{n}\) to \(n\) - times larger (the representation length, though, is only a constant factor larger) than in the samplers of Gentry, Peikert, Vaikuntanathan (STOC 2008) and Micciancio, Peikert (EUROCRYPT 2012), the sampling itself can be done very efficiently. This provides a useful time/output trade-off for devices with constrained computing power. In addition, we believe that the conceptual simplicity and generality of our algorithm may lead to it finding other applications.
V. Lyubashevsky—Partially supported by the French ANR-13-JS02-0003 “CLE” Project
D. Wichs—Supported by NSF grants 1347350, 1314722, 1413964.
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Lyubashevsky, V., Wichs, D. (2015). Simple Lattice Trapdoor Sampling from a Broad Class of Distributions. In: Katz, J. (eds) Public-Key Cryptography -- PKC 2015. PKC 2015. Lecture Notes in Computer Science(), vol 9020. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46447-2_32
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