Properties of certain semigroups and their potential as platforms for cryptosystems

Kropholler, Peter H.; Pride, Stephen J.; Othman, Wan Ainun Mior; Wong, Kok Bin; Wong, P. C.

doi:10.1007/s00233-010-9248-8

Cited by 14 publications

(7 citation statements)

References 16 publications

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“…It is easy to see that encryption of single message costs standard for Multivariate Cryptography time O(n 3 ) Elements of cryptanalisis. In the case of agstract finite group X twisted key agreement protocol with input elements G ∈ X, H ∈ X and output Z ∈ X is known instrument of noncommutative cryptography (see [27][28][29][30][31][32][33][34][35][36][37][38]). It based on complexity of Power Conjugacy Problem.…”

Section: Double Schubert Graphs and Automata For The Generation Of Stable Mapsmentioning

confidence: 99%

On computations with double Schubert automaton and stable maps of multivariate cryptography

Ustimenko¹

2021

Мiждисциплiнарнi дослiдження складних систем

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The families of bijective transformations Gn of affine space K n over general commutative ring K of increasing order with the property of stability will be constructed. Stability means that maximal degree of elements of cyclic subgroup generated by the transformation of degree d is bounded by d. In the case K = Fq these transformations of K n can be of an exponential order. We introduce large groups formed by quadratic transformations and numerical encryption algorithm protected by secure protocol of Noncommutative Cryptography. The construction of transformations is presented in terms of walks on Double Schubert Graphs.

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Section: Double Schubert Graphs and Automata For The Generation Of Stable Mapsmentioning

confidence: 99%

On computations with double Schubert automaton and stable maps of multivariate cryptography

Ustimenko¹

2021

Мiждисциплiнарнi дослiдження складних систем

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“…In [22] a bi-semigroup action problem (BSAP) is proposed and using this computational hard problem a new key exchange protocol is defined. In [8] some properties of semigroups are discussed which are useful for designing the public key cryptosystem. In [3] an efficient quantum algorithm is described by using Shor's algorithm [19] for computing discrete logarithms in semigroups.…”

Section: Some Work Based On Action Of Algebraic Structurementioning

confidence: 99%

Survey on SAP and its application in public-key cryptography

Goel

Gupta

Dass

2020

Journal of Mathematical Cryptology

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AbstractThe concept of the semigroup action problem (SAP) was first introduced by Monico in 2002. Monico explained in his paper that the discrete logarithm problem (DLP) can be generalized to SAP. After defining the action problem in a semigroup, the concept was extended using different mathematical structures. In this paper, we discuss the concept of SAP and present a detailed survey of the work which has been done using it in public-key cryptography.

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“…Currently Non-commutative cryptography is essentially wider than group based cryptography. 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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Properties of certain semigroups and their potential as platforms for cryptosystems

Cited by 14 publications

References 16 publications

On computations with double Schubert automaton and stable maps of multivariate cryptography

On computations with double Schubert automaton and stable maps of multivariate cryptography

Survey on SAP and its application in public-key cryptography

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

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