New sequences from the generalized Pell $ p- $numbers and mersenne numbers and their application in cryptography

Elahe Mehraban; T. Aaron Gulliver; Salah Mahmoud Boulaaras; Kamyar Hosseini; Evren Hincal; Elahe Mehraban; T. Aaron Gulliver; Salah Mahmoud Boulaaras; Kamyar Hosseini; Evren Hincal

doi:10.3934/math.2024660

AIMS Mathematics

2024, Volume 9, Issue 5: 13537-13552. doi: 10.3934/math.2024660

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Research article Special Issues

New sequences from the generalized Pell $p-$ numbers and mersenne numbers and their application in cryptography

1.
Mathematics Research Center, Near East University TRNC, Mersin 10, 99138 Nicosia, Turkey
2.
Department of Mathematics, Near East University TRNC, Mersin 10, 99138 Nicosia, Turkey
3.
Faculty of Art and Science, University of Kyrenia, TRNC, Mersin 10, 99320 Kyrenia, Turkey
4.
Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, V8W 2Y2, Canada
5.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
6.
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

Received: 20 January 2024 Revised: 29 March 2024 Accepted: 03 April 2024 Published: 12 April 2024
MSC : 11K31, 11C20, 68P25, 68R01, 68P30, 15A15

This paper presents the generalized Pell $p-$ numbers and provides some related results. A new sequence is defined using the characteristic polynomial of the Pell $p-$ numbers and generalized Mersenne numbers. Two algorithms for Diffie-Hellman key exchange are given as an application of these sequences. They are illustrated via numerical examples and shown to be secure against attacks. Thus, these new sequences are practical for encryption and constructing private keys.

Keywords:

Citation: Elahe Mehraban, T. Aaron Gulliver, Salah Mahmoud Boulaaras, Kamyar Hosseini, Evren Hincal. New sequences from the generalized Pell $p-$ numbers and mersenne numbers and their application in cryptography[J]. AIMS Mathematics, 2024, 9(5): 13537-13552. doi: 10.3934/math.2024660

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Abstract

1. Introduction

Let $q, m, n\in \mathbf{Z}^{+}$ with $q > 2$ and $m > n\geq 1$ . For any $u, v\in\mathbf{Z}$ , we are concerned with the two-term exponential sums

$\mathcal{G}(u,v,m, n;q) = \sum\limits_{j = 1}^{q} \mathrm{e}_{q}\left(uj^m+vj^n\right),$

where $\mathrm{e}_{q}(x) = \mathrm{exp}\left(2\pi \mathrm{i}x/ q\right)$ and $\mathrm{i}^2 = -1$ .

For convenience, the following letters and symbols are commonly used in this paper and should be interpreted in the following sense unless otherwise stated.

● $\chi$ is Dirichlet character.

● $\chi_{k}$ is $k$ -order Dirichlet character.

● $\phi(a)$ is Euler function.

● $\alpha$ is uniquely determined by $4p = \alpha^2+27\beta^2$ and $\alpha\equiv 1\bmod 3$ .

● $\tau(\chi)$ is Gauss sums defined by

$\tau(\chi) = \sum\limits_{s = 1}^{q}\chi(s) \mathrm{e}_{q}\left(s\right).$

The mean value calculation and upper bound estimation of exponential sums has always been a classical problem in analytic number theory. As a special kind of exponential sums, Gauss sums have had an important effect on both cryptography and analytic number theory. Analytic number theory and cryptography will benefit greatly from any significant advancements made in this area. In this paper, we will estimate and calculate the fourth power mean value of two-term exponential sums weighted by a character $\chi_{3}$ . In this field, many scholars have investigated the results of $\mathcal{G}(u, v, m, n; q)$ in various forms, and obtained many meaningful results, see ^{[3,5,7,8,11,15]}. For instance, Zhang and Zhang ^[9] obtained the power mean about $\mathcal{G}(u, v, 3, 1; p)$

$\begin{eqnarray} \sum\limits_{u = 1}^{p-1}\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(ui^3+vi\right)\right|^4 = \left\{ \begin{array}{ll} 2p^3-p^2 & \text{ if}\quad 3\nmid p-1 , \nonumber\\ 2p^3-7p^2 & \text{ if}\quad 3\mid p-1 ,\nonumber \end{array}\right. \end{eqnarray}$

where $p$ is an odd prime and $v$ is not divisible by $p$ .

Wang and Zhang ^[6] obtained the eighth power mean of $\mathcal{G}(u, v, 3, 1; p)$

$\begin{eqnarray} \sum\limits_{u = 1}^{p-1}\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(ui^3+vi\right)\right|^8 = \left\{ \begin{array}{ll} 7(2p^5-3p^4) & \mathrm{ if}\quad 6\mid p-5 , \nonumber\\ 14p^5-75p^4-8p^3\alpha^2 & \mathrm{ if}\quad 6\mid p-1 .\nonumber \end{array}\right. \end{eqnarray}$

In addition, Zhang and Han ^[12] shown the power mean of $\mathcal{G}(1, v, 3, 1; p)$

$\begin{eqnarray} \sum\limits_{v = 1}^{p-1}\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(i^3+vi\right)\right|^6 = 5p^4-8p^3-p^2, \end{eqnarray}$

(1.1)

where $p$ is an odd prime with $3\nmid \phi(p)$ .

