Implementing Candidate Graded Encoding Schemes from Ideal Lattices

  • Martin R. Albrecht
  • Catalin Cocis
  • Fabien Laguillaumie
  • Adeline Langlois
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9453)

Abstract

Multilinear maps have become popular tools for designing cryptographic schemes since a first approximate realisation candidate was proposed by Garg, Gentry and Halevi (GGH). This construction was later improved by Langlois, Stehlé and Steinfeld who proposed GGHLite which offers smaller parameter sizes. In this work, we provide the first implementation of such approximate multilinear maps based on ideal lattices. Implementing GGH-like schemes naively would not allow instantiating it for non-trivial parameter sizes. We hence propose a strategy which reduces parameter sizes further and several technical improvements to allow for an efficient implementation. In particular, since finding a prime ideal when generating instances is an expensive operation, we show how we can drop this requirement. We also propose algorithms and implementations for sampling from discrete Gaussians, for inverting in some Cyclotomic number fields and for computing norms of ideals in some Cyclotomic number rings. Due to our improvements we were able to compute a multilinear jigsaw puzzle for \(\kappa =52\) (resp. \(\kappa =38\)) and \(\lambda = 52\) (resp. \(\lambda = 80\)).

Keywords

Algorithms Implementation Lattice-based cryptography Cryptographic multilinear maps 

Notes

Acknowledgement

We would like to thank Guilhem Castagnos, Guillaume Hanrot, Bill Hart, Claude-Pierre Jeannerod, Clément Pernet, Damien Stehlé, Gilles Villard and Martin Widmer for helpful discussions. We would like to thank Steven Galbraith for pointing out the NTRU-style attack to us and for helpful discussions. This work has been supported in part by ERC Starting Grant ERC-2013-StG-335086-LATTAC. The work of Albrecht was supported by EPSRC grant EP/L018543/1 “Multilinear Maps in Cryptography”.

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Copyright information

© International Association for Cryptologc Research 2015

Authors and Affiliations

  • Martin R. Albrecht
    • 1
  • Catalin Cocis
    • 2
  • Fabien Laguillaumie
    • 3
  • Adeline Langlois
    • 4
    • 5
  1. 1.Information Security GroupRoyal Holloway, University of LondonEghamUK
  2. 2.Technical University of Cluj-NapocaCluj-NapocaRomania
  3. 3.LIP (U. Lyon, CNRS, ENS Lyon, INRIA, UCBL)Université Claude Bernard Lyon 1VilleurbanneFrance
  4. 4.EPFLLausanneSwitzerland
  5. 5.CNRS/IRISARennesFrance

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