Theory of Cryptography

Volume 5978 of the series Lecture Notes in Computer Science pp 553-571

Efficiency Limitations for Σ-Protocols for Group Homomorphisms

  • Endre BangerterAffiliated withBern University of Applied Sciences
  • , Jan CamenischAffiliated withIBM Research
  • , Stephan KrennAffiliated withBern University of Applied SciencesUniversity of Fribourg


Efficient zero-knowledge proofs of knowledge for group homomorphisms are essential for numerous systems in applied cryptography. Especially, Σ-protocols for proving knowledge of discrete logarithms in known and hidden order groups are of prime importance. Yet, while these proofs can be performed very efficiently within groups of known order, for hidden order groups the respective proofs are far less efficient.

This paper shows strong evidence that this efficiency gap cannot be bridged. Namely, while there are efficient protocols allowing a prover to cheat only with negligibly small probability in the case of known order groups, we provide strong evidence that for hidden order groups this probability is bounded below by 1/2 for all efficient Σ-protocols not using common reference strings or the like.

We prove our results for a comprehensive class of Σ-protocols in the generic group model, and further strengthen them by investigating certain instantiations in the plain model.


Generic Group Model Σ-Protocols Proofs of Knowledge Error Bounds