Improved Algorithms for Elliptic Curve Arithmetic in GF(2n)

Abstract

Abstract. This paper describes three contributions for efficient implementation of elliptic curve cryptosystems in GF (2 n ). The first is a new method for doubling an elliptic curve point, which is simpler to implement than the fastest known method, due to Schroeppel, and which favors sparse elliptic curve coefficients. The second is a generalized and improved version of the Guajardo and Paar's formulas for computing repeated doubling points. The third contribution consists of a new kind of projective coordinates that provides the fastest known arithmetic on elliptic curves. The algorithms resulting from this new formulation lead to a running time improvement for computing a scalar multiplication of about 17% over previous projective coordinate methods.

Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three

“…There are some different types of projective coordinates which have the respective advantages in efficiency. The relationship between affine coordinates (x, y) and projective coordinates (X, Y, Z) is (x, y) = (X/Z, Y /Z), for Jacobian projective coordinates, (x, y) = (X/Z 2 , Y /Z 3 ), and for López Dahab projective coordinates [16], (x, y) = (X/Z, Y /Z 2 ).…”

Efficient Arithmetic on Hessian Curves

“…Moreover, the doubling formulas for generalized Hessian curves are faster than doubling formulas using projective coordinates in short Weierstraß form, see [2]. But, they are slower than various doubling formulas using Jacobian [2], Lopez-Dahab representations of short Weierstraß form [2,29,24,6] and projective representation of binary Edwards [6].…”

Novel Precomputation Schemes for Elliptic Curve Cryptosystems

“…Let us illustrate the latter with the addition formula due to [15] and later refined by [5]. The cost of adding two points P Q + with the latter formula takes 13M + 4S.…”