The elliptic curve E_{a,b}(F_p[e_1, e_2, e_3])

Abstract

In this paper, we study the elliptic curve E_a,b over ring F_p[e₁, e₂, e₃], where e_ie_i = e_i and e_ie_j = 0 if i ≠ j. We characterize the invertible elements of F_p[e₁, e₂, e₃], and we determine the relation between E_a,b(F_p[e₁, e₂, e₃]) and E_a,b(F_p). Finally we show that Card(E_a,b(F_p[e₁, e₂, e₃])) ≥ (Card(E_a,b(F_p))-3)³ + Card(E_a,b(F_p)).