Journal of Cryptology

, Volume 31, Issue 2, pp 610–640 | Cite as

Improved Security Proofs in Lattice-Based Cryptography: Using the Rényi Divergence Rather than the Statistical Distance

  • Shi Bai
  • Tancrède Lepoint
  • Adeline Roux-Langlois
  • Amin Sakzad
  • Damien Stehlé
  • Ron SteinfeldEmail author


The Rényi divergence is a measure of closeness of two probability distributions. We show that it can often be used as an alternative to the statistical distance in security proofs for lattice-based cryptography. Using the Rényi divergence is particularly suited for security proofs of primitives in which the attacker is required to solve a search problem (e.g., forging a signature). We show that it may also be used in the case of distinguishing problems (e.g., semantic security of encryption schemes), when they enjoy a public sampleability property. The techniques lead to security proofs for schemes with smaller parameters, and sometimes to simpler security proofs than the existing ones.


Lattice-based cryptography Rényi divergence Statistical distance Security proofs 



We thank Léo Ducas, Vadim Lyubashevsky and Fabrice Mouhartem for useful discussions. We thank Katsuyuki Takashima and Atsushi Takayasu for informing us about an error in the conference version of this work, and another one in the computations of the \(R_a\)-based analysis of Sect. 3. This work has been supported in part by ERC Starting Grant ERC-2013-StG-335086-LATTAC, an Australian Research Fellowship (ARF) from the Australian Research Council (ARC), ARC Discovery Grants DP0987734, DP110100628 and DP150100285, and the European Unions H2020 Programme under Grant Agreement Number ICT-644209.


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Copyright information

© International Association for Cryptologic Research 2017

Authors and Affiliations

  • Shi Bai
    • 1
  • Tancrède Lepoint
    • 2
  • Adeline Roux-Langlois
    • 3
  • Amin Sakzad
    • 4
  • Damien Stehlé
    • 5
  • Ron Steinfeld
    • 4
    Email author
  1. 1.Department of Mathematical SciencesFlorida Atlantic UniversityBoca RatonUSA
  2. 2.SRI InternationalNew YorkUSA
  3. 3.CNRS/IRISARennesFrance
  4. 4.Faculty of Information TechnologyMonash UniversityClaytonAustralia
  5. 5.ENS de Lyon, Laboratoire LIP (U. Lyon, CNRS, ENSL, INRIA, UCBL)LyonFrance

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