Abstract
Fuchsbauer, Kiltz, and Loss (Crypto’18) gave a simple and clean definition of an algebraic group model (AGM) that lies in between the standard model and the generic group model (GGM). Specifically, an algebraic adversary is able to exploit group-specific structures as the standard model while the AGM successfully provides meaningful hardness results as the GGM. As an application of the AGM, they show a tight computational equivalence between the computing Diffie-Hellman (CDH) assumption and the discrete logarithm (DL) assumption. For the purpose, they used the square Diffie-Hellman assumption as a bridge, i.e., they first proved the equivalence between the DL assumption and the square Diffie-Hellman assumption, then used the known equivalence between the square Diffie-Hellman assumption and the CDH assumption. In this paper, we provide an alternative proof that directly shows the tight equivalence between the DL assumption and the CDH assumption. The crucial benefit of the direct reduction is that we can easily extend the approach to variants of the CDH assumption, e.g., the bilinear Diffie-Hellman assumption. Indeed, we show several tight computational equivalences and discuss applicabilities of our techniques.
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Notes
- 1.
To construct a reduction, we solve an equation modulo an order of \(\mathbb {G}\). Hence, if the order is composite, we do not know how to solve it in general. Hence, as [FKL18] we study only a prime-order group in this paper.
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Acknowledgement
We would like to thank anonymous reviewers of CT-RSA 2019 for their helpful comments and suggestions. This research was supported by JST CREST Grant Number JPMJCR14D6, Japan.
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Mizuide, T., Takayasu, A., Takagi, T. (2019). Tight Reductions for Diffie-Hellman Variants in the Algebraic Group Model. In: Matsui, M. (eds) Topics in Cryptology – CT-RSA 2019. CT-RSA 2019. Lecture Notes in Computer Science(), vol 11405. Springer, Cham. https://doi.org/10.1007/978-3-030-12612-4_9
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