Irreducible compositions of degree two polynomials over finite fields have regular structure

Let $q$ be an odd prime power and $D$ be the set of monic irreducible polynomials in $\mathbb F_q[x]$ which can be written as a composition of monic degree two polynomials. In this paper we prove that $D$ has a natural regular structure by showing that there exists a finite automaton having $D$ as accepted language. Our method is constructive.Comment: To appear in The Quarterly Journal of Mathematic

On algebraic cellular automata

We investigate some general properties of algebraic cellular automata, i.e., cellular automata over groups whose alphabets are affine algebraic sets and which are locally defined by regular maps. When the ground field is assumed to be uncountable and algebraically closed, we prove that such cellular automata always have a closed image with respect to the prodiscrete topology on the space of configurations and that they are reversible as soon as they are bijective

Linear solutions for cryptographic nonlinear sequence generators

This letter shows that linear Cellular Automata based on rules 90/150 generate all the solutions of linear difference equations with binary constant coefficients. Some of these solutions are pseudo-random noise sequences with application in cryptography: the sequences generated by the class of shrinking generators. Consequently, this contribution show that shrinking generators do not provide enough guarantees to be used for encryption purposes. Furthermore, the linearization is achieved through a simple algorithm about which a full description is provided

Modelling Nonlinear Sequence Generators in terms of Linear Cellular Automata

In this work, a wide family of LFSR-based sequence generators, the so-called Clock-Controlled Shrinking Generators (CCSGs), has been analyzed and identified with a subset of linear Cellular Automata (CA). In fact, a pair of linear models describing the behavior of the CCSGs can be derived. The algorithm that converts a given CCSG into a CA-based linear model is very simple and can be applied to CCSGs in a range of practical interest. The linearity of these cellular models can be advantageously used in two different ways: (a) for the analysis and/or cryptanalysis of the CCSGs and (b) for the reconstruction of the output sequence obtained from this kind of generators.Comment: 15 pages, 0 figure

Efficient Black-Box Identity Testing for Free Group Algebras

Hrubes and Wigderson [Pavel Hrubes and Avi Wigderson, 2014] initiated the study of noncommutative arithmetic circuits with division computing a noncommutative rational function in the free skew field, and raised the question of rational identity testing. For noncommutative formulas with inverses the problem can be solved in deterministic polynomial time in the white-box model [Ankit Garg et al., 2016; Ivanyos et al., 2018]. It can be solved in randomized polynomial time in the black-box model [Harm Derksen and Visu Makam, 2017], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions, in general, remains open for noncommutative circuits with inverses. We solve the problem for a natural special case. We consider expressions in the free group algebra F(X,X^{-1}) where X={x_1, x_2, ..., x_n}. Our main results are the following. 1) Given a degree d expression f in F(X,X^{-1}) as a black-box, we obtain a randomized poly(n,d) algorithm to check whether f is an identically zero expression or not. The technical contribution is an Amitsur-Levitzki type theorem [A. S. Amitsur and J. Levitzki, 1950] for F(X, X^{-1}). This also yields a deterministic identity testing algorithm (and even an expression reconstruction algorithm) that is polynomial time in the sparsity of the input expression. 2) Given an expression f in F(X,X^{-1}) of degree D and sparsity s, as black-box, we can check whether f is identically zero or not in randomized poly(n,log s, log D) time. This yields a randomized polynomial-time algorithm when D and s are exponential in n