On a new generalization of Huff curves

Recently two kinds of Huff curves were introduced as elliptic curves models and their arithmetic was studied. It was also shown that they are suitable for cryptographic use such as Montgomery curves or Koblitz curves (in Weierstrass form) and Edwards curves. In this work, we introduce the new generalized Huff curves ax(y − c) = by(x−d) with abcd(ac−bd) 6= 0, which contains the generalized Huff’s model ax(y − d) = by(x − d) with abd(a − b) 6= 0 of Joye-TibouchiVergnaud and the generalized Huff curves x(ay − 1) = y(bx − 1) with ab(a− b) 6= 0 of Wu-Feng as a special case. The addition law in projective coordinates is as fast as in the previous particular cases. More generally all good properties of the previous particular Huff curves, including completeness and independence of two of the four curve parameters, extend to the new generalized Huff curves. We verified that the method of Joye-Tibouchi-Vergnaud for computing of pairings can be generalized over the new curve.