Let Fq be a finite field of q elements, where q is a power of a prime number p greater than or equal to 5. In this paper, we study the elliptic curve denoted Ea,b(Fq[e]) over the ring Fq[e], where e2 = e and (a,b) ∈ (Fq[e])2. In a first time, we study the arithmetic of this ring. In addition, using the Weierstrass equation, we define the elliptic curve Ea,b(Fq[e]) and we will show that Eπ0(a),π0(b)(Fq) and Eπ1(a),π1(b)(Fq) are two elliptic curves over the field Fq, where π0 and π1 are respectively the canonical projection and the sum projection of coordinates of X ∈Fq[e]. Precisely, we give a bijection between the sets Ea,b(Fq[e]) and Eπ0(a),π0(b)(Fq)×Eπ1(a),π1(b)(Fq).

Let $K=\mathbb{Q}(\alpha )$ be a pure number field generated by a root α of a monic irreducible polynomial $F(x)=x^{2^r·5^s}-m,$ with $m\ne \mp 1$ is a square free integer, r and s are two positive integers. In this article, we study the monogenity of K. We prove that if $m\cancel{\equiv }1(\text{mod}\mathrm{}4)$ and $\overline{m}\notin \{\overline{1},\overline{7},\overline{18},\overline{24}\}(\text{mod}\mathrm{}25),$ then K is monogenic. We give also sufficient conditions on r, s, and m for K to be not monogenic. Some illustrating examples are given too.

In this work we introduce a new method of cryptography based on the matrices over a finite field $\mathbb{F}_{q}$, were $q$ is a power of a prime number $p$. The first time we construct the matrix $M=\left( \begin{array}{cc} A_{1} & A_{2} \\ 0 & A_{3} \\ \end{array} \right) $ were \ $A_{i}$ \ with $i \in \{1, 2, 3 \}$ is the matrix of order $n$ \ in \ $\mathcal{M}(\mathbb{F}_{q})$ - the set of matrices with coefficients in $\mathbb{F}_{q}$ - and $0$ is the zero matrix of order $n$. We prove that $M^{l}=\left( \begin{array}{cc} A_{1}^{l} & (A_{2})_{l} \\ 0 & A_{3}^{l} \\ \end{array} \right) $ were $(A_{2})_{l}=\sum\limits_{k=0}^{l-1} A_{1}^{l-1-k}A_{2}A_{3}^{k}$ for all $l\in \mathbb{N}^{\ast}$. After we will make a cryptographic scheme between the two traditional entities Alice and Bob.

This paper proposes the generalization of our previous work to the ring An=F3d[X]/(Xn). All results found before in A2, A3 and A4 [1–3] hold in An; but the approach here is clearly different, and has given more interesting results, specially when 3 does not divide #Ea0,b01; the elliptic curve over the ring An is a direct sum of the elliptic curve over the field F3d and, unexpectedly its own subgroup of elements with the third projective coordinate not invertible, instead of F3dn as it was thought in the earlier works. Other results are deduced from, we cite the equivalence of the Discrete Logarithm Problem (DLP) on the elliptic curve over the ring An and the field F3d, which is beneficial for cryptanalysts and cryptographers as well, and we will set the theoretic foundations to build a cryptosystem similar to the one in [4] with more benefits, which will be specified later