200 research outputs found
A mean value formula for elliptic curves
It is proved in this paper that for any point on an elliptic curve, the mean
value of x-coordinates of its n-division points is the same as its x-coordinate
and that of y-coordinates of its n-division points is n times of its
y-coordinate
An Information-Theoretic Framework for Evaluating Edge Bundling Visualization
Edge bundling is a promising graph visualization approach to simplifying the visual result of a graph drawing. Plenty of edge bundling methods have been developed to generate diverse graph layouts. However, it is difficult to defend an edge bundling method with its resulting layout against other edge bundling methods as a clear theoretic evaluation framework is absent in the literature. In this paper, we propose an information-theoretic framework to evaluate the visual results of edge bundling techniques. We first illustrate the advantage of edge bundling visualizations for large graphs, and pinpoint the ambiguity resulting from drawing results. Second, we define and quantify the amount of information delivered by edge bundling visualization from the underlying network using information theory. Third, we propose a new algorithm to evaluate the resulting layouts of edge bundling using the amount of the mutual information between a raw network dataset and its edge bundling visualization. Comparison examples based on the proposed framework between different edge bundling techniques are presented
Number of Jacobi quartic curves over finite fields
In this paper the number of -isomorphism
classes of Jacobi quartic curves, i.e., the number of Jacobi quartic
curves with distinct -invariants, over finite field
is enumerated
Pairing computation on Edwards curves with high-degree twists
In this paper, 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 littler faster than that proposed by Arene et.al.. Finally, to improve the efficiency of pairing computation we present twists of degree 4 and 6 on twisted Edwards curves
The Pairing Computation on Edwards Curves
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
Twisted Jacobi Intersections Curves
In this paper, the twisted Jacobi intersections which contains
Jacobi intersections as a special case is introduced. We show that
every elliptic curve over the prime field with three points of order
is isomorphic to a twisted Jacobi intersections curve. Some fast
explicit formulae for twisted Jacobi intersections curves in
projective coordinates are presented. These explicit formulae for
addition and doubling are almost as fast as the Jacobi
intersections. In addition, the scalar multiplication can be more
effective in twisted Jacobi intersections than in Jacobi
intersections. Moreover, we propose new addition formulae which are
independent of parameters of curves and more effective in reality
than the previous formulae in the literature
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