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
Summary
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.
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