Linear complexity of Legendre-polynomial quotients

TL;DR

This paper analyzes the linear complexity of a binary sequence defined via Legendre symbols, extending previous results to all w in the range 1 to p-1, under certain number-theoretic conditions.

Contribution

It provides a comprehensive determination of the linear complexity for all relevant w values, generalizing earlier work limited to w=p-1.

Findings

Linear complexity values are characterized for all 1 ≤ w < p-1.

The case w ≥ p can be reduced to smaller w cases.

Results depend on the condition 2^{p-1} ≠ 1 mod p^2.

Abstract

We continue to investigate binary sequence $(f_u)$ over $\{0,1\}$ defined by $(-1)^{f_u}=\left(\frac{(u^w-u^{wp})/p}{p}\right)$ for integers $u\ge 0$, where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol and we restrict $\left(\frac{0}{p}\right)=1$. In an earlier work, the linear complexity of $(f_u)$ was determined for $w=p-1$ under the assumption of $2^{p-1}\not\equiv 1 \pmod {p^2}$. In this work, we give possible values on the linear complexity of $(f_u)$ for all $1\le w<p-1$ under the same conditions. We also state that the case of larger $w(\geq p)$ can be reduced to that of $0\leq w\leq p-1$.

Equations

$$q_p(u)\equiv \frac{u^{p-1}-1}{p}\mathrm{mod}p\mathrm{with}0\le q_p(u)\le p-1,$$

$$q_p(kp)=0,k\in \mathbb{Z}.$$

$$q_p(u)\equiv \frac{u^{p-1}-u^{p(p-1)}}{p}\mathrm{mod}p.$$

$$q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p}\mathrm{mod}p\mathrm{with}0\le q_{p,w}(u)\le p-1,u\ge 0,$$

$$q_{p,w}(u)\equiv\left\{\begin{array}[]{ll}-u^{w}wq_{p}(u)\bmod p,&\mathrm{if}~{}\gcd(u,p)=1,\\ 0,&\mathrm{if}~{}p|u~{}\mathrm{and}~{}w\geq 2,\\ k\bmod p,&\mathrm{if}~{}u\equiv kp\pmod{p^{2}}~{}\mathrm{and}~{}w=1.\end{array}\right.$$

$$q_{p,w_1+w_2(p-1)}(u)\equiv -u^{w_1}(w_1-w_2)q_p(u)\equiv w_1^{-1}(w_1-w_2)q_{p,w_1}(u)\mathrm{mod}p,$$

$$e_{u}=\left\{\begin{array}[]{ll}0,&\mathrm{if}\,\ 0\leq q_{p,w}(u)/p<\frac{1}{2},\\ 1,&\mathrm{if}\,\ \frac{1}{2}\leq q_{p,w}(u)/p<1,\end{array}\right.\quad u\geq 0.$$

$$f_{u}=\left\{\begin{array}[]{ll}0,&\mathrm{if}\,\ \left(\frac{q_{p,w}(u)}{p}\right)=1\,\ \mathrm{or}\,\ q_{p,w}(u)=0,\\ 1,&\mathrm{otherwise},\end{array}\right.\quad u\geq 0,$$

$$s_{u+L}=c_{L-1}{s}_{u+L-1}+\cdots +c_1{s}_{u+1}+c_0{s}_u\text{for}u\ge 0$$

$$M(x)=x^L+c_{L-1}{x}^{L-1}+\cdots +c_0\in {\mathbb{F}}_2[x]$$

$$s(x)=s_0+s_{1}x+s_2{x}^2+\cdots +s_{T-1}{x}^{T-1}\in {\mathbb{F}}_2[x].$$

$$M(x)=(x^T-1)/\gcd \left(x^T-1,s(x)\right),$$

$$L((s_u))=T-\mathrm{deg}\left(\gcd (x^T-1,s(x))\right),$$

$$H_{w}(u)\equiv\left\{\begin{array}[]{ll}-wq_{p}(u)\bmod p,&\mathrm{if}~{}\gcd(u,p)=1,\\ 0,&\mathrm{if}~{}p|u~{}\mathrm{and}~{}w\geq 2,\\ k\bmod p,&\mathrm{if}~{}u\equiv kp\bmod{p^{2}}~{}\mathrm{and}~{}w=1.\end{array}\right.$$

