Fast point quadrupling on elliptic curves

Ciet et al. (2006) proposed an elegant method for trading inversions for multiplications when computing [2]P+Q from two given points P and Q on elliptic curves of Weierstrass form. Motivated by their work, this paper proposes a fast algorithm for computing [4]P with only one inversion in affine coordinates. Our algorithm that requires 1I + 8S + 8M, is faster than two repeated doublings whenever the cost of one field inversion is more expensive than the cost of four field multiplications plus four field squarings (i.e. I > 4M + 4S). It saves one field multiplication and one field squaring in comparison with the Sakai-Sakurai method (2001). Even better, for special curves that allow "a = 0" (or "b = 0") speedup, we obtain [4]P in affine coordinates using just 1I + 5S + 9M (or 1I + 5S + 6M, respectively).