On the stability of periodic binary sequences with zone restriction

TL;DR

This paper introduces a zone-restricted $k$-error linear complexity measure to efficiently analyze the local and global stability of periodic binary sequences, especially for sequences with specific period structures.

Contribution

It proposes a novel zone restriction approach for $k$-error linear complexity, enabling efficient stability analysis and complete spectrum determination for certain classes of sequences.

Findings

$k$-error linear complexity equals zone-restricted complexity for specific sequence classes.

Zone length can be much smaller than the period when the complexity is large.

Complete spectrum of 1-error linear complexity is determined for $2^n$-periodic sequences.

Abstract

Traditional global stability measure for sequences is hard to determine because of large search space. We propose the $k$-error linear complexity with a zone restriction for measuring the local stability of sequences. Accordingly, we can efficiently determine the global stability by studying a local stability for these sequences. For several classes of sequences, we demonstrate that the $k$-error linear complexity is identical to the $k$-error linear complexity within a zone, while the length of a zone is much smaller than the whole period when the $k$-error linear complexity is large. These sequences have periods $2^n$, or $2^v r$ ($r$ odd prime and $2$ is primitive modulo $r$), or $2^v p_1^{s_1} \cdots p_n^{s_n}$ ($p_i$ is an odd prime and $2$ is primitive modulo $p_i$ and $p_i^2$, where $1\leq i \leq n$) respectively. In particular, we completely determine the spectrum of $1$-error linear complexity with any zone length for an arbitrary $2^n$-periodic binary sequence.

Tables

Table 1. Table 1: L 1 ; Z subscript 𝐿 1 𝑍 L_{1;Z} values at different intervals

Table 2. Table 2: ℰ N , 1 ; Z subscript ℰ 𝑁 1 𝑍 \mathcal{E}_{N,1;Z} for N = 2 8 𝑁 superscript 2 8 N=2^{8}

$Z$	${ℰ}_{N,1;Z}$
1	254.0000
2	253.5000
3	253.2500
4	253.0000
5	252.9375
6	252.8750
7	252.8125
8	252.7500

Equations

$$L_{N,k}(S)=\underset{W_H(T)\le k}{\min }\{L(S+T)\},$$

$$L_{N,k;Z}(S)=\underset{\genfrac{}{}{0ex}{}{W_H(T)\le k}{T[i]=0,Z\le i<N}}{\min }\{L(S+T)\}.$$

$$L(S+E_m^{{}^{\prime }})=L\left(S+E_m+E_1(i_m)+E_1(i_m\mathrm{mod}{2}^s)\right).$$

$$L(S+E_m)=L_k>2^n-2^s\ge L(E_1(i_m)+E_1(i_m\mathrm{mod}{2}^s)),$$

$$L_{1,Z}(S)\in \{L_1,2^n-2^{s-1},\dots ,2^n-2^a\}.$$

$$\displaystyle\left\{\begin{array}[]{ll}Z\cdot 2^{L_{1}-1},&if~{}L_{1,Z}=L_{1,Z}(S)=L_{1};\\ Z\cdot 2^{L_{1}+s-t-2},&if~{}L_{1,Z}=2^{n}-2^{t},where\;a+1\leq t\leq s-1;\\ 2^{L_{1}-1+s}-Z\cdot 2^{L_{1}+s-a-2},&if~{}L_{1,Z}=2^{n}-2^{a}.\end{array}\right.$$

$$L_{1,Z}(S)\in \{0,2^n-2^{n-1},\dots ,2^n-2^a\}.$$

$$\displaystyle\left\{\begin{array}[]{ll}Z,&if~{}L_{1,Z}=L_{1};\\ Z\cdot 2^{n-t-1},&if~{}L_{1,Z}=2^{n}-2^{t},~{}where~{}a+1\leq t\leq n-1;\\ 2^{n}-Z\cdot 2^{n-a-1},&if~{}L_{1,Z}=2^{n}-2^{a}.\end{array}\right.$$

$$\displaystyle\!\!\!\!\!\!\!\!\left\{\begin{array}[]{ll}2^{c-1}+2^{c-1+s},&\!\!\!\!\!\!\!\!\mbox{if}~{}s\leq a,2^{n}-2^{s}<c<2^{n}-2^{s-1};\\ (1+Z)2^{c-1},&\!\!\!\!\!\!\!\!\mbox{if}~{}s>a,2^{n}-2^{s}<c<2^{n}-2^{s-1};\\ \displaystyle{\sum_{s=a+1}^{n}\sum_{L_{1}=2^{n}-2^{s}+1}^{2^{n}-2^{s-1}-1}(1-\frac{Z}{2^{a+1}})\cdot 2^{L_{1}-1+s}+(1-\frac{Z}{2^{a+1}})\cdot 2^{n}+2^{2^{n}-2^{a}-1}},&\mbox{if}~{}c=2^{n}-2^{a};\\ \displaystyle{\sum_{s=t+1}^{n}\sum_{L_{1}=2^{n}-2^{s}+1}^{2^{n}-2^{s-1}-1}\frac{Z}{2^{t+1}}\cdot 2^{L_{1}-1+s}+\frac{Z}{2^{t+1}}\cdot 2^{n}+2^{2^{n}-2^{t}-1}},&\mbox{if}~{}c=2^{n}-2^{t},a<t\leq s-1;\\ 1+Z,&\mbox{if}~{}c=0.\\ \end{array}\right.$$

$$\sum _{j=0}^k\left(\genfrac{}{}{0ex}{}{N}{j}\right)<2^{{\min }_{2\le i\le h}|C_i|},$$

$$L(S)=N-c>L_{N,k}(S)\ge N-\min (2^v,r-2),$$

$$L(S+E_m)=L_{N,k}(S)\text{mbox}andSupp(E_m)\subseteq [0,Z),$$

$$\sum _{t=0}^k\left(\genfrac{}{}{0ex}{}{N}{t}\right)<\frac{2^{r-1}}{2^v},$$

$$\mathrm{Pr}(L(S)\ge N-c,L_{N,k}(S)\ge N-\min (2^v,r-2))\to 1-2^{-1-c},\text{mbox}asr\to \infty .$$

$${\mathcal{M}}_{N,k}(N-2^v-1)\le \min \left(2^N,{\mathcal{M}}_{N,0}(N-2^v-1)\sum _{t=0}^k\left(\genfrac{}{}{0ex}{}{N}{t}\right)\right).$$

$${\mathcal{M}}_{N,0}(N-2^v-1)=2^N-(2^{r-1}-1{)}^{2^v}\cdot 2^{2^v}.$$

$$\displaystyle\sum_{t=0}^{k}{N\choose t}\leq{{\color[rgb]{0,0,0}\frac{2^{N}}{\mathcal{M}_{N,0}(N-2^{v}-1)}=}}\frac{1}{1-(1-\frac{1}{2^{r-1}})^{2^{v}}},$$

$$\rho$$

$$\displaystyle\left(1-\frac{1}{2^{r-1}}\right)^{2^{v}}=\sum_{d=0\atop s=2d}^{2^{v-1}-1}\Big{(}{2^{v}\choose s}\cdot\frac{1}{2^{(r-1)\cdot s}}-{2^{v}\choose s+1}\cdot\frac{1}{2^{(r-1)\cdot(s+1)}}\Big{)}+\frac{1}{2^{(r-1)\cdot 2^{v}}}.$$

$$\frac{\left(\genfrac{}{}{0ex}{}{2^v}{s}\right)\cdot \frac{1}{2^{(r-1)\cdot s}}}{\left(\genfrac{}{}{0ex}{}{2^v}{s+1}\right)\cdot \frac{1}{2^{(r-1)\cdot (s+1)}}}=\frac{2^{r-1}(s+1)}{2^v-s}>\frac{2^r}{2^v}>1.$$

