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Algebraic Algorithm for the Alternating Trilinear Form Equivalence Problem

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Code-Based Cryptography (CBCrypto 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14311))

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Abstract

The Alternating Trilinear Form Equivalence (\(\textsf{ATFE}\)) problem was recently used as a hardness assumption in the design of a digital signature scheme by Tang et al. using the Fiat-Shamir paradigm. It is a hard equivalence problem known to be in the class of equivalence problems that includes, for instance, the Tensor Isomorphism (\(\textsf{TI}\)), Quadratic Maps Linear Equivalence (\(\textsf{QMLE}\)) and the Matrix Code Equivalence (\(\textsf{MCE}\)) problems. Due to the increased cryptographic interest, the understanding of its practical hardness has also increased in the last couple of years. Currently, there are several combinatorial and algebraic algorithms for solving it, the best of which is a graph-theoretic algorithm that also includes an algebraic subroutine.

In this paper, we take a purely algebraic approach to the \(\textsf{ATFE}\) problem, but we use a coding theory perspective to model the problem. This modelling was introduced earlier for the \(\textsf{MCE}\) problem. Using it, we improve the cost of an algebraic attack against \(\textsf{ATFE}\) compared to previously known ones.

Taking into account the algebraic structure of alternating trilinear forms, we show that the obtained system has less variables but also less equations than for \(\textsf{MCE}\) and gives rise to structural degree-3 syzygies. Under the assumption that outside of these syzygies the system behaves semi-regularly, we provide a concrete, non-asymptotic complexity estimate of the performance of our algebraic attack. Our results show that the complexity is below the estimated security levels of the signature scheme of Tang et al. and comparable to the currently best graph-theoretic attack by Beullens.

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Notes

  1. 1.

    In [BFV13] it was conjectured that this complexity is \(\mathcal {O}(n^9)\) i.e. polynomial. Later in [Bou11, RST22] this was reevaluated and shown that the conclusion was made based on some false assumptions. Nevertheless, even though there is no proof of the polynomial behavior of this step, in practice it does finish in an expected polynomial time.

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Acknowledgements

This research is supported by the NWO grant OCNW.M.21.193 (ALPaQCa) and the ERC Starting Grant 805031 (EPOQUE).

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Authors and Affiliations

  1. Radboud Universiteit, Nijmegen, The Netherlands

    Lars Ran, Simona Samardjiska & Monika Trimoska

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  1. Lars Ran
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Correspondence to Lars Ran .

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Editors and Affiliations

  1. Technology Innovation Institute, Abu Dhabi, United Arab Emirates

    Andre Esser

  2. Marche Polytechnic University, Ancona, Italy

    Paolo Santini

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Ran, L., Samardjiska, S., Trimoska, M. (2023). Algebraic Algorithm for the Alternating Trilinear Form Equivalence Problem. In: Esser, A., Santini, P. (eds) Code-Based Cryptography. CBCrypto 2023. Lecture Notes in Computer Science, vol 14311. Springer, Cham. https://doi.org/10.1007/978-3-031-46495-9_5

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  • DOI: https://doi.org/10.1007/978-3-031-46495-9_5

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