On the linear complexity profile of some sequences derived from elliptic curves

Abstract

For a given elliptic curve $$\mathbf {E}$$ over a finite field of odd characteristic and a rational function f on $$\mathbf {E}$$ we first study the linear complexity profiles of the sequences f(nG), $$n=1,2,\dots $$ which complements earlier results of Hess and Shparlinski. We use Edwards coordinates to be able to deal with many f where Hess and Shparlinski’s result does not apply. Moreover, we study the linear complexities of the (generalized) elliptic curve power generators $$f(e^nG)$$ , $$n=1,2,\dots $$ We present large families of functions f such that the linear complexity profiles of these sequences are large.