$$\displaystyle\big{(}1-\frac{1}{2^{r-1}}\big{)}^{2^{v}}>1-\frac{2^{v}}{2^{r-1}}.$$

$$\rho >1-\frac{2^v}{2^{r-1}}\sum _{t=0}^k\left(\genfrac{}{}{0ex}{}{N}{t}\right).$$

$$\rho \to 1\text{mbox}asr\to \infty .$$

$$\displaystyle\frac{|S:L(S)=N-c|}{|S:L(S)\geq N-2^{v}|}{{\color[rgb]{0,0,0}\rightarrow}}\frac{1}{2^{c+1}}.$$

$$Pr(L_{N,k}(S)<N-r+2)\le Pr(L(S)<N-r+2)\sum _{t=0}^k\left(\genfrac{}{}{0ex}{}{N}{t}\right),$$

$$\mathrm{Pr}(L(S)\ge N-c,L_{N,k}(S)\ge N-r+2)\to 1-2^{-1-c},$$

$$\sum _{t=0}^k\left(\genfrac{}{}{0ex}{}{N}{t}\right)\le 2^{NH(\frac{k}{N})},$$

$$NH(\frac{k}{N})<r-1-v,$$

$$\displaystyle{{\color[rgb]{0,0,0}1-X^{N}=(1-X^{r})^{2^{v}}=\left(\prod_{d|r}\Phi_{d}(X)\right)^{2^{v}}}},$$

$$X^N-1={\left((X-1){\Phi }_p(X){\Phi }_{p^2}(X)\cdots {\Phi }_{p^s}(X)\right)}^{2^v}.$$

Full text

Accordingly, we can efficiently determine the global stability by studying a local stability for these sequences. For several classes of sequences, we demonstrate that the $k$-error linear complexity is identical to the $k$-error linear complexity within a zone, while the length of a zone is much smaller than the whole period when the $k$-error linear complexity is large. These sequences have periods $2^{n}$, or $2^{v}r$ ($r$ odd prime and $2$ is primitive modulo $r$), or $2^{v}p_{1}^{s_{1}}\cdots p_{n}^{s_{n}}$ ($p_{i}$ is an odd prime and $2$ is primitive modulo $p_{i}$ and $p_{i}^{2}$, where $1\leq i\leq n$) respectively. In particular, we completely determine the spectrum of $1$-error linear complexity with any zone length for an arbitrary $2^{n}$-periodic binary sequence.

Keywords Stability $\cdot$ Linear complexity $\cdot$ $k$-linear complexity $\cdot$ Zone restriction

Mathematics Subject Classification (2010) 94A60 $\cdot$ 94A55 $\cdot$ 65C10 $\cdot$ 68P25

1 Introduction

Let $S=(s_{0},s_{1},s_{2},\ldots)$ be an $N$-periodic sequence with terms in the finite field $\mathbb{F}_{q}$ of $q$ elements. We note that $N$ need not be the least period of the sequence. We denote $S=(s_{0},s_{1},\ldots,s_{N-1})^{\infty}$ and define $S^{N}(x)=s_{0}+s_{1}x+\cdots+s_{N-1}x^{N-1}$. The linear complexity of a periodic sequence over $\mathbb{F}_{q}$ is the length of the shortest linear recurrence relation which the sequence satisfies. In algebraic terms the linear complexity of an $N$-periodic sequence is given by L(S)=N-\deg(\gcd(1-x^{N},{{\color[rgb]{0,0,0}S^{N}(x)}})); see for example [3, p. 28].

For an integer $k$, $0\leq k\leq N$, the minimum linear complexity of those sequences with not more than $k$ term changes in a period $N$ from the original sequence $S$ is called the $k$-error linear complexity of $S$, denoted as $L_{N,k}(S)$, i.e.,

[TABLE]

where $T$ is an $N$-periodic sequence, $W_{H}(T)$ is the Hamming weight of $T$ in one period, the addition “$+$” for two sequences is defined elementwise in $\mathbb{F}_{q}$. A sequence $T$ reaching the $L_{N,k}(S)$ is called an error vector of the $k$-error linear complexity. When $N=2^{n}$, we denote the $k$-error linear complexity of $S$ by $L_{k}(S)$.

In addition to the Berlekamp-Massey algorithm [11] for computing the linear complexity with computational complexity $O(N^{2})$, there are efficient algorithms of several types of periodic sequences with computational complexity $O(N)$, such as the Games-Chan algorithm [7] for computing the linear complexity of a $2^{n}$-periodic binary sequence; the algorithm due to Meidl [13] for computing the linear complexity of a $u2^{n}$-periodic binary sequence, where $u$ is odd ; the algorithm for computing the linear complexity of a sequence with period $p^{n}$ over ${\mathbb{F}}_{q}$ [25], where $p$ is an odd prime and $q$ is a prime and a primitive root $\bmod\;p^{2}$; and the algorithm for computing the linear complexity of a sequence with period $2p^{n}$ over ${\mathbb{F}}_{q}$ [24], where $p$ and $q$ are odd primes, and $q$ is a primitive root $\bmod\;p^{2}$. These algorithms work because the factorization of $X^{N}-1$ is simple under these assumptions. Comparatively, there are also efficient algorithms of computing the $k$-error linear complexity for certain types of sequences such as the Stamp-Martin algorithm [19] for computing the $k$-error linear complexity of a $2^{n}$-periodic binary sequence, the algorithm for computing the $k$-Error Linear Complexity of $p^{n}$-periodic sequences over ${\mathbb{F}}_{p^{m}}$ [9], and the algorithm for computing the $k$-error linear complexity of a sequence with period $2p^{n}$ over ${\mathbb{F}}_{q}$ [26]. We also remark that there are some studies on the properties of $k$-error linear complexity of binary sequences, see [8], [22]. Earlier, S$\breve{a}$l$\breve{a}$gean et al. studied approximation algorithms for the $k$-error linear complexity of odd-periodic binary sequences by using DFT and some relaxation [1, 18]. However, there is no efficient algorithm for calculating the $k$-error linear complexity for an arbitrary binary sequence, in particular, binary sequence with even period.

A well-designed sequence should not only have a large linear complexity, but also large $k$-error linear complexities. This means its linear complexity should not decrease a lot when $k$ errors occur; see [19] and [4]. In order to measure the stability of a given periodic sequence, we have to consider $k$ errors that can occur anywhere within the whole period $N$. This means the computational task is heavy because the capacity of search space for all possible binary errors is $\sum_{t=0}^{k}{N\choose t}$, which is very large for common $N$ and moderate $k$. Indeed, it becomes exponential of $N$ when $k$ is large, resulting in infeasible computations. This motivates us to study $k$-error linear complexity with a zone restriction. Intuitively, there can be many error vectors that reaching the $k$-error linear complexity. We show later on that for many sequences we can find a window of proper length $Z$ containing at least one error vector, no matter where we start with. For the convenience, we may assume the zone starts at position 0 and ends at position $Z-1$. Therefore we define the $k$-error linear complexity with a zone of length $Z$, denote by $(N,k;Z)$-error linear complexity, as the minimum of all $k$-error linear complexity such that these errors occur in positions from [math] to $Z-1$. That is,

[TABLE]

Obviously, $L_{N,k;Z}(S)$ is easier to compute and this provides a natural upper bound of $L_{N,k}(S)$.

