Constructing sequences with high nonlinear complexity using the Weierstrass semigroup of a pair of distinct points of a Hermitian curve

Abstract

Using the Weierstrass semigroup of a pair of distinct points of a Hermitian curve over a finite field, we construct sequences with improved high nonlinear complexity. In particular we improve the bound obtained in [15, Theorem 3] considerably and the bound in [15, Theorem 4] for some parameters.

On finite length nonbinary sequences with large nonlinear complexity over the residue ring $ \mathbb{Z}_{m} $

“…Hence subsequences s n−1 in Z 3 (n − 1, 2, 4) have more forms than them in Z 2 (n − 1, 2, 4). Furthermore, sequences in Z 3 (n, n − 4) have more forms than these in Z 2 (n, n − 4) by (2).…”

“…For finite length sequences, several constructions of sequences with high nonlinear complexity from function fields were presented in [7,11]. By applying the structure of Weierstrass pairs of distinct points of a Hermitian curve, the bounds of nonlinear complexity in [11] were improved [2]. With a combinatorial method, all sequences over a residue ring Z m (m ≥ 2) of length n having nonlinear complexity n − j were completely characterized, where 2 ≤ j ≤ 3 and n ≥ 2j.…”

Construction of sequences with high nonlinear complexity from a generalization of the Hermitian function field

“…In recent years the study of sequences over finite fields has increased due to its applications in cryptography and pseudorandom number generation, see e.g. [5,8,10] and [12]. In cryptography, to assess the suitability of a pseudorandom sequence, the complexitytheoretic and statistical requirements must be tested.…”

“…In this paper we focus on the construction of sequences with large nonlinear complexity. Recently, constructions of sequences from function fields with high nonlinear complexity arose in work of H. Niederreiter and C. Xing [12]; Y. Luo, C. Xing, and L. You [8]; and O. Geil, F. Özbudak and D. Ruano [5]. In [8] the authors construct sequences over F q using the rational and cyclotomic function fields.…”

“…In [8] the authors construct sequences over F q using the rational and cyclotomic function fields. In [12,5] the authors work over the Hermitian function field H to construct sequences over F q 2 with large nonlinear complexity. Both constructions make use of the action of a certain subgroup of the authomorphism group of the function field on the F q 2 -rational places on H and certain functions with poles only in two rational places P 1 , P 2 of small degree.…”