Paper 2020/682

Succinct Diophantine-Satisfiability Arguments

Patrick Towa and Damien Vergnaud


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.

Available format(s)
Cryptographic protocols
Publication info
A minor revision of an IACR publication in ASIACRYPT 2020
Zero KnowledgeDiophantine Equations
Contact author(s)
patrick towa @ gmail com
damien vergnaud @ lip6 fr
2020-09-29: last of 3 revisions
2020-06-09: received
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      author = {Patrick Towa and Damien Vergnaud},
      title = {Succinct Diophantine-Satisfiability Arguments},
      howpublished = {Cryptology ePrint Archive, Paper 2020/682},
      year = {2020},
      note = {\url{}},
      url = {}
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