But if $3\mid \phi(p)$ , whether there exists an exact formula for (1.1). Consider the mean of the simplest

$\begin{eqnarray} \sum\limits_{v = 1}^{p-1}\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(i^3+vi\right)\right|^4. \end{eqnarray}$

(1.2)

It is worth mentioning that, Zhang and Zhang ^[10] studied the power mean of the exponential sums weight by $\chi_{2}$ , one has the identities

$\begin{eqnarray} \sum\limits_{u = 1}^{p-1}\chi_{2}(u)\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(ui^3+i\right)\right|^4 = \left\{ \begin{array}{ll} p^2(\zeta+3) & \mathrm{ if}\quad 6\mid p-5 , \nonumber\\ p^2(\zeta-3) & \mathrm{ if}\quad 6\mid p-1 ,\nonumber \end{array}\right. \end{eqnarray}$

where $\zeta = \sum_{t = 1}^{p-1}\left(\frac{t-1+\overline{t}}{p}\right)$ with $\zeta\in\mathbf{Z}$ satisfies inequality $|\zeta|\leq2\sqrt{p}$ .

Cao and Wang (see Lemma 3 in ^[2]) proved the following conclusion, that is, if $p$ is a prime with $3\mid \phi(p)$ , then for any $\chi_{3}\bmod p$ , one has the identity

$\begin{eqnarray*} \sum\limits_{u = 1}^{p-1}\chi_{3}(u)\left(\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(ui^3+i\right)\right)^4 = \big(\overline{\chi_{3}}(3)-3p-\overline{\chi_{3}}(3)p\big) \tau^2\left(\overline{\chi_{3}}\right)-\alpha p\tau(\chi_{3}). \end{eqnarray*}$

Unfortunately, this lemma is incorrect, there is a calculation error in it. It is precisely because of the computational error in this lemma that the main result in the whole text is wrong.

The following year, Zhang and Meng ^[16] studied the power mean of $\mathcal{G}(u, 1, 3, 1; p)$ weighted by $\chi_{2}$ . In this paper, We intend to correct the error in ^[2] and give a correct conclusion. At the same time, as an application, we give an exact result for (1.2). That is, it will prove these two conclusions:

Theorem 1. If $p$ is a prime with $3\mid \phi(p)$ , then we have

$\begin{eqnarray*} \sum\limits_{u = 1}^{p-1}\chi_{3}(u)\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(ui^3+i\right)\right|^4 = -\alpha p\tau(\chi_{3})-3p\cdot\tau^2(\overline{\chi_{3}}). \end{eqnarray*}$

Theorem 2. If $p$ is a prime with $3\mid \phi(p)$ , then we have

$\begin{eqnarray*} \sum\limits_{v = 1}^{p-1}\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(i^3+vi\right)\right|^4 = 2p^3-p^2-3pA^2_k-p\alpha A_k, \end{eqnarray*}$

where $A_k = \omega^k\left[{\frac{\alpha p}{2}+\left(\left(\frac{\alpha p}{2}\right)^{2}-p^{3}\right)^{\frac{1}{2}}}\right]^{\frac{1}{3}}+\omega^{-k}\left[{\frac{\alpha p}{2}-\left(\left(\frac{\alpha p}{2}\right)^{2}-p^{3}\right)^{\frac{1}{2}}}\right]^{\frac{1}{3}}$ , $k = 1, 2$ or $3$ is dependent on $p$ , and $\omega = \frac{-1+\sqrt{3}i}{2}$ .

Corollary 1. If $p$ is a prime with $3\mid \phi(p)$ , then we have the asymptotic formula

$\begin{eqnarray*} &&\sum\limits_{v = 1}^{p-1}\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(i^3+vi\right)\right|^4 = 2p^3+O\left(p^2\right). \end{eqnarray*}$

Corollary 2. If $p$ is a prime with $3\mid \phi(p)$ , then for any integer $l$ , we have recursive formula

$\begin{eqnarray*} &&V_l(p) = \sum\limits_{u = 1}^{p-1}\vartheta^{l}(u)\left|\sum\limits_{j = 1}^{p} \mathrm{e}_{p}\left(uj^3+j\right)\right|^4\\ & = &\alpha p\sum\limits_{u = 1}^{p-1}\vartheta^{l-3}(u)\left|\sum\limits_{j = 1}^{p} \mathrm{e}_{p}\left(uj^3+j\right)\right|^4+3p\sum\limits_{u = 1}^{p-1}\vartheta^{l-2}(u)\left|\sum\limits_{j = 1}^{p} \mathrm{e}_{p}\left(uj^3+j\right)\right|^4\\ & = & \alpha p V_{l-3}(p)+ 3pV_{l-2}(p), \end{eqnarray*}$

when $l$ take $1$ – $3$ , the following equations hold

$\begin{eqnarray*} V_1(p) = \sum\limits_{u = 1}^{p-1}\vartheta(u)\left|\sum\limits_{j = 1}^{p} \mathrm{e}_{p}\left(uj^3+j\right)\right|^4& = &-5\alpha p^2,\\ V_2(p) = \sum\limits_{u = 1}^{p-1}\vartheta^{2}(u)\left|\sum\limits_{j = 1}^{p} \mathrm{e}_{p}\left(uj^3+j\right)\right|^4& = &4p^4-20p^3-\alpha^2p^2,\\ V_3(p) = \sum\limits_{u = 1}^{p-1}\vartheta^{3}(u)\left|\sum\limits_{j = 1}^{p} \mathrm{e}_{p}\left(uj^3+j\right)\right|^4& = &2\alpha p^4-22\alpha p^3, \end{eqnarray*}$

where $\vartheta(u) = \sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(ui^3\right)$ .

In fact, with the third-order linear recursive formula in Corollary 2 and its three initial values $V_1(p)$ , $V_2(p)$ and $V_3(p)$ , we can easily give the general term formula for the sequence $\{V_l(p)\}$ .

