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\title{\textbf{FHE scheme based on a non-commutative Ring $\mathbb{F}_{q}[\varepsilon], \varepsilon^4=\varepsilon^3$}} 
\author[1,*]{Bilel Selikh}
\author[2]{Abdelhakim Chillali}
\author[1]{Douadi Mihoubi}
\author[1]{Nacer Ghadbane}
\affil[1]{{\small Laboratory of Pures and Applied Mathematics, Department of Mathematics, Mohamed Boudiaf University of Msila, Algeria}}
\affil[2]{{\small Sidi Mohamed Ben Abdellah University, FP, LSI, Taza, Morocco}\vspace{0.2cm} 

{\small \textbf{*} 
\email{bilel.selikh@univ-msila.dz}  \hspace{2.8cm} \email{abdelhakim.chillali@usmba.ac.ma} \hspace{2.3cm} 
\email{douadi.mihoubi@univ-msila.dz} \hspace{2.3cm} 
\email{nacer.ghadbane@univ-msila.dz}}}
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\begin{abstract}
Let $R:=\mathbb{F}_{q}[\varepsilon]$ be a finite quotient ring, where $\varepsilon^4=\varepsilon^3$, with $\mathbb{F}_{q}$ a finite field, where 
$q=p^r,p\geq 5$, $p$ prime number and $r\in \mathbb{N}^{*}$. The conjugal classical problem(CCP) is one of the famous problems in a non commutative ring. 
In this work, we introduce a new method of cryptography based on this problem in the ring R. In the fisrt time, we define a non commutative ring $R$. Furthermore, we study the problem of conjugal classical over the ring $R$. In generally this problem is very difficult to solve. Precisely, we will give a cryptogaphic application over this a non commutative ring based on CCP and disrete logarithm problem(DLP). \\
\keyword: Elliptic curve, Non commutative ring, Cryptography, Conjugal problem, Fully homomorphic cryptosystems.\\
\msc: 14H52, 11T55, 94A60, 11T71.
\end{abstract}

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\bibitem{1}W. DIFFIE and M. HELLMAN, \textit{New directions in cryptography}, IEEE Transactions on Information Theory. 22(6), pp. 644-654, (1976).
DOI: 10.1109/TIT.1976.1055638 
\bibitem{2}M. SAHMOUDI and A. CHILLALI, \textit{Key exchange over particular algebraic 
closure ring}, TATRA MOUNTAINS Mathematical Publications, 70, 151–162,(2017). 
DOI: 10.1515/tmmp-2017-0024

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