# Information Security and Cryptography Research Group

## Efficient Zero-Knowledge Proofs of Knowledge Without Intractability Assumptions

### Ronald Cramer, Ivan Damgård, and Phillip MacKenzie

Public Key Cryptography — PKC 2000, Lecture Notes in Computer Science, Springer-Verlag, vol. 1751, pp. 354–372, Jan 2000.

We initiate the investigation of the class of relations that admit extremely efficient perfect zero knowledge proofs of knowledge: constant number of rounds, communication linear in the length of the statement and the witness, and negligible knowledge error. In its most general incarnation, our result says that for relations that have a particular three-move honest-verifier zero-knowledge (HVZK) proof of knowledge, and which admit a particular three-move HVZK proof of knowledge for an associated commitment relation, perfect zero knowledge (against a general verifier) can be achieved essentially for free, even when proving statements on several instances combined under under monotone function composition. In addition, perfect zero-knowledge is achieved with an optimal 4-moves. Instantiations of our main protocol lead to efficient perfect ZK proofs of knowledge of discrete logarithms and RSA-roots, or more generally, q-one-way group homomorphisms. None of our results rely on intractability assumptions.

## BibTeX Citation

@inproceedings{CrDaMa00b,
author       = {Ronald Cramer and Ivan Damgård and Phillip MacKenzie},
title        = {Efficient Zero-Knowledge Proofs of Knowledge Without Intractability Assumptions},
booktitle    = {Public Key Cryptography --- PKC 2000},
pages        = 354--372,
series       = {Lecture Notes in Computer Science},
volume       = 1751,
year         = 2000,
month        = 1,
publisher    = {Springer-Verlag},
}