Çeşmelioğlu, Ayça
(2008)
*On the cycle structure of permutation polynomials.*
[Thesis]

PDF

Cesmeliogluayca.pdf

Download (342kB)

Cesmeliogluayca.pdf

Download (342kB)

Official URL: http://192.168.1.20/record=b1225695 (Table of Contents)

## Abstract

L. Carlitz observed in 1953 that for any a € F*q, the transposition (0 a) can be represented by the polynomial
Pa(x) = -a[2](((x - a)[q-2] + a-[1])[q-2] - a)[q-2]
which shows that the group of permutation polynomials over Fq is generated by the linear polynomials ax + b, a, b € Fq, a≠0, and x[q-2].
Therefore any permutation polynomial over Fq can be represented as
Pn = (...((a[0]x + a[1])[q-2] +a[2]) [q-2] ... + a[n])[q-2] + a[n+1], for some n ≥ 0.
In this thesis we study the cycle structure of permutation polynomials Pn, and we count the permutations Pn, n ≤ 3, with a full cycle. We present some constructions of permutations of the form Pn with a full cycle for arbitrary n. These constructions are based on the so called binary symplectic matrices.
The use of generalized Fibonacci sequences over Fq enables us to investigate a particular subgroup of Sq, the group of permutations on Fq. In the last chapter we present results on this special group of permutations.

Item Type: | Thesis |
---|---|

Uncontrolled Keywords: | Finite fields. -- Permutation polynomials. -- Cycle structure. -- Sonlu cisimler. -- Permütasyon polinomları. -- Çevrim yapısı. -- Symmetric group on q letters. -- Monomials. -- Dickson polynomials. -- Cycle decomposition. -- Nonlinear pseudorandom number generators. |

Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Engineering and Natural Sciences > Basic Sciences > Mathematics Faculty of Engineering and Natural Sciences |

Depositing User: | IC-Cataloging |

Date Deposited: | 25 Feb 2009 15:36 |

Last Modified: | 26 Apr 2022 09:50 |

URI: | https://research.sabanciuniv.edu/id/eprint/11345 |