### SNARGs and PPAD Hardness from the Decisional Diffie-Hellman Assumption

##### Abstract

We construct succinct non-interactive arguments (SNARGs) for bounded-depth computations assuming that the decisional Diffie-Hellman (DDH) problem is sub-exponentially hard. This is the first construction of such SNARGs from a Diffie-Hellman assumption. Our SNARG is also unambiguous: for every (true) statement $x$, it is computationally hard to find any accepting proof for $x$ other than the proof produced by the prescribed prover strategy. We obtain our result by showing how to instantiate the Fiat-Shamir heuristic, under DDH, for a variant of the Goldwasser-Kalai-Rothblum (GKR) interactive proof system. Our new technical contributions are (1) giving a $TC^0$ circuit family for finding roots of cubic polynomials over a special family of characteristic $2$ fields (Healy-Viola, STACS '06) and (2) constructing a variant of the GKR protocol whose invocations of the sumcheck protocol (Lund-Fortnow-Karloff-Nisan, STOC '90) only involve degree $3$ polynomials over said fields. Along the way, since we can instantiate Fiat-Shamir for certain variants of the sumcheck protocol, we also show the existence of (sub-exponentially) computationally hard problems in the complexity class $\mathsf{PPAD}$, assuming the sub-exponential hardness of DDH. Previous $\mathsf{PPAD}$ hardness results all required either bilinear maps or the learning with errors assumption.

Note: Fixed typos. Added funding information.

Available format(s)
Category
Foundations
Publication info
Preprint.
Keywords
Contact author(s)
yael @ microsoft com
alexlombardi @ alum mit edu
vinodv @ mit edu
History
2022-10-26: last of 2 revisions
See all versions
Short URL
https://ia.cr/2022/1409

CC BY-SA

BibTeX

@misc{cryptoeprint:2022/1409,
author = {Yael Tauman Kalai and Alex Lombardi and Vinod Vaikuntanathan},
title = {SNARGs and PPAD Hardness from the Decisional Diffie-Hellman Assumption},
howpublished = {Cryptology ePrint Archive, Paper 2022/1409},
year = {2022},
note = {\url{https://eprint.iacr.org/2022/1409}},
url = {https://eprint.iacr.org/2022/1409}
}

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