Corollary 3. If $p$ is a prime with $3\mid \phi(p)$ , then we have

$\begin{eqnarray*} \sum\limits_{u = 1}^{p-1}\left|\frac{ \sum\limits_{j = 1}^{p} \mathrm{e}_{p}\left(uj^3+j\right)}{ \sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(ui^3\right)}\right|^4 = \frac{54p^3}{\alpha^4}-\frac{p^2}{\alpha^2}-\frac{27p^2}{\alpha^4}+\frac{2p}{\alpha^2}-1. \end{eqnarray*}$

2. Some lemmas

Before starting our proofs of main results, we present the proofs of several key equations in preparation for the next chapter. The properties of Gauss sums and reduced (complete) residue systems are used repeatedly in the proof. In addition, we will refer to the basic contents of number theory in references ^[1] and ^[14].

Lemma 1. If $p$ is a prime with $3\mid \phi(p)$ , then

$\begin{eqnarray} \tau^3(\chi_{3})+ \tau^3\left(\overline{\chi_{3}}\right) = \alpha p. \end{eqnarray}$

(2.1)

Proof. This is consequence of ^[4] or ^[13], herein we omit it. □

Lemma 2. If $p$ is a prime with $3\mid \phi(p)$ , then

$\begin{eqnarray*} \sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3-1\right) = p(\alpha-3)+3 \tau^3\left(\overline{\chi_{3}}\right). \end{eqnarray*}$

Proof. Recall that $\tau(\chi_{3})\cdot \tau\left(\overline{\chi_{3}}\right) = p$ and (2.1), we have

$\begin{eqnarray*} &&\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3-1\right)\\ & = &\frac{1}{\tau(\chi_{3})}\sum\limits_{t = 1}^{p-1}\chi_{3}(t)\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p} \mathrm{e}_{p}\left(t\left(i^3+j^3-s^3-1\right)\right)\nonumber\\ & = & \frac{1}{\tau(\chi_{3})}\sum\limits_{t = 1}^{p-1}\chi_{3}(t) \mathrm{e}_{p}\left(-t\right) \left(\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(it^3\right)\right)^2 \left(\sum\limits_{s = 1}^{p} \mathrm{e}_{p}\left(-st^3\right)\right) \nonumber\\ & = & \frac{1}{\tau(\chi_{3})}\sum\limits_{t = 1}^{p-1}\chi_{3}(t) \mathrm{e}_{p}\left(-t\right) \left( 1+ \sum\limits_{i = 1}^{p-1} \left(1+ \chi_{3}(i)+ \overline{\chi_{3}}(i)\right) \mathrm{e}_{p}\left(it\right)\right)^3 \nonumber\\ & = &\frac{1}{\tau(\chi_{3})}\sum\limits_{t = 1}^{p-1}\chi_{3}(t) \mathrm{e}_{p}\left(-t\right) \big( \overline{\chi_{3}}(t)\cdot\tau(\chi_{3}) +\chi_{3}(t)\cdot\tau\left(\overline{\chi_{3}}\right)\big)^3 \nonumber\\ & = &\frac{1}{\tau(\chi_{3})}\sum\limits_{t = 1}^{p-1}\chi_{3}(t) \mathrm{e}_{p}\left(-t\right)\left[\tau^3(\chi_{3})+ \tau^3\left(\overline{\chi_{3}}\right)+ 3p\big( \overline{\chi_{3}}(t)\cdot\tau(\chi_{3}) +\chi_{3}(t)\cdot\tau\left(\overline{\chi_{3}}\right)\big)\right]\nonumber\\ & = &\alpha p + \frac{3p}{\tau(\chi_{3})}\cdot \tau(\chi_{3})\cdot \sum\limits_{t = 1}^{p-1} \mathrm{e}_{p}\left(-t\right)+\frac{3p}{\tau(\chi_{3})}\cdot \tau\left(\overline{\chi_{3}}\right)\cdot \sum\limits_{t = 1}^{p-1}\chi_{3}^2(t) \mathrm{e}_{p}\left(-t\right)\nonumber\\ & = &p(\alpha-3)+ \frac{3p\cdot \tau^2\left(\overline{\chi_{3}}\right)}{\tau(\chi_{3})} = p(\alpha-3)+3 \tau^3\left(\overline{\chi_{3}}\right). \end{eqnarray*}$

This completes the proof. □

Lemma 3. If $p$ is a prime with $3\mid \phi(p)$ , then

$\tau(\overline{\chi_{3}}\chi_{2}) = \frac{\overline{\chi_{3}}(2)\cdot\tau^2(\chi_{3})\cdot\tau(\chi_2)}{p}.$

Proof. Recall that $\tau(\chi_{3})\cdot\tau\left(\overline{\chi_{3}}\right) = p$ , we obtain

$\begin{eqnarray} \sum\limits_{i = 1}^{p}\chi_{3}(i^2-1)& = &\sum\limits_{i = 1}^{p}\chi_{3}(i^2+2i)\\ & = &\frac{1}{\tau(\overline{\chi_{3}})}\sum\limits_{j = 1}^{p-1}\overline{\chi_{3}}(j)\sum\limits_{i = 1}^{p-1}\chi_{3}(i) \mathrm{e}_{p}\big(j(i+2)\big)\\ & = &\frac{\tau(\chi_{3})}{\tau(\overline{\chi_{3}})}\sum\limits_{j = 1}^{p-1}\chi_{3}(j) \mathrm{e}_{p}\left(2j\right)\\ & = &\frac{\overline{\chi_{3}}(2)\cdot\tau^2(\chi_{3})}{\tau(\overline{\chi_{3}})}\\ & = &\frac{\overline{\chi_{3}}(2)\cdot\tau^3(\chi_{3})}{p}. \end{eqnarray}$

(2.2)

