Cryptology ePrint Archive: Report 2020/682

Succinct Diophantine-Satisfiability Arguments

Patrick Towa and Damien Vergnaud

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 Hilbertís 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 Fujisakiís 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.

Category / Keywords: cryptographic protocols / Zero Knowledge, Diophantine Equations

Date: received 8 Jun 2020, last revised 24 Jun 2020

Contact author: patrick towa at gmail com,damien vergnaud@lip6 fr

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Version: 20200624:161300 (All versions of this report)

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