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Password-Authenticated Key Exchange from Group Actions

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Advances in Cryptology – CRYPTO 2022 (CRYPTO 2022)

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Abstract

We present two provably secure password-authenticated key exchange (PAKE) protocols based on a commutative group action. To date the most important instantiation of isogeny-based group actions is given by CSIDH. To model the properties more accurately, we extend the framework of cryptographic group actions (Alamati et al., ASIACRYPT 2020) by the ability of computing the quadratic twist of an elliptic curve. This property is always present in the CSIDH setting and turns out to be crucial in the security analysis of our PAKE protocols.

Despite the resemblance, the translation of Diffie-Hellman based PAKE protocols to group actions either does not work with known techniques or is insecure (“How not to create an isogeny-based PAKE”, Azarderakhsh et al., ACNS 2020). We overcome the difficulties mentioned in previous work by using a “bit-by-bit” approach, where each password bit is considered separately.

Our first protocol \(\textsf{X}\text {-}\textsf{GA}\text {-}\textsf{PAKE}_\ell \) can be executed in a single round. Both parties need to send two set elements for each password bit in order to prevent offline dictionary attacks. The second protocol \(\mathsf {Com\text {-}GA\text {-}PAKE}_\ell \) requires only one set element per password bit, but one party has to send a commitment on its message first. We also discuss different optimizations that can be used to reduce the computational cost. We provide comprehensive security proofs for our base protocols and deduce security for the optimized versions.

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Notes

  1. 1.

    The original (three-round) version of this protocol was later found to have a (fixable) bug, cf. https://iacr.org/submit/files/slides/2021/eurocrypt/eurocrypt2021/20/slides.pdf.

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Acknowledgments

Thorsten Eisenhofer, Eike Kiltz, Sabrina Kunzweiler and Doreen Riepel were supported by the DFG under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972.

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Authors and Affiliations

  1. DFINITY, Zürich, Switzerland

    Michel Abdalla

  2. DIENS, École normale supérieure, CNRS, PSL University, Paris, France

    Michel Abdalla

  3. Ruhr-Universität Bochum, Bochum, Germany

    Thorsten Eisenhofer, Eike Kiltz, Sabrina Kunzweiler & Doreen Riepel

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  1. Michel Abdalla
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Correspondence to Doreen Riepel .

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Editors and Affiliations

  1. New York University, New York, NY, USA

    Yevgeniy Dodis

  2. University of Florida, Gainesville, FL, USA

    Thomas Shrimpton

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Abdalla, M., Eisenhofer, T., Kiltz, E., Kunzweiler, S., Riepel, D. (2022). Password-Authenticated Key Exchange from Group Actions. In: Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022. CRYPTO 2022. Lecture Notes in Computer Science, vol 13508. Springer, Cham. https://doi.org/10.1007/978-3-031-15979-4_24

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