Cross-correlations of geometric sequences in characteristic two

Abstract

Abstract. Cross-correlation functions are determined for a large class of geometric sequences based on m-sequences in characteristic two. These sequences are shown to have low cross-correlation values in certain cases. They are also shown to have significantly higher linear complexities than previously studied geometric sequences. These results show that geometric sequences are candidates for use in spread-spectrum communications systems in which cryptographic security is a factor.

Quadratic form Approach for the Number of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

“…Let ν 2 (x) denote the 2-adic valuation of x. Consider the trace form (Klapper, 1993), Q(x) is of Type II with rank Q = n − d + 1 where r = 0 and Λ(Q) = 0.…”

“…In section 2, we will describe some known results on one-term trace forms over finite fields of even and odd characteristics by Klapper (1993Klapper ( , 1997 and include proofs for the simpler formulation of Klapper's results which were stated by Mullen and Panario (2013)7.2 without proofs.…”

Multi-Valued Sequences Generated by Power Residue Symbols over Odd Characteristic Fields

“…According to previous works on geometric sequences [4], [9], [10], a binary sequence is generated with a primitive element ω, trace function Tr (·), and some binarizing function f (·) as Eq. (11), where the trace function maps an extension field element to a prime field element:…”

d-form sequences: families of sequences with low correlation values and large linear spans

“…It follows from Proposition 3.4 of [7] that for some values of τ , the number of solutions to equation (9) is 0 for µ = 0 and is 1 for µ = 0. For these values of τ we have w = 1, so, again, Θ S γ ,S δ (τ ) = −q k − 1.…”

“…These have been completely analyzed [7] and this analysis will allow us to count the number of times each value of the cross-correlation occurs. To make use of this analysis, we first observe that the quadratic form Bx T is zero only for x = 0, hence in the terminology of [7], is a Type III quadratic form, and has rank two.…”