Powers of tensors and fast matrix multiplication

Gall, François Le

doi:10.1145/2608628.2608664

Cited by 767 publications

(624 citation statements)

References 21 publications

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“…We let ω be a feasible exponent for linear algebra, in the sense that matrices of size n can be multiplied in O(n ω ) base ring operations over any ring; the best bound to date is ω < 2.38 [16,31]. We will have to compute rank and rank pro le of dense matrices; the cost reduces to that of matrix multiplication [25].…”

Section: Basic Resultsmentioning

confidence: 99%

Algorithms for Structured Linear Systems Solving and Their Implementation

Hyun

Lebreton

Schost

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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ABSTRACTere exists a vast literature dedicated to algorithms for structured matrices, but relatively few descriptions of actual implementations and their practical performance. In this paper, we consider the problem of solving Cauchy-like systems, and its application to mosaic Toeplitz systems, in two contexts: rst in the unit cost model (which is a good model for computations over nite elds), then over Q. We introduce new variants of previous algorithms and describe an implementation of these techniques and its practical behavior. We pay a special a ention to particular cases such as the computation of algebraic approximants.

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Section: Basic Resultsmentioning

confidence: 99%

Algorithms for Structured Linear Systems Solving and Their Implementation

Hyun

Lebreton

Schost

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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“…The best known value for ω is ω < 2.38 [32]; in the often discussed case where p is constant, the cost is then O(n 4.76p ). For the CDT problem, we have p = 3, so that generic instances of it can be solved using O(n 14.28 ) arithmetic operations.…”

Section: Minimization Problemsmentioning

confidence: 99%

Bit complexity for multi-homogeneous polynomial system solving—Application to polynomial minimization

Din

Schost

2018

Journal of Symbolic Computation

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Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input system, under some genericity assumptions. The assumptions essentially imply that the Jacobian matrix of the system under study has maximal rank at the solution set and that this solution set is finite. The algorithm is probabilistic and a probability analysis is provided.Next, we apply these results to the problem of optimizing a linear map on the real trace of an algebraic set. Under some genericity assumptions, we provide bit complexity estimates for solving this polynomial minimization problem.

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“…For a ⊕-Feistel, the cost of solving the linear system is 2 3n with Gaussian elimination, but can be improved to O(2 2.81n ) with Strassen's Algorithm (the currently best known algorithm [25] has complexity only O(2 2.3729n ) but is probably more expensive for practical values of n.)…”

Section: Integral Attackmentioning

confidence: 99%

Cryptanalysis of Feistel Networks with Secret Round Functions

Biryukov

Leurent

Perrin

2016

Lecture Notes in Computer Science

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Abstract. Generic distinguishers against Feistel Network with up to 5 rounds exist in the regular setting and up to 6 rounds in a multi-key setting. We present new cryptanalyses against Feistel Networks with 5, 6 and 7 rounds which are not simply distinguishers but actually recover completely the unknown Feistel functions. When an exclusive-or is used to combine the output of the round function with the other branch, we use the so-called yoyo game which we improved using a heuristic based on particular cycle structures. The complexity of a complete recovery is equivalent to O(2 2n ) encryptions where n is the branch size. This attack can be used against 6-and 7-round Feistel Networks in time respectively O(2 n2 n−1 +2n ) and O(2 n2 n +2n ). However when modular addition is used, this attack does not work. In this case, we use an optimized guess-and-determine strategy to attack 5 rounds with complexity O(2 n2 3n/4 ). Our results are, to the best of our knowledge, the first recovery attacks against generic 5-, 6-and 7-round Feistel Networks.

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Powers of tensors and fast matrix multiplication

Cited by 767 publications

References 21 publications

Algorithms for Structured Linear Systems Solving and Their Implementation

Algorithms for Structured Linear Systems Solving and Their Implementation

Bit complexity for multi-homogeneous polynomial system solving—Application to polynomial minimization

Cryptanalysis of Feistel Networks with Secret Round Functions

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