From another perspective, we have

$\begin{eqnarray} \sum\limits_{i = 1}^{p}\chi_{3}(i^2-1)& = &\frac{1}{\tau(\overline{\chi_{3}})}\sum\limits_{j = 1}^{p-1}\overline{\chi_{3}}(j)\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(j(i^2-1)\right)\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \\ & = &\frac{1}{\tau(\overline{\chi_{3}})}\sum\limits_{j = 1}^{p-1}\overline{\chi_{3}}(j) \mathrm{e}_{p}\left(-j\right)\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(i^2j\right)\\ & = &\frac{1}{\tau(\overline{\chi_{3}})}\sum\limits_{j = 1}^{p-1}\overline{\chi_{3}}(j) \mathrm{e}_{p}\left(-j\right)\left[1+\sum\limits_{i = 1}^{p-1}\big(1+\chi_{2}(i)\big) \mathrm{e}_{p}\left(ij\right)\right]\\ & = &\frac{1}{\tau(\overline{\chi_{3}})}\sum\limits_{j = 1}^{p-1}\overline{\chi_{3}}(j) \mathrm{e}_{p}\left(-j\right)\sum\limits_{i = 1}^{p-1}\chi_{2}(i) \mathrm{e}_{p}\left(ij\right)\\ & = &\frac{\tau(\chi_{2})}{\tau(\overline{\chi_{3}})}\sum\limits_{j = 1}^{p-1}\overline{\chi_{3}}\chi_{2}(j) \mathrm{e}_{p}\left(-j\right)\\ & = &\frac{\chi_{2}(-1)\cdot\tau(\chi_{2})\cdot\tau(\overline{\chi_{3}}\chi_{2})}{\tau(\overline{\chi_{3}})}\\ & = &\frac{\chi_{2}(-1)\cdot\tau(\chi_{2})\cdot\tau(\overline{\chi_{3}}\chi_{2})\cdot\tau(\chi_{3})}{p}. \end{eqnarray}$

(2.3)

Combining (2.2) and (2.3), we determine the relationship equation between $\tau(\overline{\chi_{3}}\chi_{2})$ , $\tau(\chi_{3})$ and $\tau(\chi_2)$

$\tau(\overline{\chi_{3}}\chi_{2}) = \frac{\overline{\chi_{3}}(2)\cdot\tau^2(\chi_{3})\cdot\tau(\chi_2)}{p}.$

This completes the proof. □

Lemma 4. If $p$ is a prime with $3\mid \phi(p)$ , then

$\begin{eqnarray*} \mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}}_{i+j-s-1\equiv 0\;{\rm mod}\; p }\overline{\chi_{3}}\left(i^3+j^3-s^3-1\right) = \frac{-\overline{\chi_{3}}(3)\cdot\tau^3\left(\overline{\chi_{3}}\right)}{p}. \end{eqnarray*}$

Proof. Using Lemma 3, we have

$\begin{eqnarray*} &&\mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}}_{i+j-s-1\equiv 0{\rm mod} p}\overline{\chi_{3}}\left(i^3+j^3-s^3-1\right)\nonumber\\ & = &\mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}}_{i+j\equiv 1\;{\rm mod}\; p }\overline{\chi_{3}}\left(i^3+j^3 +3j^2s +3js^2-1\right)\nonumber\\ & = &\chi_{3}(4) \mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}}_{i+j\equiv 1\;{\rm mod}\; p }\overline{\chi_{3}}\left(4i^3+j^3 +3j\left(2s +j\right)^2-4\right)\nonumber\\ & = &\frac{\chi_{3}(4)}{\tau(\chi_{3})}\mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{t = 1}^{p-1}}_{i+j\equiv 1\;{\rm mod}\; p}\chi_{3}(t) \sum\limits_{s = 1}^{p} \mathrm{e}_{p}\left(t\left(4i^3+j^3 +3js^2-4\right)\right)\nonumber\\ & = &\frac{\chi_{3}(4)}{\tau(\chi_{3})}\mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{t = 1}^{p-1}}_{i+j\equiv 1\;{\rm mod}\; p }\chi_{3}(t) \mathrm{e}_{p}\big(t(4i^3+j^3-4)\big)\left[1+\sum\limits_{s = 1}^{p-1}\big(1+\chi_{2}(s)\big) \mathrm{e}_{p}\left(3jst\right)\right]\nonumber\\ & = &\frac{\chi_{3}(4)}{\tau(\chi_{3})}\mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{t = 1}^{p-1}}_{i+j\equiv 1\;{\rm mod}\; p }\chi_{3}(t) \mathrm{e}_{p}\left(t(4i^3+j^3-4)\right)\chi_2(3jt)\tau(\chi_2)\nonumber\\ & = &\frac{\chi_{3}(4)\cdot \chi_2(3)\cdot \tau(\chi_2)}{\tau(\chi_{3})}\mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}}_{i+j\equiv 1\;{\rm mod}\; p }\chi_2(j) \tau\left(\chi_{3}\chi_2\right) \overline{\chi_{3}}\chi_2\left(4i^3+j^3-4\right)\nonumber\\ & = &\frac{\chi_{3}(4)\cdot \chi_2(3)\cdot \tau(\chi_2)\cdot \tau\left(\chi_{3}\chi_2\right) }{\tau(\chi_{3})}\sum\limits_{i = 1}^{p}\chi_2(1-i)\overline{\chi_{3}}\chi_2\left(3i^3+3i^2-3i-3\right)\nonumber\\ & = &\frac{\overline{\chi_{3}}(6)\cdot \chi_2(-1)\cdot \tau(\chi_2)\cdot \tau\left(\chi_{3}\chi_2\right) }{\tau(\chi_{3})}\sum\limits_{i = 1}^{p-1}\overline{\chi_{3}}\left((i+2)^2i\right)\nonumber\\ & = &\frac{\overline{\chi_{3}}(6)\cdot \chi_2(-1)\cdot \tau(\chi_2)\cdot \tau\left(\chi_{3}\chi_2\right) }{\tau(\chi_{3})} \sum\limits_{i = 1}^{p-1}\overline{\chi_{3}}(i)\chi_{3}\left(i+2\right)\nonumber\\ & = &\frac{\overline{\chi_{3}}(6)\cdot \chi_2(-1)\cdot \tau(\chi_2)\cdot \tau\left(\chi_{3}\chi_2\right) }{\tau(\chi_{3})}\sum\limits_{i = 1}^{p-1}\chi_{3}\left(1+2\cdot \overline{i}\right)\nonumber\\ & = &\frac{\overline{\chi_{3}}(6)\cdot \chi_2(-1)\cdot \tau(\chi_2)\cdot \tau\left(\chi_{3}\chi_2\right) }{\tau(\chi_{3})}\left(-1+ \sum\limits_{i = 1}^{p}\chi_{3}\left(i\right)\right)\nonumber\\ & = &\frac{-\chi_2(-1) \cdot \overline{\chi_{3}}(6)\cdot \tau(\chi_2)\cdot \tau\left(\overline{\chi_{3}}\right)\cdot \tau\left(\chi_{3}\chi_2\right) }{p}\nonumber\\ & = &\frac{-\overline{\chi_{3}}(3)\cdot\tau^3\left(\overline{\chi_{3}}\right)}{p}. \end{eqnarray*}$

