Graphs associated with the map X bar right arrow X+X-1 in finite fields of characteristic three and five

Text. In a previous paper the graphs associated with the iterations of the map theta which takes an element x of a finite field of characteristic two to x + x(-1) were studied, exploiting the relation between theta and the duplication map over Koblitz curves. While in odd characteristic the graphs associated with theta seem not to present notable symmetries, these are present in characteristic three and five. In fact, while in characteristic three the map theta is conjugated to the inverse of the square mapping, in characteristic five it is related to an endomorphism of a certain elliptic curve. Relying on these considerations we describe the structure of the graphs in finite fields of characteristic three and five and present a computational procedure for constructing examples in any characteristic. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=nnH53jawJaQ (C) 2012 Elsevier Inc. All rights reserved.