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On the complexity of the generalized MinRank problem

Jean-Charles Faugère, Mohab Safey El Din, Pierre-Jean Spaenlehauer
2013 Journal of symbolic computation  
We study the complexity of solving the generalized MinRank problem, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most r.  ...  We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem.  ...  Acknowledgments This work was supported in part by the HPAC grant and the GeoLMI grant (ANR 2011 BS03 011 06) of the French National Research Agency.  ... 
doi:10.1016/j.jsc.2013.03.004 fatcat:neopu3qjx5eb5pqxreyuebbmam

On the Complexity of the Generalized MinRank Problem [article]

Jean-Charles Faugère and Mohab Safey El Din and Pierre-Jean Spaenlehauer
2013 arXiv   pre-print
We study the complexity of solving the generalized MinRank problem, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most r.  ...  We show that its complexity can be bounded by using the complexity bounds for the generalized MinRank problem.  ...  Acknowledgments This work was supported in part by the HPAC grant and the GeoLMI grant (ANR 2011 BS03 011 06) of the French National Research Agency.  ... 
arXiv:1112.4411v2 fatcat:n7bs3yofpzanfjlqvnuswghevu

The complexity of MinRank [article]

Alessio Caminata, Elisa Gorla
2022 arXiv   pre-print
In this note, we leverage some of our results from arXiv:1706.06319 to produce a concise and rigorous proof for the complexity of the generalized MinRank Problem in the under-defined and well-defined case  ...  Acknowledgements We are grateful to an anonymous referee for a detailed reading and comments which helped us improve the clarity of the proof of the main theorem.  ...  In this paper, we take another look at the complexity of solving the generalized MinRank Problem with the minors modeling.  ... 
arXiv:1905.02682v3 fatcat:u4f2zpwbyffv5lz62bp7jqn6rm

Computing loci of rank defects of linear matrices using Gröbner bases and applications to cryptology

Jean-Charles Faugère, Mohab Safey El Din, Pierre-Jean Spaenlehauer
2010 Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation - ISSAC '10  
In both cases, under genericity assumptions on the entries of the considered matrix, we give new bounds on the degree of regularity of the considered ideal which allows us to estimate the complexity of  ...  For instance, the exact degree of regularity of the determinantal ideal formulation of a generic well-defined MinRank problem is r(n − r) + 1.  ...  When k = 1 the MinRank problem can be reduced to the well known EigenValue problem. Therefore, the MinRank problem can be seen as a generalized nonlinear eigenvalue problem.  ... 
doi:10.1145/1837934.1837984 dblp:conf/issac/FaugereDS10 fatcat:44jtyqxqyrbrnai5oadnffoqw4

Efficient Zero-Knowledge Authentication Based on a Linear Algebra Problem MinRank [chapter]

Nicolas T. Courtois
2001 Lecture Notes in Computer Science  
MinRank is one of the most efficient schemes based on NP-complete problems. It can be used to prove in Zero-knowledge a solution to any problem described by multivariate equations.  ...  We study a more general problem called MinRank that generalizes SD and contains also other well known hard problems.  ...  Related Problems This version of the MinRank, is a generalized version of one among many NPcomplete rank problems studied in [23] and [10] . In our scheme R will be a finite field GF (q).  ... 
doi:10.1007/3-540-45682-1_24 fatcat:koxjsf7fznfh7lwfntwr2xbmt4

HybridRAM: The first quantum approach for key recovery attacks on Rainbow [article]

Seong-Min Cho, Aeyoung Kim, Seung-Hyun Seo
2022 arXiv   pre-print
We also design a Grover oracle quantum circuit suitable for the rectangular MinRank attack and then propose a Hybrid Rectangular MinRank attack that recovers the keys of Rainbow using the designed quantum  ...  A rectangular MinRank attack, proposed by Ward Beullens in 2021, reduced the security of Rainbow below the security requirements set out by NIST.  ...  reduced the complexity of the MinRank attacks on the Rainbow to 1/3 of existing attacks in the parameter sets III and V of Rainbow [8] .  ... 
arXiv:2206.10898v1 fatcat:emtqi2p7dvgrbervdp72jb3w6i

Improvements of Algebraic Attacks for solving the Rank Decoding and MinRank problems [article]

Magali Bardet, Maxime Bros, Daniel Cabarcas, Philippe Gaborit, Ray Perlner, Daniel Smith-Tone, Jean-Pierre Tillich, Javier Verbel
2020 arXiv   pre-print
Based on this problem and quasi-cyclic versions of it, very efficient schemes have been proposed recently, such as those in the ROLLO and RQC submissions, which have reached the second round of the NIST  ...  For the other case, called underdetermined, we also improve the results from the previous attack by combining the Ourivski-Johansson modeling together with a new modeling for a generic MinRank instance  ...  We thank the Facultad de Ciencias of the Universidad Nacional de Colombia sede Medelln for granting us access to the Enlace server, where we ran some of the experiments.  ... 
arXiv:2002.08322v3 fatcat:qt3zmk5imjc6neytawezi7j6vq

Analysis on the MinRank Attack using Kipnis-Shamir Method Against Rainbow [article]