This completes the proof. □

Lemma 5. If $p$ is a prime and $3\mid \phi(p)$ , then

$\begin{eqnarray*} \sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3\right) \mathrm{e}_{p}\left(i+j-s\right) = -3p+\overline{\chi_{3}}(3)\cdot\tau^3(\overline{\chi_{3}}). \end{eqnarray*}$

Proof. Note that $\tau(\chi_{3})\cdot\tau\left(\overline{\chi_{3}}\right) = p$ and $\overline{\chi_{3}}\left(i^3\right) = 1$ with $i$ is an integer relatively prime to $p$ . Therefore we have

$\begin{eqnarray*} &&\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3\right) \mathrm{e}_{p}\left(i+j-s\right)\nonumber\\ & = &\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3\right) \mathrm{e}_{p}\left(i+j\right)+\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}(i^3+j^3-1) \mathrm{e}_{p}\left(s(i+j-1)\right)\\ &&-\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\overline{\chi_{3}}(i^3+j^3-1)\nonumber\\ & = &\sum\limits_{i = 1}^{p}\overline{\chi_{3}}(i^3) \mathrm{e}_{p}\left(i\right)+\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\overline{\chi_{3}}\left(i^3+1\right) \mathrm{e}_{p}\left(j(i+1)\right)-\sum\limits_{i = 1}^{p}\overline{\chi_{3}}(i^3+1)\\ &&+p\mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}}_{i+j\equiv1\;{\rm mod}\; p}\overline{\chi_{3}}(i^3+j^3-1)-\frac{1}{\tau(\chi_{3})}\sum\limits_{t = 1}^{p-1}\chi_{3}(t)\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p} \mathrm{e}_{p}\left(t(i^3+j^3-1)\right)\nonumber\\ & = &\sum\limits_{i = 1}^{p-1} \mathrm{e}_{p}\left(i\right)+p\mathop{\sum\limits_{i = 1}^{p}}_{i+1\equiv0{\rm mod} p}\overline{\chi_{3}}\left(i^3+1\right)-1-\sum\limits_{i = 1}^{p-1}\big(1+\chi_{3}(i)+\overline{\chi_{3}}(i)\big)\overline{\chi_{3}}(i+1)\\ &&+p\mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}}_{i+j\equiv0{\rm mod} p}\overline{\chi_{3}}(i^3+j^3+3j^2+3j)-\frac{1}{\tau(\chi_{3})}\sum\limits_{t = 1}^{p-1}\chi_{3}(t) \mathrm{e}_{p}\left(-t\right)\left(\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(it^3\right)\right)^{2}\nonumber\\ & = &-1-\sum\limits_{i = 1}^{p}\overline{\chi_{3}}(i+1)-\sum\limits_{i = 1}^{p-1}\chi_{3}(i)\overline{\chi_{3}}(i+1)-\sum\limits_{i = 1}^{p-1}\overline{\chi_{3}}(i)\overline{\chi_{3}}(i+1)+p\sum\limits_{j = 1}^{p}\overline{\chi_{3}}(3j^2+3j)\\ &&-\frac{1}{\tau(\chi_{3})}\sum\limits_{t = 1}^{p-1}\chi_{3}(t) \mathrm{e}_{p}\left(-t\right)\left[1+\sum\limits_{i = 1}^{p-1}\big(1+\chi_{3}(i)+\overline{\chi_{3}}(i)\big) \mathrm{e}_{p}\left(it\right)\right]^{2}\nonumber\\ & = &-1-\sum\limits_{i = 1}^{p-1}\overline{\chi_{3}}(1+\overline{i})-\sum\limits_{i = 1}^{p-1}\overline{\chi_{3}}(i^2+i)+p\overline{\chi_{3}}(3)\sum\limits_{j = 1}^{p-1}\overline{\chi_{3}}(j^2+j)\\ &&-\frac{1}{\tau(\chi_{3})}\sum\limits_{t = 1}^{p-1}\chi_{3}(t) \mathrm{e}_{p}\left(-t\right)\left(\overline{\chi_{3}}(t)\tau(\chi_{3})+\chi_{3}(t)\tau(\overline{\chi_{3}})\right)^{2}\nonumber\\ & = &-\frac{1}{\tau(\chi_{3})}\left(\tau^2(\chi_{3})\sum\limits_{t = 1}^{p-1}\overline{\chi_{3}}(t) \mathrm{e}_{p}\left(-t\right)+\tau^2(\overline{\chi_{3}})\sum\limits_{t = 1}^{p-1} \mathrm{e}_{p}\left(-t\right)+2p\sum\limits_{t = 1}^{p-1}\chi_{3}(t) \mathrm{e}_{p}\left(-t\right)\right)\nonumber\\ &&+\left(-1+p\overline{\chi_{3}}(3)\right)\frac{1}{\tau(\chi_{3})}\sum\limits_{s = 1}^{p-1}\chi_{3}(s)\sum\limits_{i = 1}^{p-1}\overline{\chi_{3}}\left(i\right) \mathrm{e}_{p}\left(s(i+1)\right)\\ & = &-\frac{1}{\tau(\chi_{3})}\left(\tau^2(\chi_{3})\cdot\tau(\overline{\chi_{3}})-\tau^2(\overline{\chi_{3}})+2p\cdot\tau(\chi_{3})\right)-\frac{\tau^2(\overline{\chi_{3}})}{\tau(\chi_{3})}+p\cdot\overline{\chi_{3}}(3)\cdot\frac{\tau^2(\overline{\chi_{3}})}{\tau(\chi_{3})}\\ & = &\overline{\chi_{3}}(3)\cdot\tau^3(\overline{\chi_{3}})-3p. \end{eqnarray*}$

