Cryptology ePrint Archive: Report 2012/029

On the Exact Security of Schnorr-Type Signatures in the Random Oracle Model

Yannick Seurin

Abstract: The Schnorr signature scheme has been known to be provably secure in the Random Oracle Model under the Discrete Logarithm (DL) assumption since the work of Pointcheval and Stern (EUROCRYPT '96), at the price of a very loose reduction though: if there is a forger making at most $q_h$ random oracle queries, and forging signatures with probability $\varepsilon_F$, then the Forking Lemma tells that one can compute discrete logarithms with constant probability by rewinding the forger $\mathcal{O}(q_h/\varepsilon_F)$ times. In other words, the security reduction loses a factor $\mathcal{O}(q_h)$ in its time-to-success ratio. This is rather unsatisfactory since $q_h$ may be quite large. Yet Paillier and Vergnaud (ASIACRYPT 2005) later showed that under the One More Discrete Logarithm (OMDL) assumption, any \emph{algebraic} reduction must lose a factor at least $q_h^{1/2}$ in its time-to-success ratio. This was later improved by Garg~\emph{et al.} (CRYPTO 2008) to a factor $q_h^{2/3}$. Up to now, the gap between $q_h^{2/3}$ and $q_h$ remained open. In this paper, we show that the security proof using the Forking Lemma is essentially the best possible. Namely, under the OMDL assumption, any algebraic reduction must lose a factor $f(\varepsilon_F)q_h$ in its time-to-success ratio, where $f\le 1$ is a function that remains close to 1 as long as $\varepsilon_F$ is noticeably smaller than 1. Using a formulation in terms of expected-time and queries algorithms, we obtain an optimal loss factor $\Omega(q_h)$, independently of $\varepsilon_F$. These results apply to other signature schemes based on one-way group homomorphisms, such as the Guillou-Quisquater signature scheme.

Category / Keywords: public-key cryptography / Schnorr signatures, discrete logarithm, Forking Lemma, Random Oracle Model, meta-reduction, one-way group homomorphism

Publication Info: Abridged version at EUROCRYPT 2012

Date: received 20 Jan 2012

Contact author: yannick seurin at m4x org

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Version: 20120122:043204 (All versions of this report)

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