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A Pairing-Free Signature Scheme from Correlation Intractable Hash Function and Strong Diffie-Hellman Assumption

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Topics in Cryptology – CT-RSA 2022 (CT-RSA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13161))

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Abstract

Goh and Jarecki (Eurocrypt 2003) showed how to get a signature scheme from the computational Diffie-Hellman assumption, and they introduced the name EDL for signatures of this type. The corresponding EDL family of signature schemes is remarkable for several reasons: elegance, simplicity and tight security. However, EDL security proofs stand in the random oracle model, and, to the best of our knowledge, extending this family without using an idealization of hash functions has never been successful.

In this paper, we propose a new signature scheme belonging to the EDL family, which is simple, natural and efficient, without using the random oracle model. Our scheme is based on the very same assumption than the Boneh-Boyen scheme, namely the strong Diffie-Hellman assumption, with the precision that our groups are not bound to being bilinear. We also make use of a correlation-intractable hash function, for a particular relation related to discrete-logarithm.

In addition to the theoretical interest of extending the EDL family without the random oracle model, our scheme is also one of the very few schemes which achieve discrete-log security properties without relying on pairings.

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Notes

  1. 1.

    We remark that other hash function idealizations or instantiations have also been investigated, e.g., in [34, 45].

  2. 2.

    Whether DL and CDH are equivalent is actually an open question.

  3. 3.

    The name EDL was proposed in [38], based upon the fact that the scheme is a proof of equality of discrete-logarithms.

  4. 4.

    A remarkable unification of [38, 51] papers appeared in [39].

  5. 5.

    Notably, the very same idea can be applied on Probabilistic RSA-FDH to get a tight signature scheme with a single extra bit.

  6. 6.

    In other words, in EDL, signing few times the same message would result in different random numbers r, while doing the same with Katz-Wang scheme would give always the same bit b.

  7. 7.

    Strictly, the equality is for Chevallier-Mames variant; for EDL or Katz-Wang, m is not an input of \(\mathcal {G}\), without changing anything to the analysis.

  8. 8.

    The reader is referred to original papers [18, 38, 39, 51] for more formal proofs.

  9. 9.

    Notably, \(\mathcal {F}\) and \(\mathcal {G}\) definitions cannot suppose the generator g to be already defined.

  10. 10.

    One may read [13, 21, 50] for state of the art on this area of research.

  11. 11.

    If \(\mathcal {F}\) and g were chosen by the solver, this latter could trivially pick any \((m, g, g^{a_1},g^{a_2},...,g^{a_{n}})\), precompute \(f = \mathcal {G}(m,g,g^{a_1},g^{a_2},...,g^{a_{n}})\), and choose \(\mathcal {F}(a_1, a_2,..., a_{n})\) as the constant function f.

  12. 12.

    Hence, our scheme does not ensure strong existential unforgeability, but only existential unforgeability, which is sufficient in most usages.

  13. 13.

    Remind that \(m \ne m_i\), since the forgery is assumed to be valid.

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Acknowledgements

The author would like to thank the anonymous referees and Marc Fischlin for their useful remarks, Pascal Paillier and Marc Joye for their careful reading of this paper and finally Jeremy Bradley-Silverio Donato for his edits. More personal thanks go to Amal and Mathieu for their continuous support.

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    Benoît Chevallier-Mames

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Chevallier-Mames, B. (2022). A Pairing-Free Signature Scheme from Correlation Intractable Hash Function and Strong Diffie-Hellman Assumption. In: Galbraith, S.D. (eds) Topics in Cryptology – CT-RSA 2022. CT-RSA 2022. Lecture Notes in Computer Science(), vol 13161. Springer, Cham. https://doi.org/10.1007/978-3-030-95312-6_2

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