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Signatures with Tight Multi-user Security from Search Assumptions

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Computer Security – ESORICS 2020 (ESORICS 2020)

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Abstract

We construct two tightly secure signature schemes based on the computational Diffie-Hellman (CDH) and factoring assumptions in the random oracle model. Our schemes are proven secure in the multi-user setting, and their security loss is constant and does not depend on the number of users or signing queries. They are the first schemes that achieve this based on standard search assumptions, as all existing schemes we are aware of are either based on stronger decisional assumptions, or proven tightly secure in the less realistic single-user setting. Under a concrete estimation, in a truly large scale, the cost of our CDH-based scheme is about half of Schnorr and DSA (in terms of signature size and running time for signing).

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Notes

  1. 1.

    Usually, “\(k\)-bit security” means that there is no adversary can break the scheme with success ratio larger than \(2^{-k}\) (see discussions in [7, 17]).

  2. 2.

    Nowadays many applications involve billions of users. For instance, Facebook has about 2 billion active users daily, according to https://about.fb.com/company-info/.

  3. 3.

    Taken from the benchmarks in https://github.com/miracl/MIRACL/blob/master/docs/miracl-explained/benchmarks.md (2020-03-26).

  4. 4.

    \(\mathsf {SIG[ID]} \)is the signature scheme constructed from a three-move identification scheme \(\textsf {ID}\) via the Fiat-Shamir transformation.

  5. 5.

    We use the result derived in the reduction, not the statement of the lemma, as they are not the same.

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Authors and Affiliations

  1. Department of Mathematical Sciences, NTNU – Norwegian University of Science and Technology, Trondheim, Norway

    Jiaxin Pan & Magnus Ringerud

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  1. Jiaxin Pan
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  2. Magnus Ringerud
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Editors and Affiliations

  1. University of Surrey, Guildford, UK

    Liqun Chen

  2. Purdue University, West Lafayette, IN, USA

    Ninghui Li

  3. Delft University of Technology, Delft, The Netherlands

    Kaitai Liang

  4. University of Surrey, Guildford, UK

    Steve Schneider

A On the Multi-user Security of DSA

A On the Multi-user Security of DSA

We show why it is difficult to show tight implication from the single-user security to the multi-user security for \(\mathsf {DSA}\). We first recall the scheme. Let p be an L-bit prime, and q be an N-bit prime such that \(q\mid (p-1)\). For specifications on L and N, see the \(\mathsf {DSA}\) documentation [36]. Let g be a generator of a subgroup of order q in \(\mathbb {Z}_p^*\). The \({\mathsf {Gen}},{\mathsf {Sign}}\) and \({\mathsf {Ver}}\) can then be described as follows.

figure g

Different to the \(\mathsf {Schnorr}\) signature, given a valid signature \(\sigma :=(R,s)\) under public key X, it is not possible to convert it to a valid signature under public key \(X\cdot g^{a_i}\) for using methods in [12, 38], since we do not have the discrete log of R, namely, \(r\in \mathbb {Z}_q^*\).

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Pan, J., Ringerud, M. (2020). Signatures with Tight Multi-user Security from Search Assumptions. In: Chen, L., Li, N., Liang, K., Schneider, S. (eds) Computer Security – ESORICS 2020. ESORICS 2020. Lecture Notes in Computer Science(), vol 12309. Springer, Cham. https://doi.org/10.1007/978-3-030-59013-0_24

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