文档介绍：有限域 (Finite Fields)
汤学明 xmtang126@
信息安全实验室
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安全椭圆曲线的构造
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The Arc Length of an Ellipse
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The arc length of a (semi)circle
-a
a
x2+y2=a2
is given by the familiar integral
is plicated
The arc length of a (semi)ellipse
x2/a2 + y2/b2 = 1
-a
b
a
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An Elliptic Curve!
The Arc Length of an Ellipse
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Let k2 = 1 – b2/a2 and change variables x ax. Then the arc length of an ellipse is
with y2 = (1 – x2) (1 – k2x2) = quartic in x.
An elliptic integral is an integral , where R(x,y) is a rational function of the coordinates (x,y) on an “elliptic curve” E : y2 = f(x) = cubic or quartic in x.
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Elliptic Integrals and Elliptic Functions
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Doubly periodic functions are called elliptic functions.
Its inverse function w = sin(z) is periodic with period 2.
The circular integral is equal to sin-1(w).
The elliptic integral has an inverse
w = (z) with two plex periods 1 and 2.
(z + 1) = (z + 2) = (z) for all z C.
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Elliptic Functions and Elliptic Curves
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This equation looks familiar
The -function and its derivative satisfy an algebraic relation
The double periodicity of (z) means that it is a function on the quotient space C/L, where L is the lattice
L = { n1w1 + n2w2 : n1,n2 Z }.
1
2
1+ 2
L
(z) and ’(z) are functions on a fundamental parallelogram
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plex Points on an Elliptic Curve
E(C) =
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The -function gives plex analytic isomorphism
Thus the points of E with coordinates in plex numbers C form a torus, that is, the surface of a donut.
E(C)
Parallelogram with opposite sides identified = a torus
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Elliptic curves over R
Elliptic curves over R of form y2=x3+ax+b
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Elliptic curves over R
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