This proves Lemma 5. □

3. Proofs of main results

Proof of Theorem 1. Recall that $\overline{\chi_{3}}\left(i^3\right) = 1$ with $(i, p) = 1$ . Hence

$\begin{eqnarray*} &&\sum\limits_{u = 1}^{p-1}\chi_{3}(u)\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(ui^3+i\right)\right|^4\nonumber\\ & = &\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\sum\limits_{t = 1}^{p}\sum\limits_{u = 1}^{p-1}\chi_{3}(u) \mathrm{e}_{p}\left(u\left(i^3+j^3-s^3-t^3\right)+i+j-s-t\right)\nonumber\\ & = &\tau(\chi_{3})\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\sum\limits_{t = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3-t^3\right) \mathrm{e}_{p}\left(i+j-s-t\right)\nonumber\\ & = &\tau(\chi_{3})\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\sum\limits_{t = 1}^{p-1}\overline{\chi_{3}}\left(i^3t^3+j^3t^3-s^3t^3-t^3\right) \mathrm{e}_{p}\left(it+jt-st-t\right)\nonumber\\ &&+ \tau(\chi_{3})\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3\right) \mathrm{e}_{p}\left(i+j-s\right)\nonumber\\ & = &\tau(\chi_{3})\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3-1\right)\sum\limits_{t = 1}^{p} \mathrm{e}_{p}\left(t(i+j-s-1)\right)\nonumber\\ &&+\tau(\chi_{3})\sum _{i = 0}^{p-1}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3\right) \mathrm{e}_{p}\left(i+j-s\right)\\ &&-\tau(\chi_{3})\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3-1\right) \nonumber\\ & = &p\tau(\chi_{3}) \mathop{\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}}_{i+j-s\equiv 1\;{\rm mod}\; p }\overline{\chi_{3}}\left(i^3+j^3-s^3-1\right)-\tau(\chi_{3})\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3-1\right) \nonumber\\ &&+\tau(\chi_{3})\sum\limits_{i = 1}^{p}\sum\limits_{j = 1}^{p}\sum\limits_{s = 1}^{p}\overline{\chi_{3}}\left(i^3+j^3-s^3\right) \mathrm{e}_{p}\left(i+j-s\right).\nonumber\\ \end{eqnarray*}$

Applying Lemmas 2, 4 and 5 we obtain

$\begin{eqnarray*} &&\sum\limits_{u = 1}^{p-1}\chi_{3}(u)\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(ui^3+i\right)\right|^4\nonumber\\ & = &-\tau(\chi_{3})\cdot\overline{\chi_{3}}(3)\cdot\tau^3\left(\overline{\chi_{3}}\right)-\tau(\chi_{3})\left(p(\alpha-3)+3 \tau^3\left(\overline{\chi_{3}}\right)\right)+\tau(\chi_{3})\left(\overline{\chi_{3}}(3)\cdot\tau^3(\overline{\chi_{3}})-3p\right)\nonumber\\ & = &-\alpha p\cdot\tau(\chi_{3})-3p\cdot\tau^2(\overline{\chi_{3}}).\nonumber \end{eqnarray*}$

□

Proof of Theorem 2. Based on Theorem 1 and the identities obtained in ^[9]

$\begin{eqnarray*} \sum\limits_{u = 1}^{p-1}\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(ui^3+vi\right)\right|^4 = \left\{ \begin{array}{ll} 2p^3-p^2 & \text{ if}\quad 3\nmid p-1 , \\ 2p^3-7p^2 & \mathrm{ if\quad 3\mid p-1 .} \end{array}\right. \end{eqnarray*}$