$$H_w(uv)\equiv H_w(u)+H_w(v)(\mathrm{mod}p),\gcd (uv,p)=1.$$

$$D_l=\{u:0\le u<p^2,\gcd (u,p)=1,H_w(u)\equiv l(\mathrm{mod}p)\}$$

$$D_0=\{g^{kp}\mathrm{mod}{p}^2:0\le k<p\}$$

$$D_l(x)=\sum _{u\in D_l}{x}^u\in {\mathbb{F}}_2[x],$$

$$L((f_{u}))=\left\{\begin{array}[]{ll}p^{2}-p,&\mathrm{if}\,\ p\equiv 1\pmod{4},\\ p^{2}-1,&\mathrm{if}\,\ p\equiv 3\pmod{4}.\end{array}\right.$$

$$(-1)^{f_{u}}=\left\{\begin{array}[]{ll}\left(\frac{H_{w}(u)}{p}\right),&\mathrm{if}~{}(u\bmod{p^{2}})\in\mathbb{Z}_{p^{2}}^{*},~{}\gcd(H_{w}(u),p)=1,\\ 1,&\mathrm{otherwise},\end{array}\right.$$

$$f_{u}=\left\{\begin{array}[]{ll}0,&\mathrm{if}\,\ (u\bmod{p^{2}})\in\cup_{l\in\mathcal{Q}}D_{l}\cup D_{0}\cup P,\\ 1,&\mathrm{if}\,\ (u\bmod{p^{2}})\in\cup_{l\in\mathcal{N}}D_{l},\end{array}\right.$$

$${\Lambda }_{\ell }(x)=\sum _{l\in \mathcal{N}}{D}_{l+\ell }(x)\in {\mathbb{F}}_2[x],\ell =0,\dots ,p-1.$$

$$0={\left({\Lambda }_{{\ell }_0}(\beta )\right)}^{2^j}={\Lambda }_{{\ell }_0}({\beta }^{2^j})={\Lambda }_{{\ell }_0+j\mu }(\beta ),$$

$$\Phi (x)=1+x^p+x^{2p}+\dots +x^{(p-1)p}\in {\mathbb{F}}_2[x],$$

$$\Phi (x)|{\Lambda }_0(x)\mathrm{in}{\overline{\mathbb{F}}}_2[x].$$

$${\Lambda }_0(x)\equiv \Phi (x)\pi (x)(\mathrm{mod}{x}^{p^2}-1).$$

$$x^p\Phi (x)\equiv \Phi (x)(\mathrm{mod}{x}^{p^2}-1),$$

$$\Lambda_{\ell}(\beta^{kp})=\left\{\begin{array}[]{ll}0,&\mathrm{if}~{}k=0,\\ (p-1)/2,&\mathrm{if}~{}1\leq k<p,\end{array}\right.$$

$$Q_l=\left\{u\in D_l:\left(\frac{u}{p}\right)=1\right\}\text{and}{N}_l=\left\{u\in D_l:\left(\frac{u}{p}\right)=-1\right\}.$$

$$Q_l(x)=\sum _{u\in Q_l}{x}^u\in {\mathbb{F}}_2[x],N_l(x)=\sum _{u\in N_l}{x}^u\in {\mathbb{F}}_2[x]$$

Full text

Keywords: polynomial quotients, Fermat quotients, Legendre symbol, linear complexity.

2010 MSC: 11K45, 11L40, 94A60.

1 Introduction

For an odd prime $p$ and an integer $u$ with $\gcd(u,p)=1$, the *Fermat quotient $q_{p}(u)$ * is defined as the unique integer

[TABLE]

and

[TABLE]

An equivalent definition is

[TABLE]

For all positive integers $w$, Chen and Winterhof extended the equation above to define

[TABLE]

which is called a polynomial quotient in [7]. In fact $q_{p,p-1}(u)=q_{p}(u)$. We have the following relation between $q_{p,w}(u)$ and $q_{p}(u)$:

[TABLE]

Many number theoretic and cryptographic questions have been studied for polynomial quotients [1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13].

