An asymptotic formula for the number of irreducible transformation shift registers

@article{Cohen2015AnAF,
  title={An asymptotic formula for the number of irreducible transformation shift registers},
  author={Stephen D. Cohen and Sartaj Ul Hasan and Daniel Panario and Qiang Wang},
  journal={ArXiv},
  year={2015},
  volume={abs/1506.02548}
}
View PDF on arXiv
9 Citations
Highly Influential Citations
1
Background Citations
4

9 Citations

Citation Type

Citation Type

On the number of irreducible linear transformation shift registers
TLDR
This work finds a bijection between Ram’s set to another set of irreducible polynomials which is easier to count, and gives a conjecture about the number of irReducible TSRs of any order.
Irreducible polynomials from a cubic transformation
Let R(x) = g(x)/h(x) be a rational expression of degree three over the finite field Fq. We count the irreducible polynomials in Fq[x], of a given degree, which have the form h(x) f · f ( R(x) ) for…
Primitive transformation shift registers over finite fields
TLDR
This work considers systems which are efficient generalizations of LFSRs and produce pseudorandom vector sequences and gives certain results in this direction.
A note on the multiple-recursive matrix method for generating pseudorandom vectors
Cubic rational expressions over a finite field
We classify the cubic rational expressions g(x)/h(x) over a finite field, having at most three ramification points, under an equivalence relation given by preand post-composition with independent…
Unimodular polynomial matrices over finite fields
We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory, we give a proof of a result of Lieb, Jordan and Helmke on the number of linear…
Nonlinear vectorial primitive recursive sequences
TLDR
The nonlinearly filtered multiple-recursive matrix generator for producing pseudorandom vectors based on some nonlinear schemes is considered and lower bounds for their componentwise linear complexity are given.
Скрученные $\sigma$-разделимые линейные рекуррентные последовательности максимального периода
Пусть $p$ - простое число, $R=\mathrm{GR}(q^d,p^d)$ - кольцо Галуа мощности $q^d$ и характеристики $p^d$, где $q = p^r$, $S=\mathrm{GR}(q^{nd},p^d)$ - расширение степени $n$ кольца $R$, $\sigma$ -…
Simple Linear Transformations 3 3 . Splitting Subspaces 7 4 . Density of Unimodular Polynomial Matrices 9
Abstract. We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of…

References

SHOWING 1-10 OF 30 REFERENCES
Enumeration of linear transformation shift registers
TLDR
This work deduces a theorem of Carlitz on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field from explicit formulae derived from results on TSRs.
Word-Oriented Transformation Shift Registers and Their Linear Complexity
TLDR
The notion of word-oriented nonlinearly filtered primitive transformation shift registers based on a Langford arrangement and their linear complexity are introduced.
Generalization of a theorem of Carlitz
  • O. Ahmadi
  • Mathematics
    Finite Fields Their Appl.
  • 2011
Efficient linear feedback shift registers with maximal period
TLDR
An efficient family of linear feedback shift registers (LFSR's) with maximal period is introduced and is suitable for implementation in software, thus providing a solution to a recent challenge 8.
Mutual Irreducibility of Certain Polynomials
TLDR
The authors recently proved that over \(\mathbb{F}_2\), irreducible TSRs occur in pairs, now the results are generalized and extended for arbitrary finite fields.
Primitive polynomials, singer cycles and word-oriented linear feedback shift registers
Using the structure of Singer cycles in general linear groups, we prove that a conjecture of Zeng et al. (Word-Oriented Feedback Shift Register: σ-LFSR, 2007) holds in the affirmative in a special…
Linear transformation shift registers
TLDR
It is proved that the characteristic polynomials of a pair of LFSRs are either both irreducible or both reducible for all transformations, which allows some time improvement when finding primitive TSRs.
Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields
The Splitting Subspace Conjecture
UNIFORM DISTRIBUTION OF POLYNOMIALS OVER FINITE FIELDS
1. Let k be the finite field of order q = p. A polynomial Q(x) in M(n), the set of polynomials in k[x] of degree n, is defined to have factor pattern X = \...n" if Q factorises in k[x] into a product…
...
...