In this paper, we study the relation between $L_{N,k;Z}(S)$ and $L_{N,k}(S)$ and prove that $L_{N,k;Z}(S)=L_{N,k}(S)$ for several classes of sequences and the zone length $Z$ can be very small comparing to the period $N$. This means that we can efficiently determine the global stability via a local stability. We focus on binary sequences with even period and large linear complexity, in particular, several classes of sequences with periods $2^{n}$, or $2^{v}r$ ($r$ odd prime and $2$ is primitive modulo $r$), or $2^{v}p_{1}^{s_{1}}\cdots p_{n}^{s_{n}}$ ($p_{i}$ is an odd prime and $2$ is primitive modulo $p_{i}$ and $p_{i}^{2}$, where $1\leq i\leq n$) respectively.

Sequences with period $2^{n}$ have attracted a lot of attention [5]; one typical example is the de Bruijin sequence of maximal $2^{n}$-periodic sequence generated by NFSR of stage $n$ [2]. Despite that there is an efficient algorithm to compute $k$-error linear complexity of these sequences, we still demonstrate our method by showing that there exists a small zone of length $Z=2^{\lceil\log_{2}(2^{n}-L_{k}(S))\rceil}$ containing the support (positions of nonzero entries) of an error vector reaching the $k$-error linear complexity for any $2^{n}$-periodic binary sequence. This means that we can indeed reduce the global stability to a local stability. Furthermore, we completely describe the spectrum of 1-error linear complexity with any given zone length. This can help us to obtain the exact counting functions for any $2^{n}$-periodic binary sequence.

Afterwards, we found two more classes of binary sequences such that their global stability can be reduced to a local stability. The first class of sequences has large linear complexity and $k$-error linear complexity with period $2^{v}r$, such that $r$ is an odd prime and $2$ is a primitive root modulo $r$. The length of a zone is $Z=2^{\lceil\log_{2}(N-L_{N,k}(S))\rceil}$. More details can be found in Theorem 4.1. We want to emphasize that our result applies to quite a lot of sequences. By Artin’s conjecture, approximately 37% of all primes satisfy that $2$ is a primitive root modulo $r$. We also justify that there are high proportion of sequences who have the required large linear complexity and $k$-error linear complexity, among those sequences with period $2^{v}r$ where $2$ is primitive modulo $r$. In particular, we show that if the growth of $p$ is polynomial in terms of $r$ and $\sum_{t=0}^{k}{2^{v}r\choose t}<\frac{2^{r-1}}{2^{v}}$, then almost all these sequences have desired properties so that their global stability can be reduced to local stability. The second class of sequences has the period $N=2^{v}p_{1}^{s_{1}}p_{2}^{s_{2}}\ldots p_{n}^{s_{n}}$, where $p_{i}$ is odd prime and $2$ is a primitive root modulo $p_{i}$ and $p_{i}^{2}$ for all $1\leq i\leq n$. For any such $N$-periodic binary sequence $S$ such that $L(S)>L_{N,k}(S)\geq N-\min(2^{v},p_{1}-2,\ldots,p_{n}-2)$, we show in Theorem 4 that there exists a zone of length $Z=2^{\lceil\log_{2}(N-L_{N,k}(S))\rceil}$ such that $L_{N,k}(S)=L_{N,k;Z}(S)$.

The rest of this paper is organized as follows. In Section 2, we study the $(k;Z)$-error linear complexity for any periodic binary sequence with period $2^{n}$, and find a proper zone of length $Z=2^{\lceil\log_{2}(2^{n}-L_{k}(S))\rceil}$ such that $L_{N,k}=L_{N,k;Z}$. The larger $L_{N,k}$, the smaller zone length $Z$. In Section 3, we study the linear complexity affected by 1-error occurrence within a zone of length $Z$, and give the exact counting functions of the $1$-error linear complexity with a restriction on zone length $Z$ for a random $2^{n}$-periodic binary sequence. In Section 4, we prove Theorems 3 and Theorem 4.

2 Reduction from global stability to local stability with a zone

restriction for any binary sequence of period $2^{n}$

In this section, we show that the global stability can be reduced to local stability with zone restriction for any binary sequence of period $2^{n}$. We denote the binary sequence with the only nonzero entry ‘$1$’ at position $j$ by $E_{1}(j)$, $0\leq j<2^{n}$, in each period $2^{n}$, and the expected $(k;Z)$ linear complexity of $N$-periodic sequences by $\mathcal{E}_{N,k;Z}$. Without causing any confusion, we denote $E_{k;Z}$ by the expected $(k;Z)$ linear complexity of $2^{n}$-periodic binary sequences, and $\mathcal{N}_{k}(c)$ by the number of sequences achieving $k$-error linear complexity value $c$ of $2^{n}$-periodic binary sequences. Because the sequence $S$ has period $2^{n}$, we only need to consider the multiplicity of $x=1$ as a root of $S^{2^{n}}(x)$ when we compute the linear complexity $L(S)=2^{n}-deg(gcd(1-X^{2^{n}},S^{2^{n}}(X))$. It is straightforward to derive the following useful result.

Lemma 1.

$L(S+S^{\prime})=\max\{L(S),L(S^{\prime})\}$* for two $2^{n}$-periodic sequences $S,S^{\prime}$ if $L(S^{\prime})\neq L(S)$, and $L(S+S^{\prime})<L(S)=L(S^{\prime})$ if $L(S)=L(S^{\prime})$. In particular $L(E_{1}(j)+E_{1}(j+i2^{s}))\leq 2^{n}-2^{s}$, where $1\leq s<n$.*

Proof.

Obviously, we can write $S^{2^{n}}(x)=(1-x)^{2^{n}-L(S)}g_{S}(x)$ for the sequence $S$, where $g_{S}(1)=1$. Simlarly, $S^{\prime 2^{n}}(x)=(1-x)^{2^{n}-L(S)}g_{S^{\prime}}(x)$, where $g_{S^{\prime}}(1)=1$. If $L(S^{\prime})\neq L(S)$, then $S^{2^{n}}(x)+s^{\prime 2^{n}}(x)=(1-x)^{2^{n}-\max\{L(S),L(S^{\prime})\}}\tilde{g}(x)$, and $\tilde{g}(1)=1$. Therefore we have $L(S+S^{\prime})=\max\{L(S),L(S^{\prime})\}$. If $L(S^{\prime})=L(S)$, we obtain $S^{2^{n}}(x)+S^{\prime 2^{n}}(x)=(1-x)^{2^{n}-L(S)}(g_{S}(x)+g_{S^{\prime}}(x))$, and $g_{S}(1)+g_{S^{\prime}}(1)=0$. Therefore, $L(S+S^{\prime})<L(S)=L(S^{\prime})$. In particular, the degree of $\gcd(x^{j}+x^{j+i2^{s}},x^{2^{n}}-1)$ is at least $2^{s}$ because $x^{j}+x^{j+i2^{s}}=x^{j}(1+x^{i})^{2^{s}}$. ∎

It is well known from [12] and [22] that $L_{k}\neq 2^{n}-2^{s}$ for any integer $s<n$, $k>0$. In particular, when $k=1$, we can determine the number of error vectors $E_{1}(j)$ such that $L(S+E_{1}(j))=L_{1}(S)$.

Lemma 2.

For a sequence $S$ satisfying $L(S)=2^{n}$ and $2^{n}-2^{s}<L_{1}(S)<2^{n}-2^{s-1}$, we have exact $2^{n-s}$ error vectors with Hamming weight $1$ in one period achieving $L_{1}(S)$.

Proof.

Suppose there is $E_{1}(j)$ such that $0\leq j<2^{s}$ and $L(S+E_{1}(j))=L_{1}(S)$, we claim that we must have a set of error vectors at positions $j+i2^{s}$, where $0\leq i<2^{n-s}$ and the addition $+$ is performed modulo $2^{n}$, such that $L(S+E_{1}(j+i2^{s}))=L(S+E_{1}(j)+E_{1}(j)+E_{1}(j+i2^{s}))=L(S+E_{1}(j))$. Since L(E_{1}(j)+E_{1}(j+i2^{s})){{\color[rgb]{0,0,0}\leq}}2^{n}-2^{s}<L_{1}(S)=L(S+E_{1}(j)), we conclude the above claim by Lemma 1.

