Twisted Jacobi intersections curves

In this paper, the twisted Jacobi intersections curve which contains Jacobi intersections curve as a special case is introduced. We show that every elliptic curve over a field of odd characteristic with three points of order 2 is isomorphic to a twisted Jacobi intersections curve. The isomorphism classes of twisted Jacobi intersections curves over finite fields are enumerated. Some fast explicit formulae for twisted Jacobi intersections curves in projective coordinates are presented. These explicit formulae for addition and doubling are almost as fast as the Jacobi intersections. Moreover, we propose new addition formulae which are independent of parameters of curves and are more effective in reality than the previous formulae in the literature. In addition, an example shows that the scalar multiplication can be more effective in twisted Jacobi intersections than in Jacobi intersections.