For any positive $w$ with $p|w$, we have $q_{p,w}(u)=0$ for all $u\geq 0$ by (1). For $w$ with $p\nmid w$, write $w=w_{1}+w_{2}(p-1)$ with $1\leq w_{1}\leq p-1$ and $w_{2}\geq 0$. By (1) again one can get for $\gcd(u,p)=1$

[TABLE]

which implies that $q_{p,w}$ for $w>p$ can be reduced to a polynomial quotient with $w<p$. Hence, we may restrict ourselves to $1\leq w\leq p-1$ from now on.

Two families of binary sequences have been considered in the literature. The first one is $(e_{u})$ defined by

[TABLE]

The second one is $(f_{u})$ defined by

[TABLE]

where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol.

Both $(e_{u})$ and $(f_{u})$ have interesting cryptographic properties. In particular, when $2^{p-1}\not\equiv 1\pmod{p^{2}}$, the linear complexity (see below for the notion) of $(e_{u})$ has been determined if $w\in\{(p-1)/2,p-1\}$ in [4, 3] and of $(f_{u})$ if $w=p-1$ in [2]. While in [5], the $k$-error linear complexity of $(e_{u})$ and $(f_{u})$ has been determined for any $w$ under the assumption of 2 being a primitive root modulo $p^{2}$. We note that ,if $2$ is a primitive root modulo $p^{2}$, then we always have $2^{p-1}\not\equiv 1\pmod{p^{2}}$. But the converse is not true, because there do exist such primes $p$, e.g., $p=43$. On the other hand, the primes $p$ satisfying $2^{p-1}\equiv 1\pmod{p^{2}}$ are very rare. To date the only known such primes are $p=1093$ and $p=3511$ and it was reported that there are no new such primes $p<4\times 10^{12}$, see [9].

In this work, we will continue to investigate the linear complexity of $(f_{u})$ for all $1\leq w<p-1$ under the assumption of $2^{p-1}\not\equiv 1\pmod{p^{2}}$. Equation (1) is a key tool for our purpose. But, it seems that the technique here does not work for $(e_{u})$ when $w\not\in\{(p-1)/2,p-1\}$.

We finally recall that the linear complexity $L((s_{u}))$ of a $T$-periodic sequence $(s_{u})$ over the binary field $\mathbb{F}_{2}$ is the least order $L$ of a linear recurrence relation over $\mathbb{F}_{2}$

[TABLE]

which is satisfied by $(s_{u})$ and where $c_{0}=1,c_{1},\ldots,c_{L-1}\in\mathbb{F}_{2}.$ The polynomial

[TABLE]

is called the minimal polynomial of $(s_{u})$. The generating polynomial of $(s_{u})$ is defined by

[TABLE]

It is easy to see that

[TABLE]

hence

[TABLE]

which is the degree of the minimal polynomial, see [10] for a more detailed exposition.

2 Linear complexity

From (1), we write

[TABLE]

We will use the theory of cyclotomy, since $H_{w}:\mathbb{Z}_{p^{2}}^{*}\rightarrow\mathbb{Z}_{p}$ is a group homomorphism

[TABLE]

by the fact that $q_{p}(uv)\equiv q_{p}(u)+q_{p}(v)\pmod{p}$ for $\gcd(uv,p)=1$, see e.g. [13]. In the context, we denote by $\mathbb{Z}_{p}=\mathbb{F}_{p}=\{0,1,\ldots,p-1\}$  (respectively $\mathbb{Z}_{p^{2}}=\{0,1,\ldots,p^{2}-1\}$) the residue class ring modulo $p$  (respectively $p^{2}$) and by $\mathbb{Z}_{p^{2}}^{*}$ the unit group of $\mathbb{Z}_{p^{2}}$.

