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2 Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica

2 2014

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On New Examples of Families of Multivariate Stable Maps and their Cryptographical Applications

Wróblewska A., Ustimenko V.

Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica

2014

Vol. 14, no. 1

19--36

Let K be a general finite commutative ring. We refer to a family gn, n = 1, 2, . . . of bijective polynomial multivariate maps of Kn as a family with invertible decomposition gn = gn1 gn2 . . . g gnk, such that the knowledge of the composition of gni allows computation of gni for O(ns) (s > 0) elementary steps. A polynomial map g is stable if all non-identical elements of kind gt, t > 0 are of the same degree. We construct a new family of stable elements with invertible decomposition. This is the first construction of the family of maps based on walks on the bipartite algebraic graphs defined over K, which are not edge transitive. We describe the application of the above mentioned construction for the development of stream ciphers, public key algorithms and key exchange protocols. The absence of edge transitive group essentially complicates cryptanalysis.

On Multivariate Cryptosystems Based on Computable Maps with Invertible Decomposition

Ustimenko V.

Annales Universitatis Mariae Curie-Skłodowska. Sectio AI, Informatica

2014

Vol. 14, no. 1

7--17

Let K be a commutative ring and K^n be a space over K of dimension n. We introduce the concept of a family of multivariate maps f(n) of K^n into itself with invertible decomposition.If f(n) is computable in polynomial time then it can be used as the public rule and the invertible decomposition provides a private key in f(n) based public key infrastructure. Requirementsof polynomial ity of degree and density for f(n) allow to estimate the complexity of encryption procedurefor a public user. The concepts of a stable family and a family of increasing order are motivatedby the studies of discrete logarithm problem in the Cremona group. The statement on the existenceof families of multivariate maps of polynomial degree and polynomial density of increasing order with the invertible decomposition is proved. The proof is supported by explicite construction which canbe used as a new cryptosystem. The presented multivariate encryption maps are induced by special walks in the algebraically dened extremal graphs A(n;K) and D(n;K) of increasing girth.