**3. Proposed scheme**

Ren et al. [5] prove that in order to achieve perfect security, data sharing between neighbours is suitable way. Therefore, in our scheme, sensor node collects data *data* and breaks it to equal shares *d1, d2…, dn*. Using following process, the sensor sends signed encoded *Yi* to the randomly selected neighbours.

### **3.1 Share generation, encoding, signing and broadcasting processes**

After sensor *vi* collects data *data*, it proceeds following steps to achieve data integrity, confidentiality and also authenticity.


$$Z\_{l} = \Sigma\_{l=1}^{p}(\alpha\_{l}) (Y\_{l}) \tag{1}$$

$$
\begin{pmatrix}
a\_{11} & a\_{12} & \dots & a\_{1p} \\
\vdots & \dots & \dots & \vdots \\
a\_{l1} & a\_{l2} & \dots & a\_{lp} \\
\vdots & \dots & \dots & \vdots \\
a\_{j1} & a\_{j2} & \dots & a\_{jp}
\end{pmatrix}
\begin{pmatrix}
Y\_1 \\
\vdots \\
Y\_l \\
\vdots \\
Y\_p
\end{pmatrix} = 
\begin{pmatrix}
Z\_1 \\
Y\_2 \\
\vdots \\
Y\_{p-1} \\
Z\_f
\end{pmatrix}
$$

$$
\Lambda\_q^\perp \langle \Delta \rangle = \{ e \epsilon \mathbb{Z}^m \colon \Delta \, e = 0 \bmod q \} \tag{2}
$$

$$
\Lambda\_q^u(\Delta) = \{ e \epsilon \mathbb{Z}^m \colon \Delta \text{ } e = u \bmod q \} \tag{3}
$$

$$\Lambda\_q(\Delta) = \{ e \epsilon \mathbb{Z}^m \, : \, \exists \mathbf{s} \epsilon \mathbb{Z}\_q^n \, with \, \Delta^t. \mathbf{s} = e \bmod q \} $$




$$\mathbf{b}. \quad \mathbf{o} \bmod \mathbf{p} = \mathbf{p} \& \mathbf{y}. \text{{}$$

$$f^{\prime\prime\prime}(d\_l) + f(d\_l)f^{\prime\prime}(d\_l) - \left(f^{\prime}(d\_l)\right)^2 - idf^{\prime}(d\_l) = 0\tag{4}$$

$$f(0) = 0, f'(0) = 1, f'(\infty) = 0,\tag{5}$$

$$
\propto \frac{d^2}{dx^2} L\_n^a(\infty) + (a+1-\varkappa) \frac{d}{d\varkappa} L\_n^a(\infty) + n L\_n^a(\infty) = 0,
$$

$$
\varkappa \in I = \{0, \infty\}, n = 0, 1, 2, \dots \tag{6}
$$

$$L\_0^a(\mathbf{x}) = 1,\tag{7}$$

$$L\_1^a(\mathbf{x}) = 1 + a - \mathbf{x},$$

$$L\_n^a(\mathbf{x}) = (2n - 1 + a - \mathbf{x})L\_{n-1}^a(\mathbf{x}) - (n + a - 1)L\_{n-2}^a(\mathbf{x}),$$

$$\phi\_f(\mathbf{x}) = \exp\left(\frac{-\chi}{2L}\right) L\_f^1\left(\frac{\chi}{L}\right), L > 0. \tag{8}$$

$$
\langle \phi\_m, \phi\_n \rangle\_{\mathfrak{w}\_L} = \left( \frac{\Gamma(n+2)}{L^2 n!} \right) \delta\_{nm\prime} \tag{9}
$$

$$f(\mathbf{x}) = \sum\_{l=0}^{+\infty} a\_l \phi\_l(\mathbf{x}),\tag{10}$$

$$a\_l = \frac{\langle f, \phi\_l \rangle\_w}{\langle \phi\_l, \phi\_l \rangle\_w}. \tag{11}$$

