International Association
for Cryptologic Research

In this paper we introduce a new attack on the multivariate encryption scheme HFERP, a big field scheme including an extra variable set, additional equations of the UOV or Rainbow shape as well as additional random polynomials. Our attack brings several parameter sets well below their claimed security levels. The attack combines novel methods applicable to multivariate schemes with multiple equation types with insights from the Simple Attack that broke Rainbow in early 2022, though interestingly the technique is applied in an orthogonal way. In addition to this attack, we apply support minors techniques on a MinRank instance drawing coefficients from the big field, which was effective against other multivariate big field schemes. This work demonstrates that there exist previously unknown impacts of the above works well beyond the scope in which they were derived.

DME is a multivariate scheme submitted to the call for additional signatures recently launched by NIST. Its performance is one of the best among all the candidates. The public key is constructed from the alternation of very structured linear and non-linear components that constitute the private key, the latter being defined over an extension field. We exploit these structures by proposing an algebraic attack which is practical on all DME parameters.

We present an efficient quantum algorithm for solving the semidirect discrete logarithm problem ($\SDLP$) in \emph{any} finite group. The believed hardness of the semidirect discrete logarithm problem underlies more than a decade of works constructing candidate post-quantum cryptographic algorithms from non-abelian groups. We use a series of reduction results to show that it suffices to consider $\SDLP$ in finite simple groups. We then apply the celebrated Classification of Finite Simple Groups to consider each family. The infinite families of finite simple groups admit, in a fairly general setting, linear algebraic attacks providing a reduction to the classical discrete logarithm problem. For the sporadic simple groups, we show that their inherent properties render them unsuitable for cryptographically hard $\SDLP$ instances, which we illustrate via a Baby-Step Giant-Step style attack against $\SDLP$ in the Monster Group. Our quantum $\SDLP$ algorithm is fully constructive, up to the computation of maximal normal subgroups, for all but three remaining cases that appear to be gaps in the literature on constructive recognition of groups; for these cases $\SDLP$ is no harder than finding a linear representation. We conclude that $\SDLP$ is not a suitable post-quantum hardness assumption for any choice of finite group.

In this paper, we show how to significantly improve algebraic techniques for solving the MinRank problem, which is ubiquitous in multivariate and rank metric code based cryptography. In the case of the structured MinRank instances arising in the latter, we build upon a recent breakthrough in Bardet et al. (EUROCRYPT 2020) showing that algebraic attacks outperform the combinatorial ones that were considered state of the art up until now. Through a slight modification of this approach, we completely avoid Gr\¨obner bases computations for certain parameters and are left only with solving linear systems. This does not only substantially improve the complexity, but also gives a convincing argument as to why algebraic techniques work in this case. When used against the second round NIST-PQC candidates ROLLO-I-128/192/256, our new attack has bit complexity respectively 71, 87, and 151, to be compared to 117, 144, and 197 as obtained in Bardet et al. (EUROCRYPT 2020). The linear systems arise from the nullity of the maximal minors of a certain matrix associated to the algebraic modeling. We also use a similar approach to improve the algebraic MinRank solvers for the usual MinRank problem. When applied against the second round NIST-PQC candidates GeMSS and Rainbow, our attack has a complexity that is very close to or even slightly better than those of the best known attacks so far. Note that these latter attacks did not rely on MinRank techniques since the MinRank approach used to give complexities that were far away from classical security levels.