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Cryptographic Divergences: New Techniques and New Applications

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Security and Cryptography for Networks (SCN 2020)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 12238))

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Abstract

In the recent years, some security proofs in cryptography have known significant improvements by replacing the statistical distance with alternative divergences. We continue this line of research, both at a theoretical and practical level. On the theory side, we propose a new cryptographic divergence with quirky properties. On the practical side, we propose new applications of alternative divergences: circuit-private FHE and prime number generators. More precisely, we provide the first formal security proof of the prime number generator PRIMEINC   [8], and improve by an order of magnitude the efficiency of a prime number generator by Fouque and Tibouchi  [16, 17] and the washing machine technique by Ducas and Stehlé  [15] for circuit-private FHE.

M. Abboud—Most of this work was done while Marc Abboud was an intern at PQShield.

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Notes

  1. 1.

    Typically, additional requirements are mandated, such as \((p + 1)\) and \((p - 1)\) having a large prime factor; but these can be added on top of the sampling procedure.

  2. 2.

    https://github.com/dlitz/pycrypto/blob/master/lib/Crypto/Util/number.py.

  3. 3.

    https://github.com/openssl/openssl/blob/master/crypto/bn/bn_prime.c.

  4. 4.

    The work of [7] requires no bootstrapping, but only applies to GSW-based schemes and is restricted to \(\text {NC}^1\).

  5. 5.

    Security-efficiency trade-offs have been presented in [8], and OpenSSL implements a variant of PRIMEINC.

  6. 6.

    This is true without loss of generality; even if more primes are generated and rejected if they fail some requirements (e.g. being safe primes), the adversary only has access to the product of exactly two outputs of the generator (p and q).

  7. 7.

    As stated in the preliminaries, this section will use Vinogradov’s notation, which is common in number theory: \((f \ll _s g) \Leftrightarrow (f =_s O(g))\).

  8. 8.

    One would find it odd that we are not using the proxy amplification property here but the computations we made showed that it wouldn’t give here a significantly better result than the amplification property for this application, so we chose not to complexify the computations done in the proof.

  9. 9.

    Alternatively, one can replace Q by \(\lambda Q\) in Theorem 4 and use Lemma 4; this results in a loss of O(1) bits of security and has a negligible effect on the parameters.

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Acknowledgements

The authors are indebted to Takahiro Matsuda and Shuichi Katsumata for their insightful discussions and for pointing out a flaw in an earlier version of the paper. Thomas Prest is supported by the Innovate UK Research Grant 104423 (PQ Cybersecurity).

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Authors and Affiliations

  1. École Normale Supérieure, Paris, France

    Marc Abboud

  2. PQShield, Oxford, UK

    Thomas Prest

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  1. Università degli Studi di Salerno, Fisciano, Italy

    Clemente Galdi

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Vladimir Kolesnikov

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Abboud, M., Prest, T. (2020). Cryptographic Divergences: New Techniques and New Applications. In: Galdi, C., Kolesnikov, V. (eds) Security and Cryptography for Networks. SCN 2020. Lecture Notes in Computer Science(), vol 12238. Springer, Cham. https://doi.org/10.1007/978-3-030-57990-6_24

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