Nonnegative ranks, decompositions, and factorizations of nonnegative matrices

Cohen, Joel E.; Rothblum, Uriel G.

doi:10.1016/0024-3795(93)90224-c

Cited by 216 publications

(251 citation statements)

References 11 publications

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“…where the first relation is implied by Equation (33). The exponent ι of generators a and b in S is defined in the following way:…”

Section: Definition 12mentioning

confidence: 99%

Enhanced Matrix Power Function for Cryptographic Primitive Construction

Sakalauskas

2018

Symmetry

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Abstract:A new enhanced matrix power function (MPF) is presented for the construction of cryptographic primitives. According to the definition in previously published papers, an MPF is an action of two matrices powering some base matrix on the left and right. The MPF inversion equations, corresponding to the MPF problem, are derived and have some structural similarity with classical multivariate quadratic (MQ) problem equations. Unlike the MQ problem, the MPF problem seems to be more complicated, since its equations are not defined over the field, but are represented as left-right action of two matrices defined over the infinite near-semiring on the matrix defined over the certain infinite, additive, noncommuting semigroup. The main results are the following: (1) the proposition of infinite, nonsymmetric, and noncommuting algebraic structures for the construction of the enhanced MPF, satisfying associativity conditions, which are necessary for cryptographic applications; (2) the proof that MPF inversion is polynomially equivalent to the solution of a certain kind of generalized multivariate quadratic (MQ) problem which can be reckoned as hard; (3) the estimation of the effectiveness of direct MPF value computation; and (4) the presentation of preliminary security analysis, the determination of the security parameter, and specification of its secure value. These results allow us to make a conjecture that enhanced MPF can be a candidate one-way function (OWF), since the effective (polynomial-time) inversion algorithm for it is not yet known. An example of the application of the proposed MPF for the Key Agreement Protocol (KAP) is presented. Since the direct MPF value is computed effectively, the proposed MPF is suitable for the realization of cryptographic protocols in devices with restricted computation resources.

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“…where the first relation is implied by Equation (33). The exponent ι of generators a and b in S is defined in the following way:…”

Section: Definition 12mentioning

confidence: 99%

Enhanced Matrix Power Function for Cryptographic Primitive Construction

Sakalauskas

2018

Symmetry

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“…Nonnegative factorizations are being widely studied and used in data analysis, statistics, computational biology, clustering and numerous other applications [2]. There are still many open questions on nonnegative rank interesting for different applications, and a considerable part of them is related to providing the bounds on the nonnegative rank in terms of other matrix invariants [4,8,10].…”

Section: Theorem 12 [10 Theorem 2]mentioning

confidence: 99%

“…It is easy to show that the nonnegative rank of a matrix equals [2] the classical rank if one of them is less than 3. However, even for a rank-three m-by-n matrix, no upper bound (instead of min{m, n}, which is trivial) for the nonnegative rank has been known.…”

Section: Theorem 12 [10 Theorem 2]mentioning

confidence: 99%

An upper bound for nonnegative rank

Shitov

2014

Journal of Combinatorial Theory, Series A

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Abstract. We provide a nontrivial upper bound for the nonnegative rank of rank-three matrices, which allows us to prove that 6n 7 linear inequalities suffice to describe a convex n-gon up to a linear projection. PreliminariesConsider a convex polytope P ⊂ R n . An extension [5,8] of P is a polytope Q ⊂ R d such that P can be obtained from Q as an image under a linear projection from R d to R n . An extended formulation [8,10] of P is a description of Q by linear equations and linear inequalities (together with the projection). The size [8, 10] of the extended formulation is the number of facets of Q. The extension complexity [8, 10] of a polytope P is the smallest size of any extended formulation of P , that is, the minimal possible number of inequalities in the description of Q. The number of facets of Q can sometimes be significantly smaller [5] than that of P , and this phenomenon can be used to reduce the complexity of linear programming problems useful for numerous applications [3,5,10

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“…In numerical scenario both M and L are known while in experimental 5 The factorization problems (9) to (12) are related to the determination of nonnegative rank of nonnegative matrix and that is defined as the smallest number of rank one matrices into which original matrix can be decomposed [46]. For some matrix scenario selection of optimal (true) value of L is hard.…”

Section: Q2l+l(l-1)/2mentioning

confidence: 99%