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SM2椭圆曲线公钥密码算法(完整)
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国密局发布的SM2椭圆曲线公钥密码算法文档,pdf,完整版的。1、总则;2、数字签名算法;3、密码交换协议;4、公钥加密算法
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SM2椭圆曲线公钥密码算法
第1部分:总则
Public Key Cryptographic Algorithm SM2 Based on Elliptic Curves
Part 1: General
国家密码管理局
2010年12月
目目目 次次次
前言····················································································· IV
引言····················································································· V
1 范围······················································································ 1
2 符号和缩略语············································································· 1
3 域和椭圆曲线············································································· 2
3.1 有限域 ················································································ 2
3.1.1 概述 ··········································································· 2
3.1.2 素域F
p
········································································· 2
3.1.3 二元扩域F
2
m
···································································· 2
3.2 有限域上的椭圆曲线 ··································································· 2
3.2.1 F
p
上的椭圆曲线 ································································ 3
3.2.2 F
2
m
上的椭圆曲线 ······························································· 3
3.2.3 椭圆曲线群 ···································································· 3
3.2.4 椭圆曲线多倍点运算 ··························································· 4
3.2.5 椭圆曲线离散对数问题(ECDLP) ················································ 4
3.2.6 弱椭圆曲线 ···································································· 4
4 数据类型及其转换 ········································································ 5
4.1 数据类型 ·············································································· 5
4.2 数据类型转换 ········································································· 5
4.2.1 整数到字节串的转换 ··························································· 5
4.2.2 字节串到整数的转换 ··························································· 6
4.2.3 比特串到字节串的转换 ························································· 6
4.2.4 字节串到比特串的转换 ························································· 6
4.2.5 域元素到字节串的转换 ························································· 6
4.2.6 字节串到域元素的转换 ························································· 6
4.2.7 域元素到整数的转换 ··························································· 6
4.2.8 点到字节串的转换 ······························································ 7
4.2.9 字节串到点的转换 ······························································ 7
5 椭圆曲线系统参数及其验证································································ 8
5.1 一般要求 ·············································································· 8
5.2 F
p
上椭圆曲线系统参数及其验证························································ 8
5.2.1 F
p
上椭圆曲线系统参数 ························································· 8
5.2.2 F
p
上椭圆曲线系统参数的验证··················································· 8
5.3 F
2
m
上椭圆曲线系统参数及其验证 ······················································· 8
5.3.1 F
2
m
上椭圆曲线系统参数························································· 8
5.3.2 F
2
m
上椭圆曲线系统参数的验证 ·················································· 9
6 密钥对的生成与公钥的验证································································ 9
6.1 密钥对的生成 ········································································· 9
6.2 公钥的验证 ············································································ 9
6.2.1 F
p
上椭圆曲线公钥的验证 ······················································· 9
6.2.2 F
2
m
上椭圆曲线公钥的验证 ······················································ 10
I
附录A(资料性附录) 关于椭圆曲线的背景知识 ················································ 11
A.1 素域F
p
················································································ 11
A.1.1 素域F
p
的定义 ·································································· 11
A.1.2 F
p
上椭圆曲线的定义···························································· 11
A.1.3 F
p
上椭圆曲线的阶 ······························································ 13
A.2 二元扩域F
2
m
··········································································· 13
A.2.1 二元扩域F
2
m
的定义 ····························································· 13
A.2.2 F
2
m
上椭圆曲线的定义 ··························································· 19
A.2.3 F
2
m
上椭圆曲线的阶 ····························································· 22
A.3 椭圆曲线多倍点运算 ·································································· 22
A.3.1 概述 ··········································································· 22
A.3.2 椭圆曲线多倍点运算的实现 ····················································· 22
A.3.3 椭圆曲线多倍点运算复杂度估计 ················································ 23
A.4 求解椭圆曲线离散对数问题的方法 ····················································· 24
A.4.1 椭圆曲线离散对数求解方法 ····················································· 24
A.4.2 安全椭圆曲线满足的条件 ······················································· 25
A.5 椭圆曲线上点的压缩 ·································································· 26
A.5.1 概述 ··········································································· 26
A.5.2 F
p
上椭圆曲线点的压缩与解压缩方法 ············································ 26
A.5.3 F
2
m
上椭圆曲线点的压缩与解压缩方法 ··········································· 26
附录B(资料性附录) 数论算法 ································································ 27
B.1 有限域和模运算 ······································································· 27
B.1.1 有限域中的指数运算···························································· 27
B.1.2 有限域中的逆运算 ······························································ 27
B.1.3 Lucas序列的生成 ······························································· 27
B.1.4 模素数平方根的求解···························································· 28
B.1.5 迹函数和半迹函数 ······························································ 28
B.1.6 F
2
m
上二次方程的求解 ··························································· 28
B.1.7 整数模素数阶的检查···························································· 29
B.1.8 整数模素数阶的计算···························································· 29
B.1.9 模素数的阶为给定值的整数的构造 ·············································· 30
B.1.10 概率素性检测 ································································· 30
B.1.11 近似素性检测 ································································· 30
B.2 有限域上的多项式 ····································································· 31
B.2.1 最大公因式····································································· 31
B.2.2 F
2
上不可约多项式在F
2
m
中根的求解 ·············································· 31
B.2.3 基的转换 ······································································· 31
B.2.4 F
2
上多项式不可约性的检测 ····················································· 33
B.3 椭圆曲线算法 ········································································· 33
B.3.1 椭圆曲线阶的计算 ······························································ 33
B.3.2 椭圆曲线上点的寻找···························································· 33
附录C(资料性附录) 曲线示例 ································································ 35
C.1 一般要求 ·············································································· 35
C.2 F
p
上椭圆曲线 ········································································· 35
II
C.3 F
2
m
上椭圆曲线········································································· 35
附录D(资料性附录) 椭圆曲线方程参数的拟随机生成及验证 ··································· 37
D.1 椭圆曲线方程参数的拟随机生成 ······················································· 37
D.1.1 F
p
上椭圆曲线方程参数的拟随机生成 ············································ 37
D.1.2 F
2
m
上椭圆曲线方程参数的拟随机生成 ··········································· 37
D.2 椭圆曲线方程参数的验证 ······························································ 37
D.2.1 F
p
上椭圆曲线方程参数的验证··················································· 38
D.2.2 F
2
m
上椭圆曲线方程参数的验证 ·················································· 38
参考文献 ··················································································· 39
III
前前前 言言言
《SM2椭圆曲线公钥密码算法》分为四个部分:
──第1部分:总则
──第2部分:数字签名算法
──第3部分:密钥交换协议
──第4部分:公钥加密算法
本部分为第1部分。
本部分的附录A、附录B、附录C和附录D为资料性附录。
IV
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