Define

[TABLE]

for $l=0,1,\ldots,p-1$. Indeed, if $g$ is a (fixed) primitive root modulo $p^{2}$, we have by (4)

[TABLE]

and for $l=0,1,\ldots,p-1$, there exists an integer $0\leq l_{0}<p$ such that $D_{l}=g^{l_{0}}D_{0}$. Hence each $D_{l}$ has the cardinality $|D_{l}|=p-1$. Naturally $D_{0},D_{1},\ldots,D_{p-1}$ form a partition of $\mathbb{Z}_{p^{2}}^{*}$. Let $P=\{kp:0\leq k<p\}$.

We use the notation $aD_{l}=\{ab\pmod{p^{2}}:b\in D_{l}\}$. Using (4) we have the following basic fact

1. (I).

$aD_{l}=D_{l+l^{\prime}\pmod{p}}$ if $a\in D_{l^{\prime}}$.

Define

[TABLE]

for $l\in\{0,\ldots,p-1\}.$

2.1 The case of even $w:2\leq w<p$.

Theorem 1**.**

Let $(f_{u})$ be the $p^{2}$-periodic binary sequence defined in (2) with even $w:2\leq w<p$. Assume that $2^{p-1}\not\equiv 1\pmod{p^{2}}$ then,

[TABLE]

Proof. For even $w$, we have by (1)

[TABLE]

and define $(f_{u})$ in an equivalent way by

[TABLE]

here and hereafter $\mathcal{Q}$ denotes the set of quadratic residues modulo $p$ and $\mathcal{N}$ denotes the set of quadratic non-residues modulo $p$. We note that the cardinality $|\mathcal{Q}|=|\mathcal{N}|=(p-1)/2$.

Then the proof of [2, Th. 4] can help us to get the desired result. Here we present a detailed proof for completeness.

Let

[TABLE]

Clearly $\Lambda_{0}(x)$ is the generating polynomial of $(f_{u})$. For $0\leq\ell<p$, we only need to show $\Lambda_{\ell}(\beta)\neq 0$ for a primitive $p^{2}$-th root of unity $\beta\in\overline{\mathbb{F}}_{2}$ by (3).

We assume that $\Lambda_{\ell_{0}}(\beta)=0$ for some $0\leq\ell_{0}<p$. Since $2^{p-1}\not\equiv 1\pmod{p^{2}}$, we set $H_{w}(2)=-w2^{w}q_{p}(2)=\mu\neq 0$. Then by Fact (I) we have for $0\leq j<p$

[TABLE]

where the subscript of $\Lambda$ is reduced modulo $p$. That is, $\Lambda_{\ell}(\beta)=0$ for all $0\leq\ell<p$. We have furtherly $\Lambda_{\ell}(\beta^{u})=0$ by Fact (I) again for all $u\in\mathbb{Z}_{p^{2}}^{*}$ and $0\leq\ell<p$. On the other hand, all ($p^{2}-p$ many) elements $\beta^{u}$ for $u\in\mathbb{Z}_{p^{2}}^{*}$ are roots of

[TABLE]

which has no other roots. Hence we have

[TABLE]

Let

[TABLE]

Using the fact that

[TABLE]

we restrict $\deg(\pi(x))<p$. However, $\Lambda_{0}(x)$ has $(p-1)^{2}/2$ terms and the right hand side of (5) has $pt$ terms if $\pi(x)$ has $t$ terms, a contradiction. So we conclude that $\Lambda_{\ell}(\beta^{u})\neq 0$ for all $u\in\mathbb{Z}_{p^{2}}^{*}$ and $0\leq\ell<p$.

On the other hand, we have

[TABLE]

for $0\leq\ell<p$. We draw a conclusion that $\Lambda_{0}(x)$, the generating polynomial of $(f_{u})$, and $x^{p^{2}}-1$ have exactly $p$ many common roots $\beta^{kp}~{}(0\leq k<p)$ if $p\equiv 1\pmod{4}$ and one common root $\beta^{0}$ otherwise.   $\square$

2.2 The case of odd $w:3\leq w<p$.

For $l\in\{0,\ldots,p-1\}$, we define

[TABLE]

Using (4) aagain we have the following facts:

1. (II).