$$f(\mathbf{x}) \simeq \sum\_{l=0}^{N-1} a\_l \phi\_l(\mathbf{x}) = A^T \phi(\mathbf{x}),\tag{12}$$

$$A = \begin{bmatrix} a\_0, a\_1, a\_2, \dots, a\_{N-1} \end{bmatrix}^T,\tag{13}$$

$$\phi(\mathbf{x}) = [\phi\_0(\mathbf{x}), \phi\_1(\mathbf{x}), \dots, \phi\_{N-1}(\mathbf{x})]^T. \tag{14}$$

$$\mathfrak{R}\_N = \text{Span}\{1, \mathfrak{x}, \dots, \mathfrak{x}^{2N-1}\}\tag{15}$$

$$p\_f(\mathbf{x}) = \phi\_f(\mathbf{x}) - (\frac{j+1}{j})\phi\_{j-1}(\mathbf{x}).\tag{16}$$

$$\int\_0^{+\infty} f(\mathbf{x}) \mathbf{w}(\mathbf{x}) d\mathbf{x} = \sum\_{j=0}^{N-1} f\_j(\mathbf{x}) \mathbf{w}\_j + \left(\frac{\Gamma(N+2)}{(N)!(2N)!}\right) f^{2N}(\xi) e^{\xi},\tag{17}$$

$$\mathbf{w}\_{j} = \mathbf{x}\_{j} \frac{\Gamma(N+2)}{(L(N+1)! \, [(N+1)\phi\_{N+1}(\mathbf{x}\_{j})]^{2})}, j = 0, 1, 2, \dots, N-1$$

$$I\_N u(\mathbf{x}) = \sum\_{j=0}^{N-1} a\_j \phi\_j(\mathbf{x}),\tag{18}$$

$$I\_N u(x\_l) = u(x\_l), \\ j = 0, 1, 2, \dots, N - 1.$$

$$<\mathfrak{u}, \,\upsilon >\_{\mathfrak{w}, \mathbb{N}} = \sum\_{j=0}^{N-1} \mathfrak{u}(\mathfrak{x}\_j) \upsilon(\mathfrak{x}\_j) \mathfrak{w}\_j,\tag{19}$$

$$||u||\_{\mathcal{W}^N} = \_{\mathcal{W}^N}^{1/2} \tag{20}$$

$$\_{\psi, N} = <\iota, \upsilon >\_{\psi, N} \forall \iota. \ \upsilon \in \mathfrak{R}\_N$$

$$\_{\omega, N} = \tag{21}$$

$$I\_N f(d\_l) = \sum\_{l=0}^{N-1} a\_l \phi\_l,\tag{22}$$

$$\begin{array}{l} \text{Res}\{d\_{l}\} = \sum\_{j=0}^{N-1} a\_{j} \phi\_{j}^{\prime\prime\prime} \langle d\_{l} \rangle + \sum\_{j=0}^{N-1} a\_{j} \phi\_{j} \langle d\_{l} \rangle \sum\_{j=0}^{N-1} a\_{j} \phi\_{j}^{\prime\prime} \langle d\_{l} \rangle - \left(\sum\_{j=0}^{N-1} a\_{j} \phi\_{j}^{\prime} \langle d\_{l} \rangle\right)^{2} - \\\text{id}\,\sum\_{j=0}^{N-1} a\_{j} \phi\_{j}^{\prime} \langle d\_{l} \rangle \end{array} \tag{23}$$

$$
\Sigma\_{l=0}^{N-1} a\_l \phi\_l(0) = 0,\tag{24}
$$

$$\sum\_{j=0}^{N-1} a\_j \phi'\_j(0) = 1,\tag{25}$$

$$
\Sigma\_{l=0}^{N-1} a\_l \phi\_l(\infty) = 0. \tag{26}
$$

$$f(d\_l) = \frac{1}{\sqrt{1 + id}} (1 - e^{-\sqrt{1 + (ld)d\_l}}) \tag{27}$$

$$f(d\_l) = 1 - e^{-d\_l}.\tag{28}$$


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