GBLA – Gröbner Basis Linear Algebra Package

International audienceThis is a system paper about a new GPLv2 open source C libraryGBLA implementing and improving the idea [7] of Faugère andLachartre (GB reduction). We further exploit underlying structuresin matrices generated during Gröbner basis computations in algorithmslike F4 or F5 taking advantage of block patterns by using aspecial data structure called multilines. Moreover, we discuss a neworder of operations for the reduction process. In various differentexperimental results we show that GBLA performs better than GBreduction or Magma in sequential computations (up to 40% faster)and scales much better than GB reduction for a higher number ofcores: On 32 cores we reach a scaling of up to 26. GBLA is upto 7 times faster than GB reduction. Further, we compare differentparallel schedulers GBLA can be used with. We also developed anew advanced storage format that exploits the fact that our matricesare coming from Gröbner basis computations, shrinking storage bya factor of up to 4. A huge database of our matrices is freely availablewith GBLA

Sparse Gaussian Elimination modulo p: an Update

International audienceThis paper considers elimination algorithms for sparse matrices over finite fields. We mostly focus on computing the rank, because it raises the same challenges as solving linear systems, while being slightly simpler. We developed a new sparse elimination algorithm inspired by the Gilbert-Peierls sparse LU factorization, which is well-known in the numerical computation community. We benchmarked it against the usual right-looking sparse gaussian elimination and the Wiedemann algorithm using the Sparse Integer Matrix Collection of Jean-Guillaume Dumas. We obtain large speedups (1000× and more) on many cases. In particular , we are able to compute the rank of several large sparse matrices in seconds or minutes, compared to days with previous methods

How Much can F5 Really Do

Our purpose is to compare how much the F5 algorithm can gain in efficiency compared to the F4 algorithm. This can be achieve as the F5 algorithm uses the concept of signatures to foresee potential useless computation which the F4 algorithm might make represented by zero rows in the reduction of a large matrix. We experimentally show that this is a modest increase in efficiency for the parameters we tested

Cryptanalysis of The Lifted Unbalanced Oil Vinegar Signature Scheme

In 2017, Ward Beullens \textit{et al.} submitted Lifted Unbalanced Oil and Vinegar (LUOV)\cite{beullens2017field}, a signature scheme based on the famous multivariate public key cryptosystem (MPKC) called Unbalanced Oil and Vinegar (UOV), to NIST for the competition for post-quantum public key scheme standardization. The defining feature of LUOV is that, though the public key $\mathcal{P}$ works in the extension field of degree $r$ of $\mathbb{F}_2$, the coefficients of $\mathcal{P}$ come from $\mathbb{F}_2$. This is done to significantly reduce the size of $\mathcal{P}$. The LUOV scheme is now in the second round of the NIST PQC standardization process. In this paper we introduce a new attack on LUOV. It exploits the lifted structure of LUOV to reduce direct attacks on it to those over a subfield. We show that this reduces the complexity below the targeted security for the NIST post-quantum standardization competition

A Combinatorial Commutative Algebra Approach to Complete Decoding

Esta tesis pretende explorar el nexo de unión que existe entre la estructura algebraica de un código lineal y el proceso de descodificación completa. Sabemos que el proceso de descodificación completa para códigos lineales arbitrarios es NP-completo, incluso si se admite preprocesamiento de los datos. Nuestro objetivo es realizar un análisis algebraico del proceso de la descodificación, para ello asociamos diferentes estructuras matemáticas a ciertas familias de códigos. Desde el punto de vista computacional, nuestra descripción no proporciona un algoritmo eficiente pues nos enfrentamos a un problema de naturaleza NP. Sin embargo, proponemos algoritmos alternativos y nuevas técnicas que permiten relajar las condiciones del problema reduciendo los recursos de espacio y tiempo necesarios para manejar dicha estructura algebraica.Departamento de Algebra, Geometría y Topologí