Volume 5, Issue 1

Several forms of elliptic curves are suggested for an efficient implementation of Elliptic Curve Cryptography. However, a complete description of the group law has not appeared in the literature for most popular forms. This paper presents group law in affine coordinates for three forms of elliptic curves. With the existence of the proposed affine group laws, stating the projective group law for each form becomes trivial. This work also describes an automated framework for studying elliptic curve group law, which is applied internally when preparing this work.

In this work, we are interested in arithmetic on large prime field and their extensions of small degree. We explain why it is very interesting to use RNS arithmetic for the base field 𝔽 p when computations in 𝔽 p k have to be done, essentially thanks to lazy reduction. This is for example the case for pairing computations on ordinary curves (as MNT or BN curves). We show that using RNS can considerably decrease the number of basic operations required for a pairing computation in many popular situations.

Let y 2 = x 3 + ax + b be an elliptic curve over 𝔽 p , p being a prime number greater than 3, and consider a, b ∈ [1, p ]. In this paper, we study elliptic curve isomorphisms, with a view towards reduction in the size of elliptic curves coefficients. We first consider reducing the ratio a / b . We then apply these considerations to determine the number of elliptic curve isomorphism classes. Later we work on both coefficients. We introduce the number M ( p ) as the lower bound of all M ∈ ℕ such that each isomorphism class has a representative with max( a, b ) < M . Using results from the theory of uniform distributions, we prove upper and lower bounds of the form c 1 p 1/2 < M ( p ) < c 2 p 3/4 with explicit constants c 1 , c 2 > 0.