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An Estimator for the Hardness of the MQ Problem

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Progress in Cryptology - AFRICACRYPT 2022 (AFRICACRYPT 2022)

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Abstract

The Multivariate Quadratic (\(\textsf {MQ}\)) problem consists in finding the solutions of a given system of m quadratic equations in n unknowns over a finite field, and it is an NP-complete problem of fundamental importance in computer science. In particular, the security of some cryptosystems against the so-called algebraic attacks is usually given by the hardness of this problem. Many algorithms to solve the \(\textsf {MQ}\) problem have been proposed and studied. Estimating precisely the complexity of all these algorithms is crucial to set secure parameters for a cryptosystem. This work collects and presents the most important classical algorithms and the estimates of their computational complexities. Moreover, it describes a software that we wrote and that makes possible to estimate the hardness of a given instance of the \(\textsf {MQ}\) problem.

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Notes

  1. 1.

    Every row of the Macaulay matrix can be compute on the fly, but it will introduce an overhead in the time complexity.

  2. 2.

    The factor \(8k \log n\) comes from the expected complexity of the Valiant-Vazirani isolation’s algorithm with probability \(1 - 1/n\), see [16, Sec. 2.5] for more details.

  3. 3.

    For instance, for the Type IV parameters where \(n=66\), the authors used several Spartan-6 FPGAs to break the challenge. There the authors implemented the \(\textsf {FES}\) algorithm to solve a system with 48 variables and equations. Such particular implementation allowed them to test \(2^{10}\) potential solutions per clock cycle, which means they are computing at least \(2^{10}\) bit operations per clock cycle.

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Authors and Affiliations

  1. Cryptography Research Center, Technology Innovation Institute, Abu Dhabi, UAE

    Emanuele Bellini, Rusydi H. Makarim & Javier Verbel

  2. Department of Mathematical Sciences, Politecnico di Torino, Torino, Italy

    Carlo Sanna

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  1. Emanuele Bellini
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  1. Radboud University, Nijmegen, The Netherlands

    Lejla Batina

  2. Radboud University, Nijmegen, Gelderland, The Netherlands

    Joan Daemen

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Bellini, E., Makarim, R.H., Sanna, C., Verbel, J. (2022). An Estimator for the Hardness of the MQ Problem. In: Batina, L., Daemen, J. (eds) Progress in Cryptology - AFRICACRYPT 2022. AFRICACRYPT 2022. Lecture Notes in Computer Science, vol 13503. Springer, Cham. https://doi.org/10.1007/978-3-031-17433-9_14

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