Abstract

Let K be a finite commutative ring and f = f(n) a bijective polynomial map f(n) of the Cartesian power K^n onto itself of a small degree c and of a large order. Let f^y be a multiple composition of f with itself in the group of all polynomial automorphisms, of free module K^n. The discrete logarithm problem with the pseudorandom base f(n) (solvef^y = b for y) is a hard task if n is sufficiently large. We will use families of algebraic graphs defined over K and corresponding dynamical systems for the explicit constructions of such maps f(n) of a large order with c = 2 such that all nonidentical powers f^y are quadratic polynomial maps. The above mentioned result is used in the cryptographical algorithms based on the maps f(n) – in the symbolic key exchange protocols and public keys algorithms.

Highlights

  • The sequence of subgroups Gl of Cremona group C(Kl), l → ∞ is a family of stable groups if the degree of each g, g ∈ Gl, is bounded by the constant c independent of l

  • In the case of "pseudorandom" polynomials fl, such that max(fl, fl−1) is bounded by constant, we obtain a stable family with c ≥ 4

  • G = fl−1τ fl looks as the appropriate base for the hidden symbolic discrete logarithm problem

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Summary

Introduction

The sequence of subgroups Gl of Cremona group C(Kl), l → ∞ is a family of stable groups if the degree of each g, g ∈ Gl, is bounded by the constant c independent of l. Construct stable subgroups via conjugation of AGLl(K) with the nonlinear polynomial maps fl ∈ C(Kl). In the case of "pseudorandom" polynomials fl, such that max(fl, fl−1) is bounded by constant, we obtain a stable family with c ≥ 4. The family of large stable subgroups of C(Kl) over the general commutative ring K containing at least 3 regular elements In this paper we propose a similar result for the case of c = 2 Those results are based on the construction of the family D(n, q) of graphs with.

On the discrete logarithm problem for the Cremona groups
Graph theoretical base
Explicit construction of families of quadratic polynomials
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