Novel Noncommutative Cryptography Scheme Using Extra Special Group

Kumar, Gautam; Saini, Hemraj

doi:10.1155/2017/9036382

Cited by 29 publications

(12 citation statements)

References 29 publications

(27 reference statements)

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“…However, this trivial (polynomial-time) reduction from Exact NMF to Equation (44) does not correspond to the MPF N 0 S M problem according to Definition 14, since Exact NMF do not include commutativity conditions on matrices X and Y in Equation (9). The Exact NMF problem complexity with commutativity constraints is not yet known and, therefore, this relation cannot tell anything about the MPF N 0 S M problem complexity.…”

Section: Proofmentioning

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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Section: Proofmentioning

confidence: 99%

Enhanced Matrix Power Function for Cryptographic Primitive Construction

Sakalauskas

2018

Symmetry

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“…It is an active area of cryptology, where the cryptographic primitives and systems are based on algebraic structures like groups, semigroups and noncommutative rings (see [26]- [33]). This direction of security research has very rapid development (see [34], [35] and further references in these publications).…”

Section: On Cryptosystems Based On New Multivariate Platforms Of Nmentioning

confidence: 99%

Expanding graphs of the Extremal Graph Theory and expanded platforms of Post Quantum Cryptography

Ustimenko

Romańczuk-Polubiec

Wróblewska

2019

Annals of Computer Science and Information Systems

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Explicit constructions in Extremal Graph Theory give appropriate lower bounds for Turan type problems. In the case of prohibited cycles, the explicit constructions can be used for various problems of Information Security. We observe recent applications of algebraic constructions of regular graphs of large girth and graphs with large cycle indicator to Coding Theory and Cryptography. In particular, we present a new multivariate platforms of postquantum Non-commutative Cryptography defined in graph theoretical terms. Index Terms-graphs of large girth, graphs of large cycle indicator, graph based stream ciphers, multivariate cryptography, non-commutative cryptography I. SOME DEFINITIONS OF EXTREMAL GRAPH THEORY T HE missing definitions of graph-theoretical concepts in the case of simple graphs which appears in this paper can be found in [1]. All graphs we consider are simple ones, i. e. undirected without loops and multiple edges. When it is convenient, we shall identify Γ with the corresponding antireflexive binary relation on V (Γ), i.e. E(Γ) is a subset of V (Γ)×V (Γ). The girth of a graph Γ, denoted by g = g(Γ), is the length of the shortest cycle in Γ. The diameter d = d(Γ) of the graph Γ is the maximal length of the shortest pass between its two vertices. Let g x = g x (Γ) be the length of the minimal cycle through the vertex x from the set V (Γ) of vertices in graph Γ. We refer to Cind(Γ) = max {g x , x ∈ V (Γ)} as cycle indicator of the graph Γ. The family Γ i of connected k-regular graphs of constant degree is a family of small world graphs, if d(Γ i) ≤ c log k (v i), for some constant c, c > 0. Recall that family of regular graphs Γ i of degree k and increasing order v i is a family of graphs of large girth, if g(Γ i) ≥ c log k (v i), for some independent constant c, c > 0. We refer to the family of regular simple graphs Γ i of degree k and order v i as a family of graphs of large cycle indicator, if Cind(Γ i) ≥ c log k (v i) for some independent constant c, c > 0. Notice that for vertex-transitive graph its girth and cycle indicator coincide. Defined above families plays an important role in Extremal Graph Theory, Theory of LDPC codes and Cryptography (see [2] and further references).

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Section: On Cryptosystems Based On New Multivariate Platforms Of Nmentioning

confidence: 99%

Position Papers of the 2019 Federated Conference on Computer Science and Information Systems

2019

Annals of Computer Science and Information Systems

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Position papers comprise two categories of contributions-challenge papers and emerging research papers. Challenge papers propose and describe research challenges in theory or practice of computer science and information systems. Papers in this category are based on deep understanding of existing research or industrial problems. Based on such understanding and experience, they define new exciting research directions and show why these directions are crucial to the society at large. Emerging research papers present preliminary research results from work-in-progress based on sound scientific approach but presenting work not completely validated as yet. They describe precisely the research problem and its rationale. They also define the intended future work including the expected benefits from solution to the tackled problem. Subsequently, they may be more conceptual than experimental. FedCSIS 2019 was Chaired by prof. Bogdan Franczyk, while prof. Rainer Unland acted as the Chair of the Organizing Committee. This year, FedCSIS was organized by the Polish Information Processing Society (Mazovia Chapter

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Novel Noncommutative Cryptography Scheme Using Extra Special Group

Cited by 29 publications

References 29 publications

Enhanced Matrix Power Function for Cryptographic Primitive Construction

Enhanced Matrix Power Function for Cryptographic Primitive Construction

Expanding graphs of the Extremal Graph Theory and expanded platforms of Post Quantum Cryptography

Position Papers of the 2019 Federated Conference on Computer Science and Information Systems

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