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Elementary Attestation of Cryptographically Useful Composite Moduli

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Innovative Security Solutions for Information Technology and Communications (SecITC 2020)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12596))

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Abstract

This paper describes a non-interactive process allowing a prover to convince a verifier that a modulus n is the product of two primes (p, q) of about the same size. A further heuristic argument conjectures that \(p-1\) and \(q-1\) have sufficiently large prime factors for cryptographic applications.

The new protocol relies upon elementary number-theoretic properties and can be implemented efficiently using very few operations. This contrasts with state-of-the-art zero-knowledge protocols for RSA modulus proper generation assessment.

The heuristic argument at the end of our construction calls for further cryptanalysis by the community and is, as such, an interesting research question in its own right.

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Notes

  1. 1.

    This is the case if and only if \(\gcd (g-1, n) = \gcd (g+1, n) = 1\), which happens with high probability.

  2. 2.

    In particular, the Golomb–Dickman constant \(\lambda \approx 0.624\) asymptotically governs the relative size of the largest prime factor of an integer [KP76, Dic30, Gol64].

  3. 3.

    e.g. the binary digits of \(\pi =3.14159265\ldots \).

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Acknowledgements

The authors are grateful to Arjen Lenstra for his pertinent remarks on an earlier version of this article.

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Authors and Affiliations

  1. QPSI, Qualcomm Technologies Incorporated, San Diego, USA

    Rémi Géraud-Stewart

  2. ÉNS (DI), Information Security Group, CNRS, PSL Research University, 75005, Paris, France

    Rémi Géraud-Stewart & David Naccache

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  1. Rémi Géraud-Stewart
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Correspondence to Rémi Géraud-Stewart .

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Editors and Affiliations

  1. Advanced Technologies Institute, Bucharest, Romania

    Diana Maimut

  2. Advanced Technologies Institute, Bucharest, Romania

    Andrei-George Oprina

  3. Faculté des Sciences et Techniques, University of Limoges, Limoges, France

    Damien Sauveron

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Géraud-Stewart, R., Naccache, D. (2021). Elementary Attestation of Cryptographically Useful Composite Moduli. In: Maimut, D., Oprina, AG., Sauveron, D. (eds) Innovative Security Solutions for Information Technology and Communications. SecITC 2020. Lecture Notes in Computer Science(), vol 12596. Springer, Cham. https://doi.org/10.1007/978-3-030-69255-1_1

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  • DOI: https://doi.org/10.1007/978-3-030-69255-1_1

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