A generalization of the Hall’s sextic residue sequences

Abstract

Let p be a prime with 6∣(p − 1) and integer m ⩾ 1. As a generalization of the Hall’s sextic residue sequences, we introduce some families of generalized cyclotomic binary sequences over the residue class ring modulo p^m by defining sextic generalized cyclotomic residue classes. We determine the values of the linear complexity, which are large enough to resist security attacks using the Berlekamp–Massey algorithm. We also investigate the trace function representation for the resulting sequences when m = 2.

Introduction

Let p be an odd prime and integer m ⩾ 1. We identify ${\mathbb{Z}}_{p^m}$, the residue ring modulo p^m, with the set {0, 1, … , p^m − 1}. We also denote by ${\mathbb{Z}}_{p^m}^{\ast }$ the set of unit elements of ${\mathbb{Z}}_{p^m}$. Since ${\mathbb{Z}}_{p^m}^{\ast }$ is a cyclic group under the multiplicative operation, let g be one of its generators (or primitive elements). Then we have

$${\mathbb{Z}}_{p^m}^{\ast }=\{g^i:i=0\text{,}1\text{,}\dots \text{,}\phi (p^m)\}\text{,}$$

where φ(−) is the Euler totient function. By defining different partitions of ${\mathbb{Z}}_{p^m}^{\ast }$, the ring ${\mathbb{Z}}_{p^m}$ is extensively applied to constructing pseudorandom sequences in the literature. A typical partition is the (generalized) cyclotomic classes, see below for the definition.

In this paper, we always suppose that g is a primitive element modulo p². We note that in this case g is a primitive element modulo pⁿ for any n ⩾ 1 [16]. That is, the order of g in ${\mathbb{Z}}_{p^n}^{\ast }$ is ${\mathrm{ord}}_{p^n}(g)=\phi (p^n)=p^{n-1}(p-1)$. Let d be an even integer satisfying d∣(p − 1). Now for each n, we get the (generalized) cyclotomic classes of ${\mathbb{Z}}_{p^n}^{\ast }$ by defining

$$D_0^{(n)}=(g^d)=\left\{g^{\mathit{dk}}(\mathrm{mod}{p}^n):k=0\text{,}1\text{,}\dots \text{,}\frac{\phi (p^n)}{d}-1\right\}\text{,}$$

$$D_l^{(n)}=g^l{D}_0^{(n)}=\left\{g^{l}x(\mathrm{mod}{p}^n):x\in D_0^{(n)}\right\}\text{,}l=1\text{,}\dots \text{,}d-1\text{,}$$

which give a partition of ${\mathbb{Z}}_{p^n}^{\ast }$. Let

$$R^{(n)}=p{\mathbb{Z}}_{p^{n-1}}=\{0\text{,}p\text{,}2p\text{,}\dots \text{,}(p^{n-1}-1)p\}\text{,}$$

we have

$${\mathbb{Z}}_{p^n}={\mathbb{Z}}_{p^n}^{\ast }\cup R^{(n)}=\left(D_0^{(n)}\cup D_1^{(n)}\cup \cdots \cup D_{d-1}^{(n)}\right)\cup R^{(n)}\text{.}$$

Thus we represent the ring ${\mathbb{Z}}_{p^m}$ as follows

$$\begin{array}{cc}\hfill {\mathbb{Z}}_{p^m}& =D_0^{(m)}\cup D_1^{(m)}\cup \cdots \cup D_{d-1}^{(m)}\cup p{\mathbb{Z}}_{p^{m-1}}\hfill \\ \hfill & =\left(D_0^{(m)}\cup {\mathit{pD}}_0^{(m-1)}\right)\cup \cdots \cup \left(D_{d-1}^{(m)}\cup {\mathit{pD}}_{d-1}^{(m-1)}\right)\cup p^2{\mathbb{Z}}_{p^{m-2}}\hfill \\ \hfill & \dots \hfill \\ \hfill & =\left({\displaystyle \bigcup _{n=1}^m}{p}^{m-n}{D}_0^{(n)}\right)\bigcup \cdots \bigcup \left({\displaystyle \bigcup _{n=1}^m}{p}^{m-n}{D}_{d-1}^{(n)}\right)\bigcup \{0\}\text{.}\hfill \end{array}$$

Assume that

$$C_l={\displaystyle \bigcup _{n=1}^m}{p}^{m-n}{D}_l^{(n)}\text{,}l=0\text{,}1\dots \text{,}d-1\text{,}$$

then we have

$${\mathbb{Z}}_{p^m}={\displaystyle \bigcup _{i=0}^{d-1}}{C}_i\cup \{0\}\mathrm{and}{C}_i\cap C_j=\varnothing \text{,}i\ne j\text{,}i\text{,}j=0\text{,}\dots \text{,}d-1\text{,}$$

where ∅ denotes the empty set.