For any other error vector $E_{1}(j^{\prime})$ such that $j^{\prime}-j\equiv d~{}(mod~{}2^{s})$ and $0<d<2^{s}$, the largest nonnegative integer $t$ such that $2^{t}\mid(j^{\prime}-j)$ must satisfy $t<s$. Hence the degree of $\gcd(x^{j}+x^{j^{\prime}},x^{2^{n}}-1)$ is exactly $2^{t}\leq 2^{s-1}$ and thus $L(E_{1}(j)+E_{1}(j^{\prime}))=2^{n}-2^{t}\geq 2^{n}-2^{s-1}>L_{1}(S)$. Therefore $L(S+E_{1}(j^{\prime}))=L(S+E_{1}(j)+E_{1}(j)+E_{1}(j^{\prime}))=L(E_{1}(j)+E_{1}(j^{\prime}))>L_{1}(S)$. This shows that there is exactly one error vector $E_{1}(j)$ such that $0\leq j<2^{s}$ and $L(S+E_{1}(j))=L_{1}(S)$. Hence there are exactly $2^{n-s}$ error vectors in the whole period achieving $L_{1}(S)$. ∎

For any positive integer $m$, we let $E_{m}$ denote a binary vector of length $2^{n}$ with Hamming weight $m$. For example, assume $E_{m}$ has ‘1’ at positions $i_{1},i_{2},\ldots,i_{m}$, where $0\leq i_{1}<i_{2}<\ldots<i_{m}\leq 2^{n}-1$. Then we define the support of $E_{m}$ as $Supp(E_{m})=\{i_{1},i_{2},\ldots,i_{m}\}$. Now we can show that there exist at least one error vector whose support is contained in a smaller zone.

Lemma 3.

Let $S$ be a binary sequence with period $2^{n}$. Suppose $2^{n}-2^{s}<L_{k}(S)<2^{n}-2^{s-1}$ for some integer $s$. Then there exists at least one error vector $E_{m}$ of weight $m$, $m\leq k$ such that $L(S+E_{m})=L_{k}(S)$ and $supp(E_{m})\subseteq[0,2^{s})$.

Proof.

There exists an error vector $E_{m}$, $m\leq k$ of Hamming weight $m$ so that $L(S+E_{m})=L_{k}$. If $i_{m}\geq 2^{s}$, then we can define a new vector $E_{m}^{{}^{\prime}}:=E_{m}+E_{1}(i_{m})+E_{1}(i_{m}\bmod{2^{s}})$ so that

[TABLE]

Because

[TABLE]

we get $L(S+E_{m^{\prime}})=L_{k}$, where $Supp(E_{m}^{\prime})=Supp(E_{m})\setminus\{i_{m}\}\cup\{i_{m}\bmod{2^{s}}\}$. Therefore, we can consecutively adjust those entries of $E_{m}$ so that we can find $\bar{E}_{m}$ such that $supp(\bar{E}_{m})\subseteq[0,2^{s})$ and $L(S+\bar{E}_{m})=L_{k}$. ∎

Remark 1.

Actually, Lemma 3 can be extended as the following: there exists at least one error vector $E_{m}$ such that $L(S+E_{m})=L_{k}(S)$ and $supp(E_{m})\subseteq[a,a+2^{s})$, for any $0\leq a\leq N-2^{s}$. In Definition (1), we just set the default zone starting at 0.

Because of the assumption $2^{n}-2^{s}<L_{k}(S)<2^{n}-2^{s-1}$, we derive $2^{s-1}<2^{n}-L_{k}(S)<2^{s}$. Let $Z=2^{\lceil\log_{2}(2^{n}-L_{k}(S))\rceil}$. Then $L_{k;Z}(S)\leq L_{k}(S)$ by Lemma 3. Conversely, it is always true that $L_{k}(S)\leq L_{k;Z}(S)$. Therefore we obtain the following theorem.

Theorem 2.1.

Let $S$ be any $2^{n}$-periodic binary sequence. For any positive integer $k$, there always exists $k^{\prime}\leq k$ such that $k^{\prime}\leq Z=2^{\lceil\log_{2}(2^{n}-L_{k}(S))\rceil}$ and $L_{k}(S)=L_{k^{\prime}}(S)=L_{k^{\prime};Z}(S)$.

Proof.

If $k\leq Z$, then $L_{k}(S)=L_{k;Z}(S)$. Otherwise if $k>Z$, then we can find an error vector $E_{k^{\prime}}$ with the support in $[0,Z)$ such that $L_{k}(S)=L(S+E_{k^{\prime}})$. The proof is complete because $L_{k^{\prime}}(S)=L_{k^{\prime};Z}(S)$. ∎

Theorem 2.1 shows that we can efficiently verify the global stability of a binary sequence of period $2^{n}$ with large $k$-error linear complexity via a local stability. If $L_{k}(S)$ is big, then $Z$ can be very small. If $k$ is large, then we can reduce $k$ so that it is bounded by the zone length as well. Of course, there is the degenerated case when $L_{k}(S)=0$, in this case, we have to set $Z=2^{n}$. However, as we commented earlier, we focus on sequences with large $k$-error linear complexity and thus the zone length is significantly reduced.

3 Spectrum of $1$-error linear complexity with arbtrary zone length

In this section we assume $N=2^{n}$ and $n\geq 4$. It is well known that the linear complexity of a $2^{n}$-periodic sequence $S$ is $2^{n}$ if and only if it has odd Hamming weight. The 1-error linear complexity of a $2^{n}$-periodic sequence can be any integer between [math] and $2^{n}-1$. However, if $S$ has an odd Hamming weight, then $L_{1}(S)$ can not be any integer of the form $2^{n}-2^{s}$ where $1\leq s\leq n$. For more details we refer the reader to [6, 12, 20, 21, 23].

If $S$ is a $2^{n}$-periodic sequence with even Hamming weight, then $L_{1}(S)=L_{0}(S)$. In this case, $L_{1,Z}(S)=L_{0}(S)$ for any zone of length $Z$. In order to study the distribution of $L_{1,Z}$, we only need to consider $2^{n}$-periodic sequences with odd Hamming weight.

Theorem 3.1.

Let $S$ be a $2^{n}$-periodic sequence with odd Hamming weight and $L_{1}$ be its $1$-error linear complexity. Let $0<Z\leq 2^{n}$, and $a=\lfloor\log_{2}(Z)\rfloor$.

For $2^{n}-2^{s}<L_{1}<2^{n}-2^{s-1}$ with some integer $s$, we have the following

• (i)

if $s\leq a$, then $L_{1;Z}(S)=L_{1}$, and the number of such sequences $S$ is $2^{L_{1}-1+s}$;

• (ii)

if $s>a$, then we have

[TABLE]

The number of all the sequences $S$ that achieve these values equals

[TABLE] 2. 2.

For $L_{1}=0$, we have

[TABLE]

The number of all the sequences $S$ that achieve these values equals

[TABLE]

Proof.

For any $2^{n}$-periodic sequence $S$ with odd Hamming weight, we have $2^{n}-2^{s}<L_{1}(S)<2^{n}-2^{s-1}$ for some positive integer $s$ or $L_{1}=0$. Let us first assume that $2^{n}-2^{s}<L_{1}(S)<2^{n}-2^{s-1}$ for some positive integer $s$. From the proof of Lemma 2, there exists exactly one $j$, $0\leq j<2^{s}$, such that $L_{1}(S)=L(S+E(j))$.

