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 \(\mathcal{O}\left( \ln\left( q\right) \,n^{3\,r^{\prime2}}\right) \). For the challenge C [8], we obtain a theoretical bound of 266.3 operations.
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Faugère, JC., Levy-dit-Vehel, F., Perret, L. (2008). Cryptanalysis of MinRank. In: Wagner, D. (eds) Advances in Cryptology – CRYPTO 2008. CRYPTO 2008. Lecture Notes in Computer Science, vol 5157. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85174-5_16
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