We have

$\begin{eqnarray} &&\sum\limits_{v = 1}^{p-1}\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(i^3+vi\right)\right|^4 = \sum\limits_{v = 1}^{p-1}\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left((\overline{v}i)^3+i\right)\right|^4\\ & = &\sum\limits_{v = 1}^{p-1}\left(1+\chi_{3}(v)+\overline{\chi_{3}}(v)\right)\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(vi^3+i\right)\right|^4\\ & = &\sum\limits_{v = 1}^{p-1}\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(vi^3+i\right)\right|^4+\sum\limits_{v = 1}^{p-1}\chi_{3}(v)\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(vi^3+i\right)\right|^4\\ &&+\sum\limits_{v = 1}^{p-1}\overline{\chi_{3}}(v)\left|\sum\limits_{i = 1}^{p} \mathrm{e}_{p}\left(vi^3+i\right)\right|^4\\ & = &2p^3-7p^2-\alpha p\tau(\chi_{3})-3p\cdot\tau^2(\overline{\chi_{3}})-\alpha p\tau(\overline{\chi_{3}})-3p\cdot\tau^2(\chi_{3})\\ & = &2p^3-p^2-3p\left(\tau(\chi_{3})+\tau(\overline{\chi_{3}})\right)^2-\alpha p\left(\tau(\chi_{3})+\tau(\overline{\chi_{3}})\right). \end{eqnarray}$

(3.1)

Now we need to determine the value of the real number $\tau(\chi_{3})+\tau(\overline{\chi_{3}})$ in (3.1). For convenience, write the $A = \tau(\chi_{3})+\tau(\overline{\chi_{3}})$ , we construct cubic equation $A^3-3pA-\alpha p = 0$ based on (2.1) and $\tau(\chi_{3})\cdot\tau(\overline{\chi_{3}}) = p$ . According to Cardans formula (formula of roots of a cubic equation), the three roots of the equation are

$\begin{eqnarray*} &A_{1} = \left[{\frac{\alpha p}{2}+\left(\left(\frac{\alpha p}{2}\right)^{2}+\left(-p\right)^{3}\right)^{\frac{1}{2}}}\right]^{\frac{1}{3}}+\left[{\frac{\alpha p}{2}-\left(\left(\frac{\alpha p}{2}\right)^{2}+\left(-p\right)^{3}\right)^{\frac{1}{2}}}\right]^{\frac{1}{3}},\; \; \; \; \; \; \nonumber\\ &A_{2} = \omega\left[\frac{\alpha p}{2}+\left(\left(\frac{\alpha p}{2}\right)^{2}+\left(-p\right)^{3}\right)^{\frac{1}{2}}\right]^{\frac{1}{3}}+\omega^{2}\left[{\frac{\alpha p}{2}-\left(\left(\frac{\alpha p}{2}\right)^{2}+\left(-p\right)^{3}\right)^{\frac{1}{2}}}\right]^{\frac{1}{3}},\nonumber\\ &A_{3} = \omega^{2}\left[{\frac{\alpha p}{2}+\left(\left(\frac{\alpha p}{2}\right)^{2}+\left(-p\right)^{3}\right)^{\frac{1}{2}}}\right]^{\frac{1}{3}}+\omega\left[{\frac{\alpha p}{2}-\left(\left(\frac{\alpha p}{2}\right)^{2}+\left(-p\right)^{3}\right)^{\frac{1}{2}}}\right]^{\frac{1}{3}},\nonumber \end{eqnarray*}$

where $\omega = \frac{-1+\sqrt{3} \mathrm{i}}{2}$ .

It is clear that all $A_k$ $(k = 1, 2\ \text{or}\ 3)$ are real numbers, So $A = A_1, A_2$ or $A_3$ . Therefore, the proof of theorem is complete. □

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors gratefully appreciates the referees and editor for their helpful and detailed comments.

This work is supported by Hainan Provincial Natural Science Foundation of China (123RC473) and Natural Science Foundation of China (12126357).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