(i) if $s\leq a$, then $2^{s}\leq 2^{a}\leq Z$ by the definition of $a=\lfloor\log_{2}(Z)\rfloor$. Hence, in the zone of length $Z$, there is at least one error vector reaching the $L_{1}(S)$, so $L_{1;Z}(S)=L_{1}(S)$.

(ii) if $s>a$, then $Z<2^{s}$. Consider $0\leq j<2^{s}$ such that $L_{1}(S)=L(S+E(j))$. Let $m\cdot 2^{a+1}\leq j<(m+1)2^{a+1}$ for some nonnegative integer $m$. Let $\bar{j}=j-m2^{a+1}$ be the positive integer less than $2^{a+1}$ such that $\bar{j}\equiv j\bmod{2^{a+1}}$. If $Z<\bar{j}<2^{a+1}$, then we take $j^{\prime}=\bar{j}-2^{a}$, which satisfies $0<j^{\prime}<2^{a}<Z$. Since $j^{\prime}-j$ is an odd multiple of $2^{a}$, we conclude that $L(E_{1}(j)+E_{1}(j^{\prime}))=2^{n}-2^{a}$ because the multiplicity of the root $1$ for the binomial $x^{j}+x^{j^{\prime}}$ is exactly $2^{a}$. Hence $L(S+E_{1}(j))=L_{1}(S)<2^{n}-2^{s-1}\leq 2^{n}-2^{a}=L(E_{1}(j)+E_{1}(j^{\prime}))$. Therefore $L(S+E_{1}(j^{\prime}))=\max\{L(S+E_{1}(j)),L(E_{1}(j)+E_{1}(j^{\prime}))\}=2^{n}-2^{a}$. And for any $\hat{j}$ satisfying $Z<\hat{j}<2^{a+1}$, $\hat{j}\neq j^{\prime}$, we have $L(E_{1}(j^{\prime})+E_{1}(\hat{j}))>2^{n}-2^{a}$ because $|\hat{j}-j^{\prime}|<2^{a}$, accordingly we have $L(S+E_{1}(\hat{j}))=\max\{L(S+E_{1}(j^{\prime})),L(E_{1}(\hat{j}))+E_{1}(j^{\prime}))\}>2^{n}-2^{a}$. Thus, $L_{1;Z}=2^{n}-2^{a}$.

On the other hand, we assume 0\leq\bar{j}{{\color[rgb]{0,0,0}\leq}}Z. For $m=0$, we must have $L_{1;Z}=L_{1}$ because $0\leq j<Z$. Let $m\geq 1$. We claim $L_{1;Z}=2^{n}-2^{a+1+\nu_{2}(m)}$ where $\nu_{2}(m)$ is the largest integer such that $2^{\nu_{2}(m)}\mid m$. Indeed, we take $j^{\prime}=j-m\cdot 2^{a+1}$. which satisfies $0<j^{\prime}<Z$. Since $j-j^{\prime}=m\cdot 2^{a+1}$, we conclude that $L(E_{1}(j)+E_{1}(j^{\prime}))=2^{n}-2^{a+1+\nu_{2}(m)}$ because the multiplicity of the root $1$ for the binomial $x^{j}+x^{j^{\prime}}$ is exactly $2^{a+1+\nu_{2}(m)}$. We note that $a+1+\nu_{2}(m)\leq s-1$ because $j-j^{\prime}=m2^{a+1}<2^{s}$. Hence $L(S+E(j))=L_{1}(S)<2^{n}-2^{s-1}\leq 2^{n}-2^{a+1+\nu_{2}(m)}=L(E_{1}(j)+E_{1}(j^{\prime}))$. Therefore $L(S+E_{1}(j^{\prime}))=\max\{L(S+E_{1}(j)),L(E_{1}(j)+E_{1}(j^{\prime}))\}=2^{n}-2^{a+1+\nu_{2}(m)}$. And for any $\hat{j}$ satisfying $0\leq\hat{j}<Z$, $\hat{j}\neq j^{\prime}$, we have $L(E_{1}(j^{\prime})+E_{1}(\hat{j}))\geq 2^{n}-2^{a}$ because $|\hat{j}-j^{\prime}|<2^{a+1}$, accordingly we have $L(S+E_{1}(\hat{j}))=\max\{L(S+E_{1}(j^{\prime})),L(E_{1}(\hat{j}))+E_{1}(j^{\prime}))\}\geq 2^{n}-2^{a}$. In this case, $L_{1;Z}=2^{n}-2^{a+1+\nu_{2}(m)}$. Hence $L_{1,Z}(S)\in\{L_{1},2^{n}-2^{s-1},\ldots,2^{n}-2^{a}\}$. We summarize these results in Table 1 and Figure 1.

Now we count the number of all these sequences having the $1$-error linear complexity $L_{1}$ and odd Hamming weight such that $L_{1;Z}(S)=2^{n}-2^{t}$ where $a+1\leq t\leq s-1$. For each sequence $S$ with $1$-error linear complexity $L_{1}$ and odd Hamming weight, we need to count the number of error positions $j$’s such that $L_{1;Z}=L(S+E_{1}(j))=2^{n}-2^{t}$. We prove that the proportion of $j$’s over an interval of length $2^{t+1}$ such that $L_{1;Z}=2^{n}-2^{t}$ is $Z/2^{t+1}$, where $t\geq a+1$.

First we show that every sub-interval $I$ of length $2^{t+1}$ in the interval $[0,2^{s})$ contains at least one interval of length $Z$ for possible $j$’s such that $L_{1;Z}=2^{n}-2^{t}$. We will construct an interval of length $Z$ for $j$’s within $I$. Because the length of $I$ is $2^{t+1}$, we can always choose an odd integer $m^{\prime}$ such that $[m^{\prime}2^{t},(m^{\prime}+1)2^{t})\subset I$. By the above proof, there exists an interval of length $Z$ within the interval $m^{\prime}2^{t}=(m^{\prime}2^{t-a-1})2^{a+1}\leq j<((m^{\prime}2^{t-a-1})+1)2^{a+1}$ such that $L_{1;Z}=2^{n}-2^{a+1+\nu_{2}(m^{\prime}2^{t-a-1})}=2^{n}-2^{a+1+t-a-1}=2^{n}-2^{t}$.

Then we show that $I$ can not contain more than one intervals of length $Z$ for possible $j$’s such that $L_{1;Z}=2^{n}-2^{t}$. We prove it by contradiction. Suppose there are $j,j^{\prime}$ such that $0<|j^{\prime}-j|<2^{t+1}$ and $L(S+E_{1}(j))=L(S+E_{1}(j^{\prime}))=2^{n}-2^{t}$. In this case, $L(E_{1}(j)+E_{1}(j^{\prime})$=$L(S+E_{1}(j)+S+E_{1}(j^{\prime}))<2^{n}-2^{t}$. However, the root $1$ of $x^{j}+x^{j\prime}$ is at most $2^{t}$ times, implying $L(E_{1}(j)+E_{1}(j^{\prime}))\geq 2^{n}-2^{t}$, a contradiction.

Therefore, the proportion of $j$’s in each interval of length $2^{t+1}$ such that $L_{1;Z}=2^{n}-2^{t}$ is $Z/2^{t+1}$, for each sequence having the $1$-error linear complexity $L_{1}$ and odd Hamming weight. Since there are $2^{L_{1}-1+s}$ sequences with odd Hamming weight such that 2^{n}-{{\color[rgb]{0,0,0}2^{s}}}<L_{1}<2^{n}-2^{s-1} (see [23][p. 2000, Theorem 3]), there are $Z2^{L_{1}-1+s}/2^{t+1}=Z2^{L_{1}+s-t-2}$ sequences having the $1$-error linear complexity $L_{1}$ and odd Hamming weight such that $L_{1;Z}(S)=2^{n}-2^{t}$. Similarly, for $L_{1;Z}=L_{1}$, there is only one internal of length $Z$ within the interval $[0,2^{s})$ and thus the proportion is $Z/2^{s}$. From Table 1 and Figure 1, the proportion of $j$’s giving $2^{n}-2^{a}$ is $1-\frac{Z}{2^{a+1}}$.

