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Polynomial time bounded distance decoding near Minkowski’s bound in discrete logarithm lattices

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Abstract

We propose a concrete family of dense lattices of arbitrary dimension n in which the lattice bounded distance decoding (BDD) problem can be solved in deterministic polynomial time. This construction is directly adapted from the Chor–Rivest cryptosystem (IEEE-TIT 1988). The lattice construction needs discrete logarithm computations that can be made in deterministic polynomial time for well-chosen parameters. Each lattice comes with a deterministic polynomial time decoding algorithm able to decode up to large radius. Namely, we reach decoding radius within \(O(\log n)\) Minkowski’s bound, for both \(\ell _1\) and \(\ell _2\) norms.

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Notes

  1. Later we relax this assumption and discuss about m.

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Acknowledgements

We would like to thank Steven Galbraith, Cong Ling, Dan Shepherd and Chaoping Xing for their interesting discussions and precious comments about this work. We are also grateful for the helpful comments of the anonymous reviewers of Design, Code and Cryptography.

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  1. Cryptology Group, CWI, Amsterdam, The Netherlands

    Léo Ducas & Cécile Pierrot

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  1. Léo Ducas
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  2. Cécile Pierrot
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Correspondence to Cécile Pierrot.

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Communicated by O. Ahmadi.

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Léo Ducas: Supported by a Veni Innovational Research Grant from NWO under Project Number 639.021.645.

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Ducas, L., Pierrot, C. Polynomial time bounded distance decoding near Minkowski’s bound in discrete logarithm lattices. Des. Codes Cryptogr. 87, 1737–1748 (2019). https://doi.org/10.1007/s10623-018-0573-3

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