Browsing by Author "Bilel Selikh"

Now showing 1 - 2 of 2
  • Thumbnail Image
    ItemOpen Access
    CLASSIFICATION OF ELEMENTS IN ELLIPTIC CURVE OVER THE RING F
    (Université de M'sila, 2022) Bilel Selikh; Douadi Mihoubi
    Let Fq["] := Fq[X]/(X4 − X3) be a finite quotient ring where "4 = "3, with Fq is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over Fq["], "4 = "3 of characteristic p 6= 2, 3 given by homogeneous Weierstrass equation of the form Y 2Z = X3 + aXZ2 + bZ3 where a and b are parameters taken in Fq["]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve Ea,b(Fq["]) and we will show that E 0(a), 0(b)(Fq) and E 1(a), 1(b)(Fq) are two elliptic curves over the finite field Fq, such that 0 is a canonical projection and 1 is a sum projection of coordinate of element in Fq["]. Precisely, we give a classification of elements in elliptic curve over the finite ring Fq["].
  • Thumbnail Image
    ItemOpen Access
    Sur les courbes elliptiques et application à la cryptographie
    (Université de M'sila, 2022) Bilel Selikh
    This thesis deals with the study of the elliptic curves over finite rings and their cryptographic applications. Firstly, we defined the elliptic curves Ea,b(Fq["]) and Ea,b(F3d ["]) over the rings Fq["] and F3d ["] respectively, with "4 = "3 by its projective equations, then we studied the classification of elements in these elliptic curves. Moreover, using the elliptic curve Ea,b(Fq["]), we introduced new non-commutative cryptography schemes on a special rings so that these cryptosystems are based on the two hard problems, the conjugal classical problem and the discrete logarithm problem, and they have strong security and very difficult to solve for decryption. At the end of the thesis we have given a numerical example of cryptography (encryption and decryption) on the elliptic curve E4 a,b over the ring F3d ["] where "4 = 0 by using two methods (with a secret key and a password).