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On the family of cubical multivariate cryptosystems based on the algebraic graph over finite commutative rings of characteristic 2

Romańczuk U., Ustimenko V.

Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica

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2012

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Vol. 12, no. 3

89--106

EN

The family of algebraic graphs A(n;K) defined over the finite commutative ring K were used for the design of different multivariate cryptographical algorithms (private and public keys, key exchange protocols). The encryption map corresponds to a special walk on this graph. We expand the class of encryption maps via the use of an automorphism group of A(n;K). In the case of characteristic 2 the encryption transformation is a Boolean map. We change finite field for the commutative ring of characteristic 2 and consider some modifications of algorithm which allow to hide a ground commutative ring.

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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.

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On the key exchange with new cubical maps based on graphs

Romańczuk U., Ustimenko V.

Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica

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2011

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Vol. 11, no. 4

11--19

EN

Families of edge transitive algebraic graphs Fn(K), over the commutative ring K were used for the graph based cryptographic algorithms. We introduce a key exchange protocol defined in terms of bipartite graph An(K), n ≥ 2 with point set Pn and line set Ln isomorphic to n-dimensional free module Kn. Graphs A(n, K) are not vertex and edge transitive. There is a well defined projective limit lim A(n, K) = A(K), n → ∞ which is an infinite bipatrtite graph with point set P = lim Pn and line set L = limLn. Let K be a commutative ring contain at least 3 regular elements (not zero divisors). For each pair of (n, d), n ≥ 2, n ≥ 1 and sequence of elements α1, α2, …, α2d, such that α1, αi+αi+1, i = 1, 2, …, 2d, i = 1, 2, … 2d-1 and α2d+α1 are regular elements of the ring K. We define polynomial automorphism hn = hn (d, α1, α2, …, α2d) of variety Ln (or Pn). The existence of projective limit lim An(K) guarantees the existence of projective limit h = h(d, α1, α2, …, α2d) = lim hn, n → ∞ which is cubical automorphism of infinite dimensional varieties L (or P). We state that the order of h is an infinity. There is a constant n0 such that hn, n ≥ n0 is a cubical map. Obviously the order of hn is growing with the growth of n and the degree of polynomial map (hn)k from the Cremona group of all polynomial automorphisms of free module Kn with operation of composition is bounded by 3. Let τ be affine automorphism of Kn i.e. the element of Cremona group of degree 1. We suggest symbolic Diffie Hellman key exchange with the use of cyclic subgroup of Cremona group generated by τ-1hnτ. In the case of K = Fp, p is prime, the order of hn is the power of p. So the order is growing with the growth of p. We use computer simulation to evaluate the orders in some cases of K = Zm, where m is a composite integer.