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On D(n; q) quotients of large girth and hidden homomorphism based cryptographic protocols

Ustymenko Vasyl, Klisowski Michał

Annals of Computer Science and Information Systems

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2022

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Vol. 32

199--206

EN

Noncommutative cryptography is based on applications of algebraic structures like noncommutative groups,semigroups, and noncommutative rings. Its intersection withMultivariate cryptography contains studies of cryptographicapplications of subsemigroups and subgroups of affine Cremona semigroups defined over finite commutative rings. Efficientlycomputed homomorphisms between stable subsemigroups of affine Cremona semigroups can be used in tame homomorphisms protocols schemes and their inverse versions. The implementationscheme with the sequence of subgroups of affine Cremona group that defines the projective limit was already suggested. We present the implementation of another scheme that uses two projective limits which define two different infinite groups and the homomorphism between them. The security of the corresponding algorithm is based on complexity of the decomposition problem for an element of affine Cremona semigroup into a product of given generators. These algorithms may be used in postquantum technologies.

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The implementation of cubic public keys based on a new family of algebraic graphs

Klisowski M., Romańczuk U., Ustimenko V.

Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica

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2011

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Vol. 11, no. 2

127--141

EN

Families of edge transitive algebraic graphs defined over finite commutative rings were used for the development of stream ciphers, public key cryptosystems and key exchange protocols. We present the results of the first implementation of a public key algorithm based on the family of algebraic graphs, which are not edge transitive. The absence of an edge transitive group of symmetries means that the algorithm can not be described in group theoretical terms. We hope that it licates cryptanalysis of the algorithm. We discuss the connections between the security of algorithms and the discrete logarithm problem.The plainspace of the algorithm is Kn, where K is the chosen commutative ring. The graph theoretical encryption corresponds to walk on the bipartite graph with the partition sets which are isomorphic to Kn. We conjugate the chosen graph based encryption map, which is a composition of several elementary cubical polynomial automorphisms of a free module Kn with special invertible affine transformation of Kn. Finally we compute symbolically the corresponding cubic public map g of Kn onto Kn. We evaluate time for the generation of g, and the number of monomial expression in the list of corresponding public rules.

3

On the application of discrete chaotic systems to cryptography : DCC method

Kotulski Z., Szczepański J.

Biuletyn Wojskowej Akademii Technicznej

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1999

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Vol. 48, nr 10

111-122

EN

In the paper we present some methods of constructing cryptosystems utilising chaotic dynamical systems that have been extensively developed last years. We start with a brief review of algorithms based on both the theory of continuous and discrete systems. Then we show our approach where the essence of chaos (that is the sensitivity of the trajectories of discrete chaotic dynamical systems to the small changes of initial conditions) is exploited for secure communication.

PL

Praca poświęcona jest omówieniu metod intensywnie rozwijanych w ostatnich latach, dotyczących konstruowania kryptosystemów wykorzystujących teorię chaotycznych układów dynamicznych. W zwięzły sposób przedstawiono w niej najnowsze algorytmy tego typu oparte zarówno na ciągłych, jak też na dyskretnych układach dynamicznych. Następnie zaprezentowano własne podejście do tego zagadnienia, w którym podstawą algorytmu jest teoria dyskretnych układów chaotycznych. Jego istotą jest silne uwzględnienie w algorytmie najważniejszej własności trajektorii chaotycznych, to znaczy ich wykładniczej wrażliwości na małe zmiany warunków początkowych.