We show that a F q 2-maximal curve of genus q(q − 3)/6 in characteristic three is unique up to F q 2-isomorphism unless an unexpected situation occurs.

Previous results on genera g of Fq2 -maximal curves are improved: (1) Either g ≤ ⌊(q − q + 4)/6⌋ , or g = ⌊(q − 1)/4⌋ , or g = q(q − 1)/2 . (2) The hypothesis on the existence of a particular Weierstrass point in [2] is proved. (3) For q ≡ 1 (mod 3), q ≥ 13, no Fq2 -maximal curve of genus (q−1)(q−2)/3 exists. (4) For q ≡ 2 (mod 3), q ≥ 11, the non-singular Fq2 -model of the plane curve of equat...

We exhibit a genus–2 curve C defined over Q(T ) which admits two independent morphisms to a rank–1 elliptic curve defined over Q(T ). We describe completely the set of Q(T )–rational points of the curve C and obtain a uniform bound for the number of Q–rational points of a rational specialization Ct of the curve C for a certain (possibly infinite) set of values t ∈ Q. Furthermore, for this set o...

We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field F q 2 whose number of F q 2-rational points reaches the Hasse-Weil upper bound. Under a hypothesis on non-gaps at a rational point, we prove that maximal curves are F q 2-isomorphic to y q + y = x m , for some m ∈ Z +. As a consequence we show that a maximal curve of genus g = (q − ...

Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...