New Formulae for Efficient Elliptic Curve Arithmetic

Hışıl, Hüseyin; Carter, Gary; Dawson, Ed

doi:10.1007/978-3-540-77026-8_11

Cited by 32 publications

(32 citation statements)

References 15 publications

(38 reference statements)

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“…The special treatment in [11] for obtaining the original coordinates is also removed since (X 3 : Y 3 : T 3 : Z 3 ) satisfies (15). The new representation can be equally fast as the representations in [3], [17], and [19]. However this is achieved by using only 4 coordinates rather than 5, 6, or 7 coordinates.…”

Section: Unified Point Addition In Q Ementioning

confidence: 99%

“…Duquesne's method converts the base point in weighted projective coordinates to a new point representation with 4 coordinates, performs the scalar multiplication within the new coordinate system, and outputs the final result in original weighted projective coordinates. Duquesne's improvement was followed by additional results in [3], [17], and [19]. However, the latter proposals tend to use more space -up to 7 coordinates per point-despite their speed advantage.…”

Section: Introductionmentioning

confidence: 99%

“…Hisil, Carter, and Dawson [17] introduced new point doubling formulae together with a fast point doubling algorithm costing only 3M + 4S in (X : Y : Z : X 2 : Z 2 ) with d = 1. Roughly at the same time essentially the same formulae were independently derived by Feng and Wu, see EFD [3].…”

Section: Introductionmentioning

confidence: 99%

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Jacobi Quartic Curves Revisited

Hışıl

Wong

Carter

et al. 2009

Information Security and Privacy

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This paper provides new results about efficient arithmetic on (extended) Jacobi quartic form elliptic curves y 2 = dx 4 + 2ax 2 + 1. Recent works have shown that arithmetic on an elliptic curve in Jacobi quartic form can be performed solidly faster than the corresponding operations in Weierstrass form. These proposals use up to 7 coordinates to represent a single point. However, fast scalar multiplication algorithms based on windowing techniques, precompute and store several points which require more space than what it takes with 3 coordinates. Also note that some of these proposals require d = 1 for full speed. Unfortunately, elliptic curves having 2-times-aprime number of points, cannot be written in extended Jacobi quartic form if d = 1. Even worse the contemporary formulae may fail to output correct coordinates for some inputs. This paper provides improved speeds using fewer coordinates without causing the above mentioned problems. For instance, our proposed point doubling algorithm takes only 2 multiplications, 5 squarings, and no multiplication with curve constants when d is arbitrary and a = ±1/2.

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Section: Unified Point Addition In Q Ementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Jacobi Quartic Curves Revisited

Hışıl

Wong

Carter

et al. 2009

Information Security and Privacy

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“…-tripling oriented Doche-Icart-Kohel curves (3DIK) [10] -Edwards curves (Edwards) [13,3] with inverted coordinates [4] -Hessian curves [6,15,16] -Extended Jacobi Quartics (ExtJQuartic) [6,12,15] -Jacobi intersections (JacIntersect) [6,17] -Jacobian coordinates (Jacobian) with the special case a 4 = −3 (Jacobian-3). Finally, some more optimizations can be found in [21,19] for the quintupling formulae.…”

Section: Elliptic Curvesmentioning

confidence: 99%

Elliptic Curve Scalar Multiplication Combining Yao’s Algorithm and Double Bases

Méloni

Hasan

2009

Lecture Notes in Computer Science

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Abstract. In this paper we propose to take one step back in the use of double base number systems for elliptic curve point scalar multiplication. Using a modified version of Yao's algorithm, we go back from the popular double base chain representation to a more general double base system. Instead of representing an integer k as P n i=1 2 b i 3 t i where (bi) and (ti) are two decreasing sequences, we only set a maximum value for both of them. Then, we analyze the efficiency of our new method using different bases and optimal parameters. In particular, we propose for the first time a binary/Zeckendorf representation for integers, providing interesting results. Finally, we provide a comprehensive comparison to state-ofthe-art methods, including a large variety of curve shapes and latest point addition formulae speed-ups.

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“…Recently, the improved formulae on this case are given in [18,13]. In [11], Hisil et al gave a new tripling formulae for Hessian curve in characteristic three. The generalized form of Hessian curves has been presented by Farashahi, Joye, Bernstein, Lange and Kohel [6,4].…”

Section: Introductionmentioning

confidence: 99%

Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three

Farashahi

Zhao

2013

Selected Areas in Cryptography

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Abstract. This paper presents new explicit formulae for the point doubling, tripling and addition for ordinary Weierstraß elliptic curves with a point of order 3 and their equivalent Hessian curves over finite fields of characteristic three. The cost of basic point operations is lower than that of all previously proposed ones. The new doubling, mixed addition and tripling formulae in projective coordinates require 3M + 2C, 8M + 1C + 1D and 4M + 4C + 1D respectively, where M, C and D is the cost of a field multiplication, a cubing and a multiplication by a constant. Finally, we present several examples of ordinary elliptic curves in characteristic three for high security levels.

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New Formulae for Efficient Elliptic Curve Arithmetic

Cited by 32 publications

References 15 publications

Jacobi Quartic Curves Revisited

Jacobi Quartic Curves Revisited

Elliptic Curve Scalar Multiplication Combining Yao’s Algorithm and Double Bases

Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three

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