Abstract In this note, we provide proven estimates for the complexity of the SupportMinors Modeling, mostly confirming the heuristic complexity estimates contained in the original article (Bardet (2020) Improvements of algebraic attacks for solving the rank decoding and MinRank problems. In: Advances in cryptology. Springer, pp 507–536).

International audience

Digital archives rely on trusted parties, such as certification authorities, to ensure authenticity, integrity and proof of existence protection for documents. In this paper, we analyse the trust assumptions that a verifier has to make in order to trust in the protection of a document. We show that trust fades out in the long term due to the ever-growing number of trusted parties. Despite such a dire prospect, current technologies such as X.509 PKI do not assess trust, thereby leaving verifiers in the dark. We present a certification scheme for documents that provides verifiers with a better assessment of trust than in X.509 PKI. In the proposed scheme, trusted parties are rated based on the correctness of their performance. From the ratings, verifiers can assess quantitatively the trust in the trusted parties for the short term, and in the protection of documents for the long term. The proposed scheme encourages trusted parties to work properly.

In this paper, we introduce a new method to avoid zero reductions in Gröbner basis computation. We call this method LASyz, which stands for Lineal Algebra to compute Syzygies. LASyz uses exhaustively the information of both principal syzygies and non-trivial syzygies to avoid zero reductions. All computation is done using linear algebra techniques. LASyz is easy to understand and implement. The method does not require to compute Gröbner bases of subsequences of generators incrementally and it imposes no restrictions on the reductions allowed. We provide a complete theoretical foundation for the LASyz method and we describe an algorithm to compute Gröbner bases for zero dimensional ideals based on this foundation. A qualitative comparison with similar algorithms is provided and the performance of the algorithm is illustrated with experimental data.