$aQ_{l}=Q_{l+l^{\prime}\pmod{p}}$ if $a\in Q_{l^{\prime}}$. 2. (III).

$aN_{l}=N_{l+l^{\prime}\pmod{p}}$ if $a\in Q_{l^{\prime}}$. 3. (IV).

$aQ_{l}=N_{l+l^{\prime}\pmod{p}}$ if $a\in N_{l^{\prime}}$. 4. (V).

$aN_{l}=Q_{l+l^{\prime}\pmod{p}}$ if $a\in N_{l^{\prime}}$. 5. (VI).

$\{u\pmod{p}:u\in D_{l}\}=\{1,2,\ldots,p-1\},~{}l\in\{0,1,\ldots,p-1\}$. Hence, $|Q_{l}|=|N_{l}|=(p-1)/2$.

Define

[TABLE]

for $l\in\{0,\ldots,p-1\}$. We see that $D_{l}(x)=Q_{l}(x)+N_{l}(x)$. Now we present the results in the following two theorems.

Theorem 2**.**

Let $(f_{u})$ be the $p^{2}$-periodic binary sequence defined in (2) with odd $3\leq w<p$. Assume that $2^{p-1}\not\equiv 1\pmod{p^{2}}$ then,

[TABLE]

Proof. For odd $w\geq 3$, we have by (1)

[TABLE]

and define $(f_{u})$ in an equivalent way by

[TABLE]

Then the generating polynomial of $(f_{u})$ is

[TABLE]

Below we will consider the common roots of $G^{\mathrm{odd}}(x)$ and $x^{p^{2}}-1$. The number of the common roots will lead to the values of linear complexity of $(f_{u})$ by (3). We need the following lemmas, which can be proved by following the way of [4].

Lemma 1**.**

[4]** Let $\beta\in\overline{\mathbb{F}}_{2}$ be a primitive $p^{2}$-th root of unity. We have

[TABLE]

Lemma 2**.**

[4]** Let $\beta\in\overline{\mathbb{F}}_{2}$ be a primitive $p^{2}$-th root of unity. For all $n\in\mathbb{Z}_{p^{2}}^{*}$, we have $\sum\limits_{l=0}^{p-1}N_{l}(\beta^{n})=0$.

Lemma 3**.**

[4]** Let $\beta\in\overline{\mathbb{F}}_{2}$ be a primitive $p^{2}$-th root of unity. If $2\in D_{\ell_{0}}$ for some $1\leq\ell_{0}\leq p-1$, we have $D_{l}(\beta^{n})\not=0$ for all $0\leq l\leq p-1$ and $n\in\mathbb{Z}_{p^{2}}^{*}$.

We use Lemmas 2 and 3 to show the following lemma.

Lemma 4**.**

[4]** Let $\beta\in\overline{\mathbb{F}}_{2}$ be a primitive $p^{2}$-th root of unity, then

1. (1).

If $2\in Q_{\ell_{0}}$ for some $1\leq\ell_{0}\leq p-1$, then there exist(s) either exactly $(p^{2}-p)/2$ many or no $n\in\mathbb{Z}_{p^{2}}^{*}$ such that $G^{\mathrm{odd}}(\beta^{n})=0$. 2. (2).

If $2\in N_{\ell_{0}}$ for some $1\leq\ell_{0}\leq p-1$, then $G^{\mathrm{odd}}(\beta^{n})\neq 0$ for all $n\in\mathbb{Z}_{p^{2}}^{*}$.

Now we continue the proof of Theorem 2.

