In this work we introduce a new method of cryptography based on the matrices over a finite field Fq, were q is a power of a prime number p. The first time we construct the matrix M = A 1 A 2 0 A 3 were A 1 , A 2 and A 3 are matrices of order n with coefficients in Fq and 0 is the zero matrix of order n. We prove thatA 2 A k 3 for all l ∈ N * . After we will make a cryptographic scheme between the two traditional entities Alice and Bob.
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).
The goal of this article is to study elliptic curves over the ring F q [ ], with F q a finite field of order q and with the relation 5 = 0. The motivation for this work came from search for new groups with intractable (DLP) discrete logarithm problem is therefore of great importance. The observation groups where the discrete logarithm problem (DLP) is believed to be intractable have proved to be inestimable building blocks for cryptographic applications.
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