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### Revision(s):

Revision #1 to TR17-099 | 21st June 2017 03:52

#### Multi-Collision Resistance: A Paradigm for Keyless Hash Functions

Revision #1
Authors: Nir Bitansky, Omer Paneth, Yael Tauman Kalai
Accepted on: 21st June 2017 03:52
Keywords:

Abstract:

We study the notion of multi-collision resistance of hash functions --- a natural relaxation of collision-resistance that only guarantees the intractability of finding many (rather than two) inputs that map to the same image. An appealing feature of such hash functions is that unlike their collision-resistant counterparts, they do not necessarily require a key. Specifically, in the keyless setting, we only require that the size of collisions an adversarial algorithm can find is not much larger than its description size, or non-uniform advice.

We show how to replace collision resistance with multi-collision resistance in several foundational applications. Relying on such keyless functions, we improve on the best known round complexity for these applications. This includes:
\begin{itemize}
\item
3-message zero-knowledge arguments for NP.

\item
3-message succinct arguments of knowledge for NP.

\item
4-message $\varepsilon$-zero-knowledge proofs for NP.
\item
5-message public-coin zero-knowledge arguments for NP.
\end{itemize}

These results are obtained in the standard model of non-uniform adversaries of arbitrary polynomial size. Our techniques can also be applied in the keyed setting, at the cost of adding another message. In this case, we relax the known complexity assumptions for the last three applications, while still matching the state of the art in terms of round complexity.

The core technical contribution behind our results is a domain extension transformation from multi-collision-resistant hash functions for a fixed input length to ones with an arbitrary input length and a local opening property.

### Paper:

TR17-099 | 5th June 2017 18:14

#### Multi-Collision Resistance: A Paradigm for Keyless Hash Functions

TR17-099
Authors: Nir Bitansky, Omer Paneth, Yael Tauman Kalai
Publication: 5th June 2017 18:18
Keywords:

Abstract:

We study multi-collision-resistant hash functions --- a natural relaxation of collision-resistant hashing that only guarantees the intractability of finding many (rather than two) inputs that map to the same image. An appealing feature of such hash functions is that unlike their collision-resistant counterparts, they do not necessarily require a key. In the unkeyed setting we only require that the size of collisions an adversarial algorithm can find is not much larger than its description size, or non-uniform advice.

We show how to replace collision resistance with multi-collision resistance in several foundational applications. Relying on such unkeyed functions, we improve on the best known round complexity for these applications. This includes:

* 3-message zero-knowledge arguments for NP.
* 3-message succinct arguments of knowledge for NP.
* 4-message $\epsilon$-zero-knowledge proofs for NP.
* 5-message public-coin zero-knowledge arguments for NP.

These results are obtained in the standard model of non-uniform adversaries of arbitrary polynomial size. Our techniques can also be applied in the keyed setting, at the cost of adding another message. In this case, we weaken the known complexity assumptions for the last three applications, while still matching the state of the art in terms of round complexity.

The core technical contribution behind our results is a domain extension transformation from multi-collision-resistant hash functions for a fixed input length to ones with an arbitrary input length and a local opening property.

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