When $L_{1}=0$, the proof is similar and thus we omit the details. ∎

The distribution of $k$-error linear complexity is provided in [22] when $k\leq 4$. In particular, the number $N_{0}(c)$ of $2^{n}$-periodic sequences with the linear complexity $c$ is $2^{c-1}$; see [14]. In Theorem 3.1, we have counted the number of sequences with odd Hamming weight achieving $(1,Z)$-error linear complexity values. In the following, we count the number $\mathcal{N}_{1;Z}(c)$ of all sequences achieving $(1,Z)$-error linear complexity value $c$, without emphasizing on their Hamming weights.

Corollary 1.

Let $a=\lfloor\log_{2}(Z)\rfloor$. The value $\mathcal{N}_{1;Z}(c)$ is equal to

[TABLE]

Proof.

We note that every $2^{n}$-periodic sequence $S$ has a linear complexity $L(S)=c<2^{n}$ if and only if it has an even Hamming weight. In this case, $L_{1}(S)=L(S)$. Hence the result follows immediately from Theorem 3.1. ∎

The exact expectation $\mathcal{E}_{1;Z}$ can be derived from the above counting functions, and may be used as a measure for determining the randomness of a $2^{n}$-periodic binary sequence, with variations on $Z$. The exact formula is too complicated. Thus we omit all the details here. Instead, we provide a concrete example for the expected values of $L_{1;Z}$ for sequences with period $N=2^{8}$ in Table 2.

4 Extension to sequences with other even periods

Now we consider stability of other periodic sequences with even period $N=2^{v}r$ such that $r,v$ are positive integer and $r$ is odd. For some types of $2^{v}r$-periodic binary sequences, we can still find a proper zone of length $Z$ so that $L_{N,k}=L_{N,k;Z}$. From the paper by Niederreiter [16, Theorem 1, P. 503], there exists $2^{v}r$-periodic binary sequence $S$ with $L_{N,k}(S)\geq N-2^{v}$ and $L(S)=N$, provided that

[TABLE]

where $C_{2},\ldots,C_{h}$ are the different cyclotomic cosets modulo $r$. In the following, we will reveal that $L_{N,k}=L_{N,k;Z}$ with certain $Z\ll N$ for some of these ‘ideal’ cryptographic sequences.

Theorem 4.1.

*Let $N=2^{v}r$, $v>0$, $r$ be an odd prime, and $2$ be a primitive root modulo $r$. If $S$ is an $N$-periodic binary sequence such that *

[TABLE]

for some nonnegative integer $c$, then there exists at least one error vector $E_{m}$, $m\leq k$ such that

[TABLE]

where $Z=2^{\lceil\log_{2}(N-L_{N,k}(S))\rceil}$. In particular, if $L(S)=N>L_{N,1}(S)\geq N-\min(2^{v},r-2)$, then there exists exactly one error vector $E_{1}$ satisfying that $Supp(E_{1})\subseteq[0,Z)$ and $L_{N,1}(S)=L_{N,1;Z}(S)$.

Proof.

Because $2$ is primitive root modulo the prime number $r$, the cyclotomic polynomial $\phi_{r}(x)$ of the order $r$ is irreducible over $\mathbb{F}_{2}$. Hence $X^{2^{v}r}-1=(X^{r}-1)^{2^{v}}=((X-1)\phi_{r}(X))^{2^{v}}$. Let $\alpha$ be a primitive $r$-th root of unity. That is, $\phi_{r}(\alpha)=0$.

If $2^{v}\leq r-2$, then $\alpha$ can not be a root of $S^{N}(x)+E_{t}(x)$ for any error polynomial $E_{t}(x)$ of Hamming weight $t\leq k$ because of the assumption $L_{N,t}(S)\geq L_{N,k}(S)\geq N-2^{v}$. Otherwise, $\phi_{r}(x)\mid s^{N}(x)+E_{t}(x)$ and thus the greatest common divisor of $s^{N}(x)+E_{t}(x)$ and $X^{N}-1$ has degree greater than $2^{v}$, a contradiction.

If $2^{v}>r-2$, then $L_{N,k}(S)\geq N-r+2$. Similarly, $\alpha$ can not be a root of $S^{N}(x)+E_{t}(x)$ for any error polynomial $E_{t}(x)$ of Hamming weight $t\leq k$.

Now we only need to consider the multiplicity of root $1$ when computing $L_{N,k}(S)$. As in the proof of Lemma 3, we can derive an error vector $E_{m}$ such that $Supp(E_{m})\subseteq[0,Z)$ and $L_{k}(S+E_{m})=L_{N,k}(S)$. Indeed, suppose there exists an error vector $E_{m}$ such that $L(S+E_{m})=L_{N,k}(S)$, where $Supp(E_{m})=\{i_{1},i_{2},\ldots,i_{m}\}$. Note that $Z=2^{\lceil\log_{2}(N-L_{N,k}(S))\rceil}\geq N-L_{N,k}(S)$. If $i_{m}\geq Z$, then we can define a new vector $E_{m}^{\prime}=E_{m}+E_{1}(i_{m})+E_{1}(i_{m}\bmod Z)$. Because $L(S+E_{m})=L_{N,k}(S)\geq N-Z\geq L(E_{1}(i_{m})+E_{1}(i_{m}\bmod Z))$, we must have $L(S+E_{m}^{\prime})\leq L_{N,k}(S)$ by counting the multiplicity of $1$’s. Hence $L(S+E_{m}^{\prime})=L_{N,k}(S)$. Continuing this process, we can derive an error vector such that the support is contained $[0,Z)$.

In particular, if $L(S)=N>L_{N,1}(S)\geq N-\min(2^{v},r-2)$, then there exists $s>0$ such that $2^{s-1}<{N-L_{N,1}(S)}\leq 2^{s}$. From the previous discussion, $Z=2^{s}$ and there exists at least one error vector $E_{m}$ such that $m\leq 1$ such that $L(S+E_{m})=L_{N,1}(S)$ and $supp(E_{m})\subseteq[0,2^{s})$. Because $L(S)>L_{N,1}(S)$, we must have $m=1$. Suppose there are $E_{1},E_{1}^{\prime}$ such that $L(S+E_{1})=L(S+E_{1}^{\prime})=L_{N,1}(S)$ and $Supp(E_{1}),Supp(E_{1}^{\prime})\subseteq[0,2^{s})$. In this case, the multiplicity of root $1$ in $S^{N}(x)+E_{1}(x)$ and $S^{N}(x)+E_{1}^{\prime}(x)$ is greater than $2^{s-1}$ respectively, however the multiplicity of root 1 of the generating polynomial corresponding to $E_{1}+E_{1}^{\prime}$ is not more than $2^{s-1}$, a contradiction. ∎

Theorem 4.1 says that if $2^{v}\geq r-1$ then $Z=2^{\lceil\log_{2}(N-L_{N,k}(S))\rceil}\leq 2^{\lceil\log_{2}(r-2)\rceil}<2r$. On the other hand, if $2^{v}<r-1$ then Theorem 4.1 gives $Z\leq 2^{v}$ for any $N$-periodic binary sequence $S$ such that $L(S)=N-c\geq L_{N,k}(S)\geq N-2^{v}$.

Example 1.