Shuhei Nakamura, Yacheng Wang, Yasuhiko Ikematsu
2020 IACR Cryptology ePrint Archive  
As this method reduces the problem to the MQ problem that asks for a solution of a system of quadratic equations, its complexity depends on the solving degree of a quadratic system deduced from the method  ...  Moreover, the termination of the y-XL algorithm on an inhomogeneous system is not clear, and its complexity depends on the existence of a non-trivial syzygy on each system and its discussion is delicate  ... 
dblp:journals/iacr/NakamuraWI20 fatcat:vvwduwchajbczi4bsnhnbvmpxq

Improvement of algebraic attacks for solving superdetermined MinRank instances [article]

Magali Bardet, Manon Bertin
2022 arXiv   pre-print
We give complexity estimates of the attack for generic random instances.We apply those results to the DAGS cryptosystem, that was submitted to the first round of the NIST standardization process.  ...  The MinRank (MR) problem is a computational problem that arises in many cryptographic applications. In Verbel et al.  ...  Introduction The MinRank Problem The MinRank problem was first mentioned in [12] where its NP-completeness was also proven.  ... 
arXiv:2208.01442v1 fatcat:hdgnajo5wnbs5pkzxwaqq4u56m

The computational complexity of some problems of linear algebra [chapter]

Jonathan F. Buss, Gudmund S. Frandsen, Jeffrey O. Shallit
1997 Lecture Notes in Computer Science  
We consider the computational complexity of some problems dealing with matrix rank. Let E, S be subsets of a commutative ring R. a 2 , . . .a t ) and minrank S (M ) = min (a 1 ,a 2 ,...  ...  Depending on E, S, and on which variant is studied, the complexity of these problems can range from polynomial-time solvable to random polynomial-time solvable to NP-complete to PSPACE-solvable to unsolvable  ...  We have good evidence that the MINRANK and SING problems do not in general have the same complexity.  ... 
doi:10.1007/bfb0023480 fatcat:c2znq7yvbzfkninppzss3iffuy

The Computational Complexity of Some Problems of Linear Algebra

Jonathan F. Buss, Gudmund Skovbjerg Frandsen, Jeffery O. Shallit
1996 BRICS Report Series  
<br />Depending on E, S, and on which variant is studied, the complexity<br />of these problems can range from polynomial-time solvable to random<br />polynomial-time solvable to NP-complete to PSPACE-solvable  ...  We consider the computational complexity of some problems dealing with matrix rank.<br /> Let E, S be subsets of a commutative ring R.<br />Let x1, x2, ..., xt be variables.  ...  We have good evidence that the MINRANK and SING problems do not in general have the same complexity.  ... 
doi:10.7146/brics.v3i33.20013 fatcat:mdqhjtrvf5fblhnjleitq7mspu

The Computational Complexity of Some Problems of Linear Algebra

Jonathan F Buss, Gudmund S Frandsen, Jeffrey O Shallit
1999 Journal of computer and system sciences (Print)  
We consider the computational complexity of some problems dealing with matrix rank. Let E, S be subsets of a commutative ring R. a 2 , . . .a t ) and minrank S (M ) = min (a 1 ,a 2 ,...  ...  Depending on E, S, and on which variant is studied, the complexity of these problems can range from polynomial-time solvable to random polynomial-time solvable to NP-complete to PSPACE-solvable to unsolvable  ...  We have good evidence that the MINRANK and SING problems do not in general have the same complexity.  ... 
doi:10.1006/jcss.1998.1608 fatcat:4omziu6rcnfbvgbakda7dbwqam

The Complexity of Tensor Rank [article]

Marcus Schaefer, Daniel Stefankovic
2016 arXiv   pre-print
We show that determining the rank of a tensor over a field has the same complexity as deciding the existential theory of that field. This implies earlier NP-hardness results by Håstad H90.  ...  The hardness proof also implies an algebraic universality result.  ...  Over the complex numbers, Koiran's result places the problem at the second level of the polynomial hierarchy assuming the Generalized Riemann hypothesis is true.  ... 
arXiv:1612.04338v1 fatcat:hswlqjlb6zdutiao2hyh6mkfcy

Improved Cryptanalysis of UOV and Rainbow [article]

Ward Beullens
2020 IACR Cryptology ePrint Archive  
Secondly, we give two new attacks against the UOV and Rainbow signature schemes; the intersection attack that applies to both UOV and Rainbow and the rectangular MinRank attack that applies only to Rainbow  ...  The contributions of this paper are twofold.  ...  to the solutions of the MinRank problem.  ... 
dblp:journals/iacr/Beullens20a fatcat:fc65bfxhzrbj3aylxb7zpaqxaq

Breaking Rainbow Takes a Weekend on a Laptop [article]

Ward Beullens
2022 IACR Cryptology ePrint Archive  
Concretely, given a Rainbow public key for the SL 1 parameters of the second-round submission, our attack returns the corresponding secret key after on average 53 hours (one weekend) of computation time  ...  This work introduces new key recovery attacks against the Rainbow signature scheme, which is one of the three finalist signature schemes still in the NIST Post-Quantum Cryptography standardization project  ...  Solving MinRank problems. Our attacks will also make use of an algorithm to solve the MinRank problem.  ... 
dblp:journals/iacr/Beullens22 fatcat:vqicd74vifctvca4zmcqftzrsa
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