Cryptanalysis of MinRank

Abstract : In this paper, we investigate the difficulty of one of the most relevant problems in multivariate cryptography - namely MinRank - about which no real progress has been reported since [9, 19]. Our starting point is the Kipnis-Shamir attack [19]. We first show new properties of the ideal generated by Kipnis-Shamir's equations. We then propose a new modeling of the problem. Concerning the practical resolution, we adopt a Gröbner basis approach that permitted us to actually solve challenges A and B proposed by Courtois in [8]. Using the multi-homogeneous structure of the algebraic system, we have been able to provide a theoretical complexity bound reflecting the practical behavior of our approach. Namely, when r ′ the dimension of the matrices minus the rank of the target matrix in the MinRank problem is constant, then we have a polynomial time attack O(ln(q)n3r′2) . For the challenge C [8], we obtain a theoretical bound of 266.3 operations.
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Submitted on : Wednesday, April 9, 2014 - 5:41:17 PM
Last modification on : Wednesday, July 3, 2019 - 10:48:04 AM

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Jean-Charles Faugère, Françoise Levy-Dit-Vehel, Ludovic Perret. Cryptanalysis of MinRank. CRYPTO 2008 - 28th Annual International Cryptology Conference, Aug 2008, Santa Barbara, CA, United States. pp.280-296, ⟨10.1007/978-3-540-85174-5_16⟩. ⟨hal-00976374⟩

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