# A New Interactive Hashing Theorem

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## Abstract

Interactive hashing, introduced by Naor, Ostrovsky, Venkatesan, and Yung (J. Cryptol. 11(2):87–108, 1998), plays an important role in many cryptographic protocols. In particular, interactive hashing is a major component in all known constructions of statistically hiding commitment schemes and of statistical zero-knowledge arguments based on general one-way permutations/functions. Interactive hashing with respect to a one-way function *f* is a two-party protocol that enables a sender who knows *y*=*f*(*x*) to transfer a random hash *z*=*h*(*y*) to a receiver such that the sender is committed to *y*: the sender cannot come up with *x* and *x*′ such that *f*(*x*)≠*f*(*x*′), but *h*(*f*(*x*))=*h*(*f*(*x*′))=*z*. Specifically, if *f* is a permutation and *h* is a two-to-one hash function, then the receiver does not learn which of the two preimages {*y*,*y*′}=*h* ^{−1}(*z*) is the one the sender can invert with respect to *f*. This paper reexamines the notion of interactive hashing, and proves the security of a variant of the Naor et al. protocol, which yields a more versatile interactive hashing theorem. When applying our new proof to (an equivalent variant of) the Naor et al. protocol, we get an alternative proof for this protocol that seems simpler and more intuitive than the original one, and achieves better parameters (in terms of how security preserving the reduction is).

## Key words

Cryptography Interactive hashing Statistically hiding and computationally binding commitments Statistical zero-knowledge arguments## Notes

### Acknowledgements

We are grateful to Moni Naor and Ronen Shaltiel for helpful conversations. We are also grateful to Oded Goldreich and the anonymous referees for their many useful comments and suggestions.

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