International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: Succinct Diophantine-Satisfiability Arguments

Patrick Towa
Damien Vergnaud
DOI: 10.1007/978-3-030-64840-4_26
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Abstract: A Diophantine equation is a multi-variate polynomial equation with integer coefficients and it is satisfiable if it has a solution with all unknowns taking integer values. Davis, Putnam, Robinson and Matiyasevich showed that the general Diophantine satisfiability problem is undecidable (giving a negative answer to Hilbert's tenth problem) but it is nevertheless possible to argue in zero-knowledge the knowledge of a solution, if a solution is known to a prover. We provide the first succinct honest-verifier zero-knowledge argument for the satisfiability of Diophantine equations with a communication complexity and a round complexity that grows logarithmically in the size of the polynomial equation. The security of our argument relies on standard assumptions on hidden-order groups. As the argument requires to commit to integers, we introduce a new integer-commitment scheme that has much smaller parameters than Damgard and Fujisaki's scheme. We finally show how to succinctly argue knowledge of solutions to several NP-complete problems and cryptographic problems by encoding them as Diophantine equations.
Video from ASIACRYPT 2020
  title={Succinct Diophantine-Satisfiability Arguments},
  booktitle={Advances in Cryptology - ASIACRYPT 2020},
  author={Patrick Towa and Damien Vergnaud},