Let $S_{1}$ be the following binary sequence with period $16*19$. Namely, $S_{1}=11100011001111011010100011011100110000001100000001101110010100000\\ 1101000110010010000100011001010010100110010111001001101011001000110100\\ 0000100100001111001000101011000010010111110101001111110101101111000110\\ 1001111101111110101011100100010001000011000011101111111111001011011011\\ 01001010011010010001001101000$. The linear complexity is $304$ and $1$-error linear complexity is $301$. The zone length is $4$ and we have $L_{304,1}(S_{1})=L_{304,1;4}(S_{1})$.

Example 2.

Let $S_{2}$ be a random generated binary sequence with period $11*16$, $S_{2}=11100111101100100110110111100100110101001010111111011011100010011\\ 0001101001110010011101010100000100011111000001000011101011101101110011\\ 10101000111010000100001101101001101110010$. Then $L(S_{2})=L_{1}(S_{2})=175$, $L_{2}(S_{2})=169$, and the length of zone is $8$. Indeed, we find an error vector of $L_{2}$ with two errors at positions $6$ and $7$ within the zone $[0,8)$.

We observe that the above result can be extended to any $N$-periodic binary sequence $S$ such that $L(S)=N-c\geq L_{N,k}(S)=N-r+1$ when $2^{v}<r-1$. In this case, we can take $Z=r$.

Proposition 1.

Let $N=2^{v}r$, $v>0$, $r$ be an odd prime, and $2$ be a primitive root. If $2^{v}<r-1$ and $S$ is an $N$-periodic binary sequence such that $L(S)=N-c\geq L_{N,k}(S)=N-r+1$ for some positve integer $c$, then there exists at least one error vector $E_{m}$, $m\leq k$ such that $L(S+E_{m})=L_{N,k}(S)$ and $Supp(E_{m})\subseteq[0,Z)$, where $Z=r$. In particular, $L_{N,k}(S)=L_{N,k;Z}(S)$.

Proof.

Let $\alpha$ be the primitive $r$-th root of unity. Since the multiplicity of 1 is at most $2^{v}<r-1$ and $L_{N,k}(S)=N-r+1$, there exists an error vector $E_{m}$, $m\leq k$ reaching $L_{N,k}$, such that the generating polynomial corresponding to $S+E_{m}$ will be divisible by $\phi_{r}(x)=1+X+\cdots+X^{r-1}$. Suppose $E_{m}$ has entry ‘1’ at positions $i_{1},i_{2},\ldots,i_{m-1},i_{m}$, where $i_{1}<i_{2}<\ldots<i_{m}$. If $i_{m}>r$, since the generating polynomial of $S+E_{m}+E_{1}(i_{m})+E_{1}(i_{m}\bmod r)$ will be also divisible by $1+X+\cdots+X^{r-1}$, implying L\big{(}S+E_{m}+E_{1}(i_{m})+E_{1}(i_{m}\bmod r)\big{)}\leq N-r+1. Moreover, $L(E_{1}(i_{m})+E_{1}(i_{m}\bmod r)=N-r$ and $L_{N,k}(S)=L(S+E_{m})\leq N-r+1$ imply that L\big{(}S+E_{m}+E_{1}(i_{m})+E_{1}(i_{m}\bmod r)\big{)}=N-r+1. Thus we get an error vector $E_{m}+E_{1}(i_{m})+E_{1}(i_{m}\bmod r)$ reaching $L_{N,k}=N-r+1$. Consequently, we obtain an error vector reaching $L_{N,k}(S)$ with support in $[0,r)$. ∎

Remark 2.

From Theorem 4.1 we obtain a small zone of length $Z\leq 2^{v}$ or $Z<2r$ such that $L_{N,k}=L_{N,k;Z}$ for the above classes of sequences with large linear complexity and $k$-error linear complexity. Our assumptions on these classes of sequences are not very restricted. By Artin’s conjecture, there are approximately $37\%$ of all primes having $2$ as a primitive root [15]. By the following corollary 2, we show that under certain conditions almost all random sequences have $k$-error linear complexity greater than or equal to $N-2^{v}$ and about $50\%$ of these sequences have linear complexity equal to the period. Therefore our result can be very useful to determine the stability of many random binary sequences with low computational cost.

Corollary 2.

Suppose $r$ is a prime, 2 is a primitive root modulo $r$ and $2^{v}\leq p(r)$, where $p(r)$ is a polynomial of the variable $r$. Let $N=2^{v}r$. If

[TABLE]

for $0\leq c<\min(2^{v},r-2)$, we have

[TABLE]

Proof.

For any positive intger $k$, we denote by $\mathcal{M}_{N,k}(c)$ the number of $N$-periodic sequences with the $k$-error linear complexity not more than $c$. Obviously,

[TABLE]

Then by Proposition 1 and Lemma 1 in [14] (page 2818), we have

[TABLE]

If $\mathcal{M}_{N,0}(N-2^{v}-1)\sum_{t=0}^{k}{N\choose t}\leq 2^{N}$, i. e.,

[TABLE]

then we have $\mathcal{M}_{N,k}(N-2^{v}-1)\leq\mathcal{M}_{N,0}(N-2^{v}-1)\sum_{t=0}^{k}{N\choose t}.$

Denote by $\rho$ the ratio of the number of periodic sequences satisfying $L_{N,k}(S)\geq N-2^{v}$ over the number of all periodic sequences with period $N$. Hence

[TABLE]

Note that

[TABLE]

For $d=0$ and $s=0$, we have ${2^{v}\choose s}\cdot\frac{1}{2^{(r-1)\cdot s}}-{2^{v}\choose s+1}\cdot\frac{1}{2^{(r-1)\cdot(s+1)}}=1-\frac{2^{v}}{2^{r-1}}$. Then for $d>0$, we have $s\geq 2$ and

[TABLE]

The last inequality holds because we have $r>v$ by the assumption. Therefore, we obtain

[TABLE]

If $\sum_{t=0}^{k}{N\choose t}<\frac{2^{r-1}}{2^{v}}$, then by (10) we have $\frac{2^{r-1}}{2^{v}}<\frac{1}{1-(1-\frac{1}{2^{r-1}})^{2^{v}}}$ and the condition (8) holds. Therefore, from (9) and (10) we derive

[TABLE]

For small $k$, we have $\sum_{t=0}^{k}{N\choose t}\leq c_{0}N^{k}$ for some constant $c_{0}$, and thus

$\frac{2^{v}}{2^{r-1}}\sum_{t=0}^{k}{N\choose t}\leq 2^{1-r}c_{0}p(r)^{k+1}r^{k}$). Hence we must have

[TABLE]

This implies that almost all sequences of period $2^{v}r$ satisfy $L(S)\geq L_{N,k}(S)\geq N-2^{v}$ as long as $r\rightarrow\infty$.

Next we prove that once $r\rightarrow\infty$, for $0\leq c<2^{v}$ we have

[TABLE]

By the relationship between the linear complexity and Günther weight of the GDFT of sequences [14][p. 2818], and almost all sequences satisfy $L(S)\geq N-2^{v}$, we only consider the first column of the GDFT matrix, and the contribution to the Günther weight of other columns are all $2^{v}$. Additionally, the elements of the first column are over ${\mathbb{F}}_{2}$, and the pattern of the first column is the transpose of $\underbrace{0\ldots 0}_{c}\;1***$, where ‘$*$’ can be 0 or 1. Hence we have (11).

If $2^{v}\leq r-2$, then we obtain \Pr(L(S)\geq N-c,L_{N,k}(S)\geq N-2^{v})=\Pr(L(S)\geq N-c,L(S)\geq N-2^{v}){{\color[rgb]{0,0,0}\rightarrow}}\frac{1}{2}+\cdots+\frac{1}{2^{c+1}}=1-2^{-1-c},\mbox{ as }r\rightarrow\infty.

