@conference {634703,
title = {Composition of zero-knowledge proofs with efficient provers},
booktitle = {Daniele Micciancio, editor, Proceedings of the 7th IACR Theory of Cryptography Conference (TCC {\textquoteleft}10), Lecture Notes on Computer Science},
volume = {5978},
year = {2010},
pages = {572-587},
publisher = {Springer-Verlag},
organization = {Springer-Verlag},
abstract = {
We revisit the composability of different forms of zero- knowledge proofs when the honest prover strategy is restricted to be polynomial time (given an appropriate auxiliary input). Our results are:
When restricted to efficient provers, the original Goldwasser{\textendash}Micali{\textendash}Rackoff (GMR) definition of zero knowledge (STOC {\textquoteleft}85), here called\ plain zero knowledge, is closed under a constant number of sequential compositions (on the same input). This contrasts with the case of unbounded provers, where Goldreich and Krawczyk (ICALP {\textquoteleft}90, SICOMP {\textquoteleft}96) exhibited a protocol that is zero knowledge under the GMR definition, but for which the sequential composition of 2 copies is not zero knowledge.
\
If we relax the GMR definition to only require that the simulation is indistinguishable from the verifier{\textquoteright}s view by uniform polynomial-time distinguishers, with no auxiliary input beyond the statement being proven, then again zero knowledge is not closed under sequential composition of 2 copies.
\
We show that auxiliary-input zero knowledge with efficient provers is not closed under\ parallel\ composition of 2 copies under the assumption that there is a secure key agreement protocol (in which it is easy to recognize valid transcripts). Feige and Shamir (STOC {\textquoteleft}90) gave similar results under the seemingly incomparable assumptions that (a) the discrete logarithm problem is hard, or (b)\ \(\mathcal{UP}\nsubseteq\mathcal{BPP}\)\ and one-way functions exist.
},
url = {https://link.springer.com/chapter/10.1007/978-3-642-11799-2_34},
author = {Eleanor Birrell and Salil Vadhan}
}