Efficient Zero-Knowledge Proofs for Commitments from Learning with Errors over Rings

  • Fabrice Benhamouda
  • Stephan Krenn
  • Vadim Lyubashevsky
  • Krzysztof Pietrzak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9326)


We extend a commitment scheme based on the learning with errors over rings (\(\mathsf{RLWE}\)) problem, and present efficient companion zero-knowledge proofs of knowledge. Our scheme maps elements from the ring (or equivalently, n elements from \(\mathbb F_q\)) to a small constant number of ring elements. We then construct \(\varSigma \)-protocols for proving, in a zero-knowledge manner, knowledge of the message contained in a commitment. We are able to further extend our basic protocol to allow us to prove additive and multiplicative relations among committed values.

Our protocols have a communication complexity of \(\mathcal {O}(Mn\log q)\) and achieve a negligible knowledge error in one run. Here M is the constant from a rejection sampling technique that we employ, and can be set close to 1 by adjusting other parameters. Previously known \(\varSigma \)-protocols for LWE-related languages only achieved a noticeable or even constant knowledge error (thus requiring many repetitions of the protocol), or relied on “smudging” out the error (which necessitates working over large fields, resulting in poor efficiency).


Commitment schemes Ring learning with errors Zero-Knowledge Proofs of Knowledge 

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Fabrice Benhamouda
    • 1
  • Stephan Krenn
    • 2
  • Vadim Lyubashevsky
    • 3
  • Krzysztof Pietrzak
    • 4
  1. 1.ENS, CNRS, INRIA, and PSLParisFrance
  2. 2.AIT Austrian Institute of Technology GmbHViennaAustria
  3. 3.ENS, INRIAParisFrance
  4. 4.IST AustriaKlosterneuburgAustria

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