So one can construct a p^m-periodic binary sequence {s_t}_t⩾0 by defining

$$s_t=\left\{\begin{array}{cc}1\text{,}\hfill & \mathrm{if}t(\mathrm{mod}{p}^m)\in C_0\cup \cdots \cup C_{\frac{d}{2}-1}\text{,}\hfill \\ 0\text{,}\hfill & \mathrm{otherwise}\text{,}\hfill \end{array}\right.t⩾0\text{,}$$

where $C_0\cup \cdots \cup C_{\frac{d}{2}-1}$ is called the characteristic set of {s_t}_t⩾0. We note that {s_t}_t⩾0 is called a cyclotomic sequence for m = 1 or a generalized cyclotomic sequence for m ⩾ 2. Some special cases have been investigated in the literature, see Table 1, in which the symbol “√” means that the trace function representation of the corresponding sequences has been investigated.

In this paper, we will consider {s_t}_t⩾0 defined in (2) and its related sequences when m ⩾ 1 and d = 6.

Definition 1

Let distinct integers $u\text{,}v\text{,}w\in {\mathbb{Z}}_6=\{0\text{,}1\text{,}\dots \text{,}5\}$, which is the residue class ring modulo 6. The triple subset {u,v,w} is admissible over ${\mathbb{Z}}_6$ if there exists an $\ell \in {\mathbb{Z}}_6$ such that

$$\{u+\ell (\mathrm{mod}6)\text{,}v+\ell (\mathrm{mod}6)\text{,}w+\ell (\mathrm{mod}6)\}={\mathbb{Z}}_6⧹\{u\text{,}v\text{,}w\}\text{.}$$

One can easily verify that all admissible triples over ${\mathbb{Z}}_6$ are

$$\{0\text{,}1\text{,}2\}\text{;}\{1\text{,}2\text{,}3\}\text{;}\{2\text{,}3\text{,}4\}\text{;}\{3\text{,}4\text{,}5\}\text{;}\{4\text{,}5\text{,}0\}\text{;}\{5\text{,}0\text{,}1\}\text{;}\{1\text{,}3\text{,}5\}\text{;}\{0\text{,}2\text{,}4\}\text{.}$$

Proposition 1

Each admissible triple {u, v, w} over ${\mathbb{Z}}_6$ satisfies

$$\{u\text{,}v\text{,}w\text{,}u+3(\mathrm{mod}6)\text{,}v+3(\mathrm{mod}6)\text{,}w+3(\mathrm{mod}6)\}={\mathbb{Z}}_6\text{.}$$

Proof

Clearly. □

Now we introduce the new sequences. Let $\mathcal{A}$ be the set of all admissible triples over ${\mathbb{Z}}_6$.

Definition 2

Let p = 6f + 1 be a prime and m ⩾ 1. Using C_l in (1) with d = 6, we construct a p^m-periodic binary sequence {e_t}_t⩾0 by defining

$$e_t=\left\{\begin{array}{cc}1\text{,}\hfill & \mathrm{if}t(\mathrm{mod}{p}^m)\in C_u\cup C_v\cup C_w\text{,}\hfill \\ 0\text{,}\hfill & \mathrm{otherwise}\text{,}\hfill \end{array}\right.t⩾0\text{,}$$

where $(u\text{,}v\text{,}w)\in \mathcal{A}⧹\{\{0\text{,}2\text{,}4\}\text{,}\{1\text{,}3\text{,}5\}\}$.

We remark that if {u, v, w} = {0, 2, 4} or {1, 3, 5}, {e_t}_t⩾0 was considered in  [18], [12]. So we exclude them here. Clearly {e_t}_t⩾0 is a generalization and modification of the Hall’s sequences[9], [10]. We will present the linear complexity and trace function representation for {e_t}_t⩾0.