The Rank Decoding problem (RD) is at the core of rank-based cryptography. This problem can also be seen as a structured version of MinRank, which is ubiquitous in multivariate cryptography. Recently, \cite{BBBGNRT20,BBCGPSTV20} proposed attacks based on two new algebraic modelings, namely the MaxMinors modeling which is specific to RD and the Support-Minors modeling which applies to MinRank in general. Both improved significantly the complexity of algebraic attacks on these two problems. In the case of RD and contrarily to what was believed up to now, these new attacks were shown to be able to outperform combinatorial attacks and this even for very small field sizes. However, we prove here that the analysis performed in \cite{BBCGPSTV20} for one of these attacks which consists in mixing the MaxMinors modeling with the Support-Minors modeling to solve RD is too optimistic and leads to underestimate the overall complexity. This is done by exhibiting linear dependencies between these equations and by considering an $\fqm$ version of these modelings which turns out to be instrumental for getting a better understanding of both systems. Moreover, by working over $\Fqm$ rather than over $\ff{q}$, we are able to drastically reduce the number of variables in the system and we (i) still keep enough algebraic equations to be able to solve the system, (ii) are able to analyze rigorously the complexity of our approach. This new approach may improve the older MaxMinors approach on RD from \cite{BBBGNRT20,BBCGPSTV20} for certain parameters. We also introduce a new hybrid approach on the Support-Minors system whose impact is much more general since it applies to any MinRank problem. This technique improves significantly the complexity of the Support-Minors approach for small to moderate field sizes.

翻译：秩译码问题是基于秩的密码学的核心，也可视为多变量密码学中普遍存在的MinRank问题的结构化版本。近期[BBBGNRT20,BBCGPSTV20]提出了基于两种新代数建模的攻击方法：专用于秩译码问题的MaxMinors建模，以及适用于一般MinRank问题的Support-Minors建模。这两种方法显著提升了针对上述问题的代数攻击复杂度。与以往认知相反，在秩译码问题情境下，这些新攻击被证明能够超越组合攻击，即便在字段尺寸极小的情况下亦如此。然而，我们证明[BBCGPSTV20]中对将MaxMinors建模与Support-Minors建模混合用于求解秩译码问题的攻击分析过于乐观，导致整体复杂度被低估。通过揭示这些方程之间的线性依赖关系，并引入能够更好理解两个系统的$\fqm$版本建模，我们验证了此结论。此外，在$\Fqm$而非$\ff{q}$上操作时，我们能够大幅减少系统中的变量数量，同时(i)保留足够多的代数方程以求解系统，(ii)严谨分析所提方法的复杂度。对于某些参数，该新方法可能改进[BBBGNRT20,BBCGPSTV20]中基于MaxMinors的秩译码旧方法。我们还提出了一种针对Support-Minors系统的新型混合方法，其影响更为广泛——可应用于任意MinRank问题。该技术显著提升了针对中小规模字段尺寸的Support-Minors方法的复杂度。