If $2^{v}>r-2$, then almost all sequences of period $2^{v}r$ satisfy $L(S)\geq L_{N,k}(S)\geq N-2^{v}$ and we obtain $Pr(L(S)\geq N-r+2)\rightarrow 1-2^{1-r}$ similarly. Because

[TABLE]

for small $k$ we have $Pr(L_{N,k}(S)<N-r+2)=O(2^{1-r}\cdot N^{k})=O(2^{1-r}{p(r)}^{k}r^{k})$, then $Pr(L_{N,k}(S)\geq N-r+2)\rightarrow 1$. Therefore, for $0\leq c<r-2$ we obtain

[TABLE]

analogously. ∎

Remark 3.

According to the result in [17][p. 25; Theorem 1.2.8], for $0<\frac{k}{N}\leq 1/2$ we have

[TABLE]

where $H(\frac{k}{N}):=-\frac{k}{N}\log(\frac{k}{N})-(1-\frac{k}{N})\log(1-\frac{k}{N})$ is the entropy function on the variable $\frac{k}{N}$, and the base of the $\log(\cdot)$ is 2.

Consequently, if $k$ satisfies

[TABLE]

then the condition (6) holds by (12) and (13). Hence we have a weaker but explicit requirement of $k$ for (7) holds. Note that the entrophy function $H(x)$ is non-decreasing when $0<x\leq 1/2$.

We provide the following example to demonstrate the usefulness of our results.

Example 3.

Let $N=32*947$, i.e., $r=947$, $v=5$. Let us consider $k=10$. The traditional method of computing $L_{N,10}(S)$ is estimated as $\sum_{t=0}^{10}{32*947\choose t}>{32*947\choose 10}\approx 2^{127}$, which is infeasible. But from (13) we require $H(\frac{k}{N})<\frac{941}{32*947}$ and thus $k\leq 96$. Hence the condition (13) holds for $k=10$. From Corollary 2 we know that almost all sequences satisfy $L_{N,10}\geq N-32$. We can take the zone length $Z=32$.

If $c=1$ and $L(S)\geq N-c$, then $3$ out of $4$ such random sequences satisfy the condition (5) in Theorem 4.1, and we have $L_{N,10}(S)=L_{N,10;32}(S)$. If $L(S)\geq N-2$, then $7$ out of $8$ such random sequences satisfy (5), and we have $L_{N,10}(S)=L_{N,10;32}(S)$. The percentage of these sequences satisfying the condition (5) grows as $c$ increases. Finally, if $L(S)\geq N-31$, then almost any random sequence satisfy (5), and $L_{N,10}(S)=L_{N,10;32}(S)$.

Now we move to other types of sequences with period $N=2^{v}r$, where $r$ is a composite. More generally, we have

[TABLE]

where $\Phi_{d}(X)$ is the $d$-th cyclotomic polynomial. We do not require that $\Phi_{d}(X)$ is irreducible over ${\mathbb{F}}_{2}$, which is required for existing fast algorithms of computing the ($k$-error) linear complexity. Then, we can similarly obtain the following result by analyzing the multiplicity of root 1 when the $k$-error linear complexity is large.

Theorem 4.2.

Suppose $N=2^{v}p_{1}^{s_{1}}p_{2}^{s_{2}}\ldots p_{n}^{s_{n}}$, $v>0$, $p_{i}$ is odd prime, and 2 is a primitive root modulo $p_{i}$ and $p_{i}^{2}$ for all $1\leq i\leq n$. For any $N$-periodic binary sequence $S$ such that $L(S)>L_{N,k}(S)\geq N-\min(2^{v},p_{1}-2,\ldots,p_{n}-2)$, there exists a zone of length $Z=2^{\lceil\log_{2}(N-L_{N,k}(S))\rceil}$ such that $L_{N,k}(S)=L_{N,k;Z}(S)$.

Proof.

First, we suppose $N=2^{v}\cdot p^{s}$, where $2^{v}<p-1$ and $2$ is a primitive root modulo $p$ and $p^{2}$ respectively. Obviously $2$ is a primitive root of $p^{s}$ for any integer $s$ (see for example [10]). From (14) we derive

[TABLE]

Because the degree of each irreducible polynomial $\Phi_{p^{i}}(X)$ is \phi(p^{i})=p^{i}-p^{i-1}{{\color[rgb]{0,0,0}\geq p-1}}, we only need to consider the multiplicity of root $1$ for estimating $L_{N,k}$. The rest of proof is similar to the proof of Theorem 4.1.

Secondly, suppose $N=2^{v}\cdot p\cdot q$, $2$ is a primitive root modulo $p$ and $q$. Let $c$ be the least integer such that $2^{c}\equiv 1\bmod~{}({pq})$, then $\Phi_{pq}(X)$ can be factorized into $\frac{(p-1)(q-1)}{c}$ irreducible polynomials, each with degree $c$. In addition, because $2$ is the primitive root modulo $p$, we have $(p-1)\mid c$ and thus $c\geq p-1$. Similarly, $c\geq q-1$. From (14) we derive $X^{N}-1=\left((X-1)\Phi_{p}(X)\Phi_{q}(X)\Phi_{pq}(X)\right)^{2^{v}}$, implying the degree of any irreducible factors except $X-1$ is greater than or equal to $\min(p-1,q-1)$. Hence we only need to consider the multiplicity of root 1. The rest of proof follows.

Finally, if $2$ is the primitive root of $p_{1}^{s_{1}}$, $\ldots$, $p_{n}^{s_{n}}$ for any integers $i_{1},\ldots,i_{n}$, then we obtain $X^{N}-1=\left((X-1)\prod_{0\leq t_{j}\leq s_{j}\atop j=1,2,\ldots,n}\Phi_{p_{1}^{t_{1}}\cdots p_{j}^{t_{j}}\cdots p_{n}^{t_{n}}}(X)\right)^{2^{v}}$. Similarly, the degree of each irreducible factor of $\Phi_{p_{1}^{t_{1}}\cdots p_{j}^{t_{j}}\cdots p_{n}^{t_{n}}}(X)$ is no less than $\min(p_{1}-1,\ldots,p_{n}-1)$. Hence we only need to consider the multiplicity of the root 1 analogously. ∎

Example 4.

From computer experiments, there are many examples satisfying the above theorem for sequences with period $N$ such as N=32*37,{{\color[rgb]{0,0,0}4*11^{2}}},8*11^{2},8*11*13; $16*11^{2},16*11*13$; 111619, 131619. The global stability can be effectively determined by local stability within a much smaller zone. For example, for the random generated $N=4*11^{2}$-periodic sequence

$S=010110001110010010101001101011100110010010001110011100010110001000\\ 0010100100100101001000011110000101011100011110000000111010011001101110\\ 0110101110100010110111000011101010100101101100001000101001111010010110\\ 1110000000011010111010000000100100001011011100000100111001110101110111\\ 0010101000100111010010100011110011110100110110001000110000010001110100\\ 0110101010110001000100000000000001011011001111000011010000100111110000\\ 00101110000000100000111011001100110100111101000000100100100110000100$, we have $L(S)=L_{1}(S)=482$, $L_{2}(S)=480$, and the zone $[0,4)$. Indeed, we can find an error vector $E_{2}$ of $L_{2}$ such that error positions are $1$ and $3$ within the zone $[0,4)$.

Remark 4.

For a binary sequence $S$ with the period $N=2^{v}m$, $m\geq 1$, $v>0$. If $L_{N,k}(S)\geq\frac{N}{2}$, then the derived length of zone in Theorems 2.1, 4.1, and 4.2

[TABLE]

so $Z$ becomes effective: the larger $L_{N,k}(S)$, the smaller $Z$.

Acknowledgment

Ming Su expresses his sincere thanks for the hospitality during his visit to School of Mathematics and Statistics, Carleton University, Canada.