Binary Edwards Curves
نویسندگان
چکیده
This paper presents a new shape for ordinary elliptic curves over fields of characteristic 2. Using the new shape, this paper presents the first complete addition formulas for binary elliptic curves, i.e., addition formulas that work for all pairs of input points, with no exceptional cases. If n ≥ 3 then the complete curves cover all isomorphism classes of ordinary elliptic curves over F2n . This paper also presents dedicated doubling formulas for these curves using 2M+ 6S+ 3D, where M is the cost of a field multiplication, S is the cost of a field squaring, and D is the cost of multiplying by a curve parameter. These doubling formulas are also the first complete doubling formulas in the literature, with no exceptions for the neutral element, points of order 2, etc. Finally, this paper presents complete formulas for differential addition, i.e., addition of points with known difference. A differential addition and doubling, the basic step in a Montgomery ladder, uses 5M+ 4S+ 2D when the known difference is given in affine form.
منابع مشابه
Fast Algorithm for Converting Ordinary Elliptic Curves into Binary Edward Form
Scalar multiplication is computationally the most expensive operation in elliptic curve cryptosystems. Many techniques in literature have been proposed for speeding up scalar multiplication. In 2008, Bernstein et al proposed binary Edwards curves on which scalar multiplication is faster than traditional curves. At Crypto 2009, Bernstein obtained the fastest implementation for scalar multiplicat...
متن کاملEdwards model of elliptic curves de ned over any nite eld
In this paper, we present an Edwards model for elliptic curves which is de ned over any perfect eld and in particular over nite elds. This Edwards model is birationally equivalent to the well known Edwards model over non-binary elds and is ordinary over binary elds. For this, we use theta functions of level four to obtain an intermediate model that we call a level 4 theta model. This model enab...
متن کاملHalving on Binary Edwards Curves
Edwards curves have attracted great interest for their efficient addition and doubling formulas. Furthermore, the addition formulas are strongly unified or even complete, i.e., work without change for all inputs. In this paper, we propose the first halving algorithm on binary Edwards curves, which can be used for scalar multiplication. We present a point halving algorithm on binary Edwards curv...
متن کاملFaster Addition and Doubling on Elliptic Curves
Edwards recently introduced a new normal form for elliptic curves. Every elliptic curve over a non-binary field is birationally equivalent to a curve in Edwards form over an extension of the field, and in many cases over the original field. This paper presents fast explicit formulas (and register allocations) for group operations on an Edwards curve. The algorithm for doubling uses only 3M+ 4S,...
متن کاملEfficient Implementation of Elliptic Curve Point Operations Using Binary Edwards Curves
This paper presents a deterministic algorithm for converting points on an ordinary elliptic curve (defined over a field of characteristic 2) to points on a complete binary Edwards curve. This avoids the problem of choosing curve parameters at random. When implemented on a large (512 bit) hardware multiplier, computation of point multiplication using this algorithm performs significantly better,...
متن کاملTwisted μ4-Normal Form for Elliptic Curves
We introduce the twisted μ4-normal form for elliptic curves, deriving in particular addition algorithms with complexity 9M+ 2S and doubling algorithms with complexity 2M + 5S + 2m over a binary field. Every ordinary elliptic curve over a finite field of characteristic 2 is isomorphic to one in this family. This improvement to the addition algorithm, applicable to a larger class of curves, is co...
متن کامل