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Title:
Succinct Diophantine-Satisfiability Arguments
Authors: Patrick Towa, 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 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.
ePrint: https://eprint.iacr.org/2020/682
Talk: https://www.youtube.com/watch?v=QB_UIsuM1MU
Slides: https://iacr.org/submit/files/slides/2020/asiacrypt/ac2020/387/slides.pdf
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