First we suppose that $\left(\frac{2}{p}\right)=1$. In this case, $p\equiv\pm 1\pmod{8}$. If $p\equiv 1\pmod{8}$, we have $G^{\mathrm{odd}}(\beta^{n})=0$ if $n\in\{kp:0\leq k\leq p-1\}$ by Lemma 1 and there are either no numbers in $\mathbb{Z}_{p^{2}}^{*}$ or $p(p-1)/2$ many $n\in\mathbb{Z}_{p^{2}}^{*}$ such that $G^{\mathrm{odd}}(\beta^{n})=0$ by Lemma 4. Then the number of the common roots of $G^{\mathrm{odd}}(x)$ and $x^{p^{2}}-1$ is either $p$ or $(p^{2}+p)/2$ and hence the linear complexity of $(f_{u})$ is $p^{2}-p$ or $(p^{2}-p)/2$. For the case of $p\equiv-1\pmod{8}$, the result follows similarly.

Under the condition of $\left(\frac{2}{p}\right)=-1$, it can be proved in a similar way.   $\square$

We calculate the linear complexity of $(f_{u})$ for all primes $p<200$. We list some examples in Table 1. The experiment results illuminate that, when $p\equiv-1\bmod 8$ the linear complexity only equals $\frac{p^{2}+p}{2}-1$. So we might ask whether there exists $p$ such that linear complexity equals $p^{2}-1$ if $p\equiv-1\bmod 8$.

[TABLE]

Table 1. Linear complexity of $(f_{u})$ for odd $3\leq w<p$.

2.3 The case of $w=1$.

Theorem 3**.**

Let $(f_{u})$ be the $p^{2}$-periodic binary sequence defined in (2) with $w=1$. Assume that $2^{p-1}\not\equiv 1\pmod{p^{2}}$ then,

[TABLE]

Proof. We have by (1)

[TABLE]

and define $(f_{u})$ in an equivalent way by

[TABLE]

Then the generating polynomial $G^{\mathrm{one}}(x)$ of $(f_{u})$ is

[TABLE]

We need to compute the number of $n\in\mathbb{Z}_{p^{2}}$ with $G^{\mathrm{one}}(\beta^{n})=0$ for a primitive $p^{2}$-th root of unity $\beta\in\overline{\mathbb{F}}_{2}$.

Lemma 5**.**

Let $\beta\in\overline{\mathbb{F}}_{2}$ be a primitive $p^{2}$-th root of unity. We have

[TABLE]

Lemma 6**.**

Let $\beta\in\overline{\mathbb{F}}_{2}$ be a primitive $p^{2}$-th root of unity. If $2\in D_{\ell_{0}}$ for some $1\leq\ell_{0}\leq p-1$, we have $D_{l}(\beta^{n})\not=1$ for all $0\leq l\leq p-1$ and $n\in\mathbb{Z}_{p^{2}}^{*}$.

(Proof of Lemma 6). Let $\widetilde{D}_{l}(x)=1+D_{l}(x)$. Suppose that there is an $n_{0}\in D_{i_{0}}$ for some $0\leq i_{0}\leq p-1$ such that $D_{l_{0}}(\beta^{n_{0}})=1$ for some $0\leq l_{0}\leq p-1$. Then we have $\widetilde{D}_{l}(\beta^{n_{0}})=0$ and drive that each polynomial $\widetilde{D}_{l}(x)$ has at least $p(p-1)$ many roots using the proof of Lemma 3. We get a contradiction since at least one $\widetilde{D}_{l}(x)$ has degree $<p^{2}-p$. We complete the proof of Lemma 6.

Lemma 7**.**

Let $\theta\in\overline{\mathbb{F}}_{2}$ be a primitive $p$-th root of unity. Then

1. (1).

we have $\sum\limits_{j\in\mathcal{N}}\theta^{j}+\sum\limits_{j\in\mathcal{Q}}\theta^{j}=1$. 2. (2).

we have

[TABLE]

and

[TABLE]

for $0\leq l<p$.

Lemma 8**.**

Let $\beta\in\overline{\mathbb{F}}_{2}$ be a primitive $p^{2}$-th root of unity. If $2\in Q_{\ell_{0}}$ or $2\in N_{\ell_{0}}$ for some $1\leq\ell_{0}\leq p-1$, then there exist(s) either exactly $(p^{2}-p)/2$ many or no $n\in\mathbb{Z}_{p^{2}}^{*}$ such that $G^{\mathrm{one}}(\beta^{n})=0$.

