scholarly journals The Pairing Computation on Edwards Curves

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hongfeng Wu ◽  
Liangze Li ◽  
Fan Zhang
Keyword(s):  
Geometric Interpretation ◽  
Tate Pairing ◽  
Quadric Surfaces ◽  
Edwards Curves ◽  
Explicit Formulae ◽  
Pairing Computation ◽  
Twisted Edwards Curves ◽  
Group Law

We propose an elaborate geometry approach to explain the group law on twisted Edwards curves which are seen as the intersection of quadric surfaces in place. Using the geometric interpretation of the group law, we obtain the Miller function for Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on twisted Edwards curves. Our formulae for the doubling step are a little faster than that proposed by Arène et al. Finally, to improve the efficiency of pairing computation, we present twists of degrees 4 and 6 on twisted Edwards curves.

An optimal Tate pairing computation using Jacobi quartic elliptic curves

2018 ◽  
Vol 35 (4) ◽  
pp. 1086-1103
Author(s):  
Srinath Doss ◽  
Roselyn Kaondera-Shava
Keyword(s):  
Elliptic Curves ◽  
Tate Pairing ◽  
Pairing Computation

scholarly journals Fast Jacobian Group Operations for C3,4 Curves over a Large Finite Field

2007 ◽  
Vol 10 ◽  
pp. 307-328 ◽  
Author(s):  
Fatima K. Abu Salem ◽  
Kamal khuri-makdisi
Keyword(s):  
Finite Field ◽  
Linear Algebra ◽  
Algebraic Curve ◽  
Vector Spaces ◽  
Explicit Formulae ◽  
Group Law

Let C be an arbitrary smooth algebraic curve of genus g over a large finite field F. The authors of this paper revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi [math.NT/0409209, to appear in Mathematics of Computation]. The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| ≫ g and which asymptotically require O(g2.376) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, the authors provide explicit formulae for performing the group law on Jacobians of C3,4 curves of genus 3. They show show that, typically, the addition of two distinct elements in the Jacobian of a C3,4 curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.

scholarly journals Compression for trace zero points on twisted Edwards curves

2016 ◽  
Vol 10 (1) ◽  
Author(s):  
Giulia Bianco ◽  
Elisa Gorla
Keyword(s):  
Elliptic Curves ◽  
Edwards Curves ◽  
Twisted Edwards Curves ◽  
Zero Points

AbstractWe propose two optimal representations for the elements of trace zero subgroups of twisted Edwards curves. For both representations, we provide efficient compression and decompression algorithms. The efficiency of the algorithm is compared with the efficiency of similar algorithms on elliptic curves in Weierstrass form.

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