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International Conference on the Theory and Application of Cryptology and Information Security

ASIACRYPT 2019: Advances in Cryptology – ASIACRYPT 2019 pp 293–322Cite as

Hard Isogeny Problems over RSA Moduli and Groups with Infeasible Inversion

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11922))

Abstract

We initiate the study of computational problems on elliptic curve isogeny graphs defined over RSA moduli. We conjecture that several variants of the neighbor-search problem over these graphs are hard, and provide a comprehensive list of cryptanalytic attempts on these problems. Moreover, based on the hardness of these problems, we provide a construction of groups with infeasible inversion, where the underlying groups are the ideal class groups of imaginary quadratic orders.

Recall that in a group with infeasible inversion, computing the inverse of a group element is required to be hard, while performing the group operation is easy. Motivated by the potential cryptographic application of building a directed transitive signature scheme, the search for a group with infeasible inversion was initiated in the theses of Hohenberger and Molnar (2003). Later it was also shown to provide a broadcast encryption scheme by Irrer et al. (2004). However, to date the only case of a group with infeasible inversion is implied by the much stronger primitive of self-bilinear map constructed by Yamakawa et al. (2014) based on the hardness of factoring and indistinguishability obfuscation (iO). Our construction gives a candidate without using iO.

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Acknowledgments

The research of Salim Ali Altuğ is supported by the grant DMS-1702176. The research of Yilei Chen was conducted at Boston University supported by the NSF MACS project and NSF grant CNS-1422965.

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Authors and Affiliations

  1. Boston University, Boston, USA

    Salim Ali Altuğ

  2. Visa Research, Palo Alto, USA

    Yilei Chen

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  1. Salim Ali Altuğ
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Correspondence to Salim Ali Altuğ .

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Editors and Affiliations

  1. University of Auckland, Auckland, New Zealand

    Steven D. Galbraith

  2. Security Fundamentals Lab, NICT, Tokyo, Japan

    Shiho Moriai

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Altuğ, S.A., Chen, Y. (2019). Hard Isogeny Problems over RSA Moduli and Groups with Infeasible Inversion. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11922. Springer, Cham. https://doi.org/10.1007/978-3-030-34621-8_11

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  • DOI: https://doi.org/10.1007/978-3-030-34621-8_11

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  • Online ISBN: 978-3-030-34621-8

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