原文: https://www.cnblogs.com/Kalafinaian/p/7392505.html
前言
ECC英文全称"Ellipse Curve Cryptography"
与传统的基于大质数因子分解困难性的加密方法不同,ECC通过椭圆曲线方程式的性质产生密钥
ECC164位的密钥产生一个安全级,相当于RSA 1024位密钥提供的保密强度,而且计算量较小,处理速度更快,存储空间和传输带宽占用较少。目前我国居民二代身份证
正在使用 256 位的椭圆曲线密码,虚拟货币比特币
也选择ECC作为加密算法。
从射影平面讲起
古希腊数学家欧几里得的《几何原本》提出了五条公设。
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1.由任意一点到任意一点可作直线。
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2.一条有限直线可以继续延长。
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3.以任意点为心及任意的距离可以画圆。
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4.凡直角都相等。
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5.同一平面内一条直线a和另外两条直线b.c相交,若在a某一侧的两个内角的和小于两直角,则b.c两直线经无限延长后在该侧相交。
《几何原本》只有在第29个命题
一条直线与两条平行直线相交,则所成的内错角相等,同位角相等,且同旁内角之和等于两直角
中才用到第五公设,即《几何原本》中可不依靠第五公设而推出前28命题。因此,一些数学家提出,第五公设能不能不作为公设,而作为定理?能不能依靠前四个公设来证明第五公设?这就是几何发展史上最著名的,争论了长达两千多年的关于“平行线理论”的讨论
1820年代,俄国喀山大学罗巴切夫斯基用“至少可以找到两条相异的直线,且都通过P点,并不与直线R相交”代替第五公设,然后与欧氏几何的前四个公设结合成一个公理系统,他经过细致深入的推理过程中,得出了一个又一个在直觉上匪夷所思,但在逻辑上毫无矛盾的几何体系。
这种几何学被称为罗巴切夫斯基几何,简称罗氏几何。从罗氏几何学中,可以得出这样一个结论:逻辑上不矛盾的一些公理都有可能提供一种几何学。现存非欧几何的类型可以概括如下:
1.坚持第五公设,引出欧几里得几何。
2.“可以引最少两条平行线”为公设,罗氏几何(双曲几何)。
3.“一条平行线也不能引”为公设,黎曼几何(椭圆几何)
左:双曲几何,即罗氏几何;中:欧几里德几何;右:椭圆几何,即黎曼几何
了解非欧式几何,就可以理解平行线的交点。
定义平行线相交于无穷远点P∞,使平面上所有直线都统一为有唯一的交点
性质:
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1.一条直线只有一个无穷远点;一对平行线有公共的无穷远点
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2.任何两条不平行的直线有不同的无穷远点(否则会造成有两个交点)
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3.平面上全体无穷远点构成一条无穷远直线
射影平面:平面上全体无穷远点与全体平常点构成射影平面
射影平面点的定义
对普通平面上点(x,y),令x=X/Z,y=Y/Z,Z≠0,则投影为射影平面上的点(X:Y:Z)
求点(1,2)在新的坐标体系下的坐标
∵X/Z=1 ,Y/Z=2(Z≠0)
∴X=Z,Y=2Z ∴坐标为(Z:2Z:Z),Z≠0
即(1:2:1)(2:4:2)(1.2:2.4:1.2)等形如(Z:2Z:Z),Z≠0的坐标都是(1,2)在新的坐标体系下的坐标
(2) 求平行线L1:X+2Y+3Z=0 与L2:X+2Y+Z=0 相交的无穷远点
∵ L1∥L2 所以有Z=0, X+2Y=0
∴坐标为(-2Y:Y:0),Y≠0
即(-2:1:0)(-4:2:0)(-2.4:1.2:0)等形如(-2Y:Y:0),Y≠0
椭圆曲线
一条椭圆曲线是在射影平面上满足威尔斯特拉斯方程(Weierstrass)所有点的集合
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1椭圆曲线方程是一个齐次方程
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2曲线上的每个点都必须是非奇异的(光滑的),偏导数FX(X,Y,Z)、FY(X,Y,Z)、FZ(X,Y,Z)不同为0
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3圆曲线的形状,并不是椭圆的。只是因为椭圆曲线的描述方程,类似于计算一个椭圆周长的方程故得名
椭圆曲线示例
非椭圆曲线示例
这两个方程都不是椭圆曲线,因为他们在(0:0:1)点处(即原点)没有切线,不满足椭圆曲线每个点都必须是非奇异的(光滑的),
椭圆曲线普通方程
椭圆曲线普通方程:
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无穷远点 (0, Y, 0)
平常点(x,y)斜率k:
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椭圆曲线阿贝尔群
我们已经看到了椭圆曲线的图象,但点与点之间好象没有什么联系。我们能不能建立一个类似于在实数轴上加法的运算法则呢?这就要定义椭圆曲线的加法群,这里需要用到近世代数中阿贝尔群。
在数学中,群是一种代数结构,由一个集合以及一个二元运算所组成。已知集合和运算(G,*)如果是群则必须满足如下要求
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封闭性:∀a,b∈G,a*b ∈ G
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结合性: ∀a,b,c∈G ,有 (ab)c = a* (b*c)
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单位元:ョe∈G, ∀a ∈G,有ea = ae = a
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逆元: ∀a ∈G ,ョb∈G 使得 ab = ba = e
阿贝尔群除了上面的性质还满足交换律公理(ab)c = a* (b*c)
同样在椭圆曲线也可以定义阿贝尔群。
任意取椭圆曲线上两点P、Q(若P、Q两点重合,则作P点的切线),作直线交于椭圆曲线的另一点R',过R'做y轴的平行线交于R,定义P+Q=R。这样,加法的和也在椭圆曲线上,并同样具备加法的交换律、结合律
同点加法
若有k个相同的点P相加,记作kP
P+P+P=2P+P=3P
有限域椭圆曲线
椭圆曲线是连续的,并不适合用于加密;所以,我们必须把椭圆曲线变成离散的点,我们要把椭圆曲线定义在有限域上。
我们给出一个有限域Fp
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Fp中有p(p为质数)个元素0,1,2,…, p-2,p-1
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Fp的加法是a+b≡c(mod p)
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Fp的乘法是a×b≡c(mod p)
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Fp的除法是a÷b≡c(mod p),即 a×b^(-1)≡c (mod p),b-1也是一个0到p-1之间的整数,但满足b×b-1≡1 (mod p)
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Fp的单位元是1,零元是 0
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Fp域内运算满足交换律、结合律、分配律
椭圆曲线Ep(a,b),p为质数,x,y∈[0,p-1]
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选择两个满足下列约束条件的小于p的非负整数a、b
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Fp上的椭圆曲线同样有加法
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1.无穷远点 O∞是零元,有O∞+ O∞= O∞,O∞+P=P
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2.P(x,y)的负元是 (x,-y mod p)= (x,p-y) ,有P+(-P)= O∞
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3.P(x1,y1),Q(x2,y2)的和R(x3,y3) 有如下关系:
x3≡k2-x1-x2(mod p)
y3≡k(x1-x3)-y1(mod p)
若P=Q 则 k=(3x2+a)/2y1mod p
若P≠Q,则k=(y2-y1)/(x2-x1) mod p
例题椭圆曲线已知E23(1,1)上两点P(3,10),Q(9,7),求(1)-P,(2)P+Q,(3) 2P
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