Methods of construction of skew linear recurrent sequences with maximal period based on the Galois polynomials factorization in the ring of matrix polynomials

M. A. Goltvanitsa

Matematičeskie voprosy kriptografii, Tome 10 (2019), pp. 25-51

Résumé

Let $p$ be a prime, $R=\mathrm{GR}(q^d,p^d)$ be a Galois ring of cardinality $q^d$ and characteristic $p^d$, where $q=p^r$, $S=\mathrm{GR}(q^{nd},p^d)$ be an $R$-extension of degree $n$ and $\check{S}$ be an endomorphism ring of the module $_RS$. A sequence $v$ over $S$ with the recursion law $$ \forall i\in\mathbb{N}_0 :\;\;\;v(i+m)= \\psi_{m-1}(v(i+m-1))+...+\psi_0(v(i)),\;\;\;\psi_0,...,\psi_{m-1}\in \check{S},$$ is called a skew LRS over $S$ with a characteristic polynomial $\Psi(x) = x^m - \sum_{j=0}^{m-1}\psi_jx^j$. The maximal period $T(v)$ of such sequence equals $\tau = (q^{mn}-1)p^{d-1}$. In this article we propose some new methods for construction the polynomials $\Psi(x)$, which define the recursion laws of skew linear recurrent sequences of maximal period. These methods are based on the search in $\check{S}[x]$ the divisors for classic Galois polynomials of period $\tau$ over $R$.