In [11, p.52-57], M.Virat has established the group law on the elliptic curve defined over the ring Fpd[ε], εn = 0, where p a prime number ≥ 5, d ∈ ℕ* and n = 2; and in [4] A.Chillali has, on his part studied the same case for p and extend it to n = 3. In this article we will study the elliptic curve over the ring A2 = F3d[ε], where ε2 = 0. More precisely we will give many various explicit formulas describing the binary operations calculus in Ea,b2, where Ea,b2 is the elliptic curve over A2, and we will reduce the cost of the complexity of the calculus found in Theorem 1.1, by proving the Lemmas 3.1, 3.2, 3.3, 3.4, 3.5 and 3.6; as it is shown in the Section 3.
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