The Shift-and-Add Property of m-Sequences

Abstract

The shift-and-add property (SAP) of a p-ary m-sequence {a_k} with period N=pⁿ-1 means that this sequence satisfies the equation {a_k+η}+{a_k+τ}={a_k+λ} for some integers η, τ and λ. For an arbitrarily-given p-ary m-sequence {a_k}, we develop an algebraic approach to determine the integer λ for the arbitrarily-given integers η and τ. And all trinomials can be given. Our calculation only depends on the reciprocal polynomial of the primitive polynomial which produces the given m-sequence {a_k}, and the cyclotomic cosets mod pⁿ-1.