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Fast Reduction of Algebraic Lattices over Cyclotomic Fields

  • Paul KirchnerEmail author
  • Thomas EspitauEmail author
  • Pierre-Alain FouqueEmail author
Conference paper
  • 486 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12171)

Abstract

We describe two very efficient polynomial-time algorithms for reducing module lattices defined over arbitrary cyclotomic fields that solve the \(\gamma \)-Hermite Module-SVP problem. They both exploit the structure of tower fields and the second one also uses the symplectic geometry existing in these fields. We conjecture that a rank-2 module over a cyclotomic field of degree n with B-bit coefficients can be heuristically reduced within approximation factor \(2^{\widetilde{\text {O}}\left( n\right) }\) in time \(\widetilde{\text {O}}\left( n^2B\right) \). In the symplectic algorithm, if the condition number C of the input matrix is large enough, this complexity shrinks to \(\widetilde{\text {O}}\left( n^{\log _2 3}C\right) \). In cryptography, matrices are well-conditioned and we can take \(C=B\), but in the worst case, C can be as large as nB. This last result is particularly striking as for some matrices, we can go below the \(n^2B\) swaps lower bound given by the analysis of LLL based on the potential. These algorithms are parallel and we provide a full implementation. We apply them on multilinear cryptographic concrete parameters by reducing matrices of dimension 4096 with 6675-bit integers in 4 days. Finally, we give a quasicubic time for the Gentry-Szydlo algorithm and run it in dimension 1024. It requires efficient ideal multiplications which need fast lattice reductions.

Notes

Acknowledgment

We would like to thank Bill Allombert for his help in the parallelization of the program and Léo Ducas and Damien Stehlé for interesting discussions. Part of this work was done while the authors were visiting the Simons Institute for the Theory of Computing in February 2020. This work is supported by the European Union H2020 program under grant agreements ERC-669891 and Prometheus Project-780701.

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Copyright information

© International Association for Cryptologic Research 2020

Authors and Affiliations

  1. 1.Rennes Univ., IRISA/CNRS FranceRennesFrance
  2. 2.NTT Corp. Secure Plateform LaboratoriesRennesFrance

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