For more information about this meeting, contact Robert Vaughan.

Title: | Reversed Dickson Polynomials and Related Polynomials |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Xiang-dong Hou, South Florida University |

Abstract: |

Let D_n(x,y) be the polynomial over Z such that x_1^n+x_2^n=D_n(x_1+x_2,x_1x_2) and fix an element a\in F_q^*. The polynomial D_n(x,a) is the well known Dickson polynomial; the polynomial D_n(a,x) is the reversed Dickson polynomial recently introduced. Reversed Dickson polynomials are a rich source of permutation polynomials over finite fields and are closely related to the ubiquitous APN (almost perfect nonlinear) functions in cryptography. The ultimate goal is to determine all pairs (q,n), called desirable pairs, for which D_n(1,x) is a permutation polynomial on F_q. We will survey the known families of desirable pairs and introduce a new family. Together, these families cover all desirable pairs (q,n) with q<200. We cautiously conjecture that all desirable pairs have been found. We also introduce a q-ary version of reversed Dickson polynomials which coincide with D_n(1,x) when q=2. The permutation properties of the q-ary version will be discussed. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 10 / 22 / 2009 |

Time: | 11:15am - 12:05pm |