[1]	N. Jiang, W. Y. Wu, L. Wang, The quantum realization of Arnold and Fibonacci image scrambling, Quantum Inf. Process., 13 (2014), 1223–1236. https://doi.org/10.1007/s11128-013-0721-7 doi: 10.1007/s11128-013-0721-7
[2]	B. Prased, Coding theory on Lucas $p$ numbers, Discrete Math. Algorithms Appl., 8 (2016), 1650074. https://doi.org/10.1142/S1793830916500749 doi: 10.1142/S1793830916500749
[3]	T. Zhang, S. Li, R. Ge, M. Yuan, Y. Ma, A novel 1D hybrid chaotic map-based image compression and encryption using compressed sensing and Fibonacci-Lucas transform, Math. Probl. Eng., 2016 (2016), 7683687. https://doi.org/10.1155/2016/7683687 doi: 10.1155/2016/7683687
[4]	S. Halici, S. Oz, On Gaussian Pell polynomials and their some properties, Palest. J. Math., 7 (2018), 251–256.
[5]	M. Hashemi, E. Mehraban, Fibonacci length and the generalized order $k$ -Pell sequences of the 2-generator $p$ -groups of nilpotency class 2, J. Algebra Appl., 22 (2023), 2350061. https://doi.org/10.1142/S0219498823500615 doi: 10.1142/S0219498823500615
[6]	J. Hiller, Y. Aküzüm, Ö. Deveci, The adjacency-Pell-Hurwitz numbers, Integers, 18 (2018), A83. https://doi.org/10.5281/zenodo.10682656 doi: 10.5281/zenodo.10682656
[7]	E. Kilic, The generalized order- $k$ Fibonacci-Pell sequence by matrix methods, J. Comput. Appl. Math., 209 (2007), 133–145. https://doi.org/10.1016/j.cam.2006.10.071 doi: 10.1016/j.cam.2006.10.071
[8]	E. Kilic, The generalized Pell $(p, i)$ -numbers and their Binet formulas, combinatorial representations, sums, Chaos Solit. Fractals, 40 (2009), 2047–2063. https://doi.org/10.1016/j.chaos.2007.09.081 doi: 10.1016/j.chaos.2007.09.081
[9]	Ö. Deveci, The $k$ -nacci sequences and the generalized order $k$ -Pell sequences in the semi-direct product of finite cyclic groups, Chiang Mai J. Sci., 40 (2013), 89–98.
[10]	Ö. Deveci, A. G. Shannon, The quaternion-Pell sequence, Commun. Algebra, 46 (2018), 5403–5409. https://doi.org/10.1080/00927872.2018.1468906 doi: 10.1080/00927872.2018.1468906
[11]	M. Hashemi, E. Mehraban, The generalized order $k$ -Pell sequences in some special groups of nilpotency class 2, Commun. Algebra, 50 (2021), 1768–1784. https://doi.org/10.1080/00927872.2021.1988959 doi: 10.1080/00927872.2021.1988959
[12]	W. M. Abd-Elhameed, N. A. Zeyada, New identities involving generalized Fibonacci and generalized Lucas numbers, Indian J. Pure Appl. Math., 49 (2018), 527–537. https://doi.org/10.1007/s13226-018-0282-7 doi: 10.1007/s13226-018-0282-7
[13]	A. K. Amin, N. A. Zeyada, Some new identities of a type of generalized numbers involving four parameters, AIMS Math., 7 (2021), 12962–12980. https://doi.org/10.3934/math.2022718 doi: 10.3934/math.2022718
[14]	W. M. Abd-Elhameed, A. N. Philippou, N. A. Zeyada, Novel results for two generalized classes of Fibonacci and Lucas polynomials and their uses in the reduction of some radicals, Mathematics, 10 (2022), 2342. https://doi.org/10.3390/math10132342 doi: 10.3390/math10132342
[15]	W. M. Abd-Elhameed, A. Napoli, New formulas of convolved Pell polynomials, AIMS Math., 9 (2024), 565–593. https://doi.org/10.3934/math.2024030 doi: 10.3934/math.2024030
[16]	P. Ochalik, A. Wloch, On generalized Mersenne numbers, their interpretations and matrix generators, Ann. Univ. Mariae Curie-Skłodowska, Sect. A, 72 (2018), 69–76. https://doi.org/10.17951/a.2018.72.1.69-76 doi: 10.17951/a.2018.72.1.69-76
[17]	P. Catarino, H. Campos, P. Vasco, On the Mersenne sequence, Ann. Math. Inform., 46 (2016), 37–53.
[18]	M. Chelgham, A. Boussayoud, On the $k$ -Mersenne-Lucas numbers, Notes Number Theory Discrete Math., 27 (2021), 7–13. https://doi.org/10.7546/nntdm.2021.27.1.7-13 doi: 10.7546/nntdm.2021.27.1.7-13
[19]	A. S. Sergeer, Generalized Mersenne matrices and Balonin's conjecture, Autom. Control Comput. Sci., 48 (2014), 214–220. https://doi.org/10.3103/S0146411614040063 doi: 10.3103/S0146411614040063
[20]	Y. Soykan, On generalized $p$ -Mersenne numbers, Earthline J. Math. Sci., 8 (2022), 83–120. https://doi.org/10.34198/ejms.8122.83120 doi: 10.34198/ejms.8122.83120
[21]	Y. Zheng, S. Shon, Exact inverse matrices of Fermat and Mersenne circulant matrix, Abstr. Appl. Anal., 2015 (2015), 760823. https://doi.org/10.1155/2015/760823 doi: 10.1155/2015/760823
[22]	Y. Akuzum, Ö. Deveci, The Hadamard-type $k$ -step Fibonacci sequences in groups, Commun. Algebra, 48 (2020), 2844–2856. https://doi.org/10.1080/00927872.2020.1723609 doi: 10.1080/00927872.2020.1723609
[23]	L. Chen, Y. Chen, The $n$ -Diffie-Hellman problem and multiple-key encryption, Int. J. Inf. Secur., 11 (2012), 305–320. https://doi.org/10.1007/s10207-012-0171-8 doi: 10.1007/s10207-012-0171-8
[24]	H. Chien, Provably secure authenticated Diffie-Hellman key exchange for resource-limited smart card, J. Shanghai Jiaotong Univ. (Sci.), 19 (2014), 436–439. https://doi.org/10.1007/s12204-014-1521-7 doi: 10.1007/s12204-014-1521-7
[25]	D. Coppersmith, A. M. Odlzyko, R. Schroeppel, Discrete logarithms in $GF(p)$ , Algorithmica, 1 (1986), 1–15. https://doi.org/10.1137/0406010 doi: 10.1137/0406010
[26]	L. Harn, C. Lin, Efficient group Diffie-Hellman key agreement protocols, Comput. Electr. Eng., 40 (2014), 1972–1980. https://doi.org/10.1016/j.compeleceng.2013.12.018 doi: 10.1016/j.compeleceng.2013.12.018
[27]	M. Eftekhari, A Diffie-Hellman key exchange protocol using matrices over noncommutative rings, Groups Complex. Cryptol., 4 (2012), 167–176. https://doi.org/10.1515/gcc-2012-0001 doi: 10.1515/gcc-2012-0001
[28]	J. Partala, Algebraic generalization of Diffie-Hellman key exchange, J. Math. Cryptol., 12 (2018), 1–21. https://doi.org/10.1515/jmc-2017-0015 doi: 10.1515/jmc-2017-0015
[29]	P. A. Grillet, Abstract Algebra, 2 Eds., Berlin: Springer, 2007.
[30]	W. Stallings, Cryptography and Network Security: Principles and Practice, 7 Eds., Harlow: Pearson, 2017.

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New sequences from the generalized Pell $p-$ numbers and mersenne numbers and their application in cryptography