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Succinct Diophantine-Satisfiability Arguments

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.
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Contributor : Damien Vergnaud <>
Submitted on : Thursday, September 3, 2020 - 8:43:07 PM
Last modification on : Tuesday, March 23, 2021 - 9:28:03 AM



Patrick Towa, Damien Vergnaud. Succinct Diophantine-Satisfiability Arguments. Asiacrypt 2020 - 26th Annual International Conference on the Theory and Application of Cryptology and Information Security, Dec 2020, Daejeon / Virtual, South Korea. pp.774-804, ⟨10.1007/978-3-030-64840-4_26⟩. ⟨hal-02929841⟩



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