(Proof of Lemma 8). It is easy to see that for all $n\in\mathbb{Z}_{p^{2}}^{*}$

[TABLE]

by Lemma 2. Let $\xi=\sum\limits_{l\in\mathcal{N}}\beta^{lp}$ and as before

[TABLE]

Then together with Facts (I)-(V), we have

[TABLE]

which indicates $G^{\mathrm{one}}(\beta^{m})\neq G^{\mathrm{one}}(\beta^{n})$ for $m\in Q_{l}$ and $n\in N_{l}$ by Lemma 6.

Case (i). Let $2\in Q_{\ell_{0}}$. We suppose that $G^{\mathrm{one}}(\beta^{n_{0}})=0$ for some $n_{0}\in D_{i_{0}},~{}0\leq i_{0}\leq p-1$. If $n_{0}\in Q_{i_{0}}$, then $2^{j}n_{0}\in Q_{j\ell_{0}+i_{0}\pmod{p}}$ for $0\leq j\leq p-1$. For all $n\in Q_{j\ell_{0}+i_{0}\pmod{p}}$, by (6) we derive

[TABLE]

which also implies $G^{\mathrm{one}}(\beta^{n})\neq 0$ for all $n\in N_{j\ell_{0}+i_{0}\pmod{p}}$. So we have for $n\in\mathbb{Z}_{p^{2}}^{*}$

[TABLE]

Similarly, if $n_{0}\in N_{i_{0}}$, we have

[TABLE]

Case (ii). For the case of $2\in N_{\ell_{0}}$, i.e., $\left(\frac{2}{p}\right)=-1$, if $G^{\mathrm{one}}(\beta^{n_{0}})=0$ for some $n_{0}\in Q_{i_{0}}$, then we have $2^{j}n_{0}\in N_{j\ell_{0}+i_{0}\pmod{p}}$ for odd $0\leq j\leq 2p-1$ and $2^{j}n_{0}\in Q_{j\ell_{0}+i_{0}\pmod{p}}$ for even $0\leq j\leq 2p-1$.

We note that $\xi^{2}=1+\xi$ by Lemma 7. So for even $j$ we have by (6)

[TABLE]

and so $G^{\mathrm{one}}(\beta^{n})=0$ iff $n\in Q_{0}\cup Q_{1}\cup\cdots\cup Q_{p-1}$.

Similarly, if $G^{\mathrm{one}}(\beta^{n_{0}})=0$ for some $n_{0}\in N_{i_{0}}$, we will get $G^{\mathrm{one}}(\beta^{n})=0$ iff $n\in N_{0}\cup N_{1}\cup\cdots\cup N_{p-1}$.

Thus we conclude that there exist $p(p-1)/2$ many $n\in\mathbb{Z}_{p^{2}}^{*}$ such that $G^{\mathrm{one}}(\beta^{n})=0$ since both $Q_{l}$ and $N_{l}$ contain $(p-1)/2$ elements. We complete the proof of Lemma 8.

Finally we finish the proof of Theorem 3 by Lemmas 5 and 8.   $\square$

We calculate the linear complexity of $(f_{u})$ for all primes $p<200$ and list some examples in Table 2. The experiment results illuminate that, no primes $p$ such that the linear complexity equals $p^{2}-p$, $\frac{p^{2}-p}{2}+1$, $p^{2}-p+1$ when $p\equiv 1,3,7\bmod 8$, respectively. We leave it open.

[TABLE]

Table 2. Linear complexity of $(f_{u})$ for $w=1$.

3 Final remarks

In this work, we have determined all possible values on the linear complexity of the binary sequences defined via computing the Legendre symbol of polynomial quotients. The achievement extends corresponding results of the Legendre-Fermat quotients studied in [2].

It is interesting to consider the linear complexity of $(e_{u})$ defined in Section 1 for $w\not\in\{p-1,(p-1)/2\}$.

Acknowledgements

The work was partially supported by the National Natural Science Foundation of China under grant No. 61373140.