International Workshop on Public Key Cryptography

PKC 2014: Public-Key Cryptography – PKC 2014 pp 556-573 | Cite as

General Impossibility of Group Homomorphic Encryption in the Quantum World

  • Frederik Armknecht
  • Tommaso Gagliardoni
  • Stefan Katzenbeisser
  • Andreas Peter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8383)

Abstract

Group homomorphic encryption represents one of the most important building blocks in modern cryptography. It forms the basis of widely-used, more sophisticated primitives, such as CCA2-secure encryption or secure multiparty computation. Unfortunately, recent advances in quantum computation show that many of the existing schemes completely break down once quantum computers reach maturity (mainly due to Shor’s algorithm). This leads to the challenge of constructing quantum-resistant group homomorphic cryptosystems.

In this work, we prove the general impossibility of (abelian) group homomorphic encryption in the presence of quantum adversaries, when assuming the IND-CPA security notion as the minimal security requirement. To this end, we prove a new result on the probability of sampling generating sets of finite (sub-)groups if sampling is done with respect to an arbitrary, unknown distribution. Finally, we provide a sufficient condition on homomorphic encryption schemes for our quantum attack to work and discuss its satisfiability in non-group homomorphic cases. The impact of our results on recent fully homomorphic encryption schemes poses itself as an open question.

Keywords

Public-Key Cryptography Homomorphic Encryption Semantic Security Quantum Algorithms Sampling Group Generators 

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Copyright information

© International Association for Cryptologic Research 2014

Authors and Affiliations

  • Frederik Armknecht
    • 1
  • Tommaso Gagliardoni
    • 2
  • Stefan Katzenbeisser
    • 2
  • Andreas Peter
    • 3
  1. 1.Universität MannheimGermany
  2. 2.CASEDTechnische Universität DarmstadtGermany
  3. 3.University of TwenteThe Netherlands

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