公钥密码加密解密（RSA，ELGamal，ECC椭圆曲线）
以下是RSA加解密的python实现
1 import math 2 3 #欧拉函数 4 def euler_phi(n): 5 return sum(math.gcd(n, i) == 1 for i in range(1, n+1)) 6 7 #扩展欧几里得 8 def exgcd(a, b): 9 if b == 0: 10 return 1, 0, a 11 else: 12 x, y, q = exgcd(b, a % b) 13 x, y = y, x - (a//b) * y 14 return x, y, q 15 16 def encrypt(plaintext): 17 c = (int(plaintext) ** e) % n 18 return c 19 20 def decrypt(cryptotext): 21 ming = (int(cryptotext) ** d) % n 22 return ming 23 24 25 p=int(input("请输入P：")) 26 q=int(input("请输入Q：")) 27 n=p*q 28 f_n=(p-1)*(q-1) 29 e=int(input("请输入公钥e：")) 30 d,a,b=exgcd(e,f_n) 31 d = (d % f_n + f_n) % f_n # 防止出现负数 32 print("私钥是："+str(d)) 33 m=input("请输入明文：") 34 cryptotext=encrypt(m) 35 plaintext=decrypt(cryptotext) 36 print("密文是：",cryptotext) 37 print("解密后的明文是：",plaintext)
下图是运行及结果
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以下是ELGamal加解密的python实现
1 p=int(input("请输入素数p：")) 2 g=int(input("请输入生成元g：")) 3 x=int(input("请输入私钥x∈(1,p-1)：")) 4 y=(g**x) % p 5 print("公钥为(" + str(y) +','+ str(g) +','+ str(p) + ')'+'\n'+"私钥为："+str(x)) 6 k=int(input("请输入随机数k∈(1,p-1)：")) 7 m=int(input("请输入明文m：")) 8 9 # 加密函数 10 def encrypt(m): 11 c1=(g**k)%p 12 c2=(m*(y**k))%p 13 return (c1,c2) 14 15 #分数模运算 16 def calculateMod(numerator, denominator, p): 17 numerator_mod_p = (numerator % p + p) % p # 处理负数情况 18 denominator_mod_p = (denominator % p + p) % p 19 20 # 计算逆元 21 denominator_inverse = 1 22 for i in range(1, p): 23 # 判断i是否为分母的逆元 24 if (denominator_mod_p * i) % p == 1: 25 denominator_inverse = i 26 break 27 28 result = (numerator_mod_p * denominator_inverse) % p 29 return result 30 31 c1,c2=encrypt(m) #获得密文 32 #解密函数 33 def decrypt(c1,c2): 34 plaintext=calculateMod(c2,c1**x,p) 35 return plaintext 36 37 print("密文为："+str(encrypt(m))) 38 print("解密后的明文为："+str(decrypt(c1,c2)))
下图是运行及结果
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下面是ECC椭圆曲线的加解密实现（C++）
1 #include <vector> 2 #include <iostream> 3 #include <cmath> 4 using namespace std; 5 6 class ellipse //定义椭圆类 7 { 8 public: 9 int a; //椭圆的长半轴 10 int b; //椭圆的短半轴 11 int p; //椭圆所在的有限域Fp 12 ellipse(int u, int i, int o) //构造函数 13 { 14 a = u; 15 b = i; 16 p = o; 17 }; 18 int count = 0; //记录点的个数,初始化为0 19 typedef struct point //定义点的数据结构 20 { 21 int x = 0; 22 int y = 0; 23 }Point; 24 Point P[100]; //储存100个点 25 void showPoint(); //显示点的函数 26 }; 27 void ellipse::showPoint() //显示点的函数定义 28 { 29 for (int x = 0; x < p; x++) //遍历Fp域中的x值 30 { 31 int temp = (x * x * x + a * x + b) % p; //计算y的平方根的值 32 for (int y = 0; y <= p; y++) //遍历Fp域中的y值 33 { 34 if ((y * y - temp) % p == 0) //如果y满足椭圆方程 35 { 36 P[count].x = x; //记录点的x坐标 37 P[count].y = y; //记录点的y坐标 38 count++; //点的个数加1 39 } 40 } 41 } 42 //输出点的坐标 43 for (int j = 0; j < count; j++) 44 { 45 cout << "(" << P[j].x << "," << P[j].y << ")"; 46 } 47 } 48 49 // 计算最大公约数 50 int get_gcd(int a, int b) { 51 if (b == 0) 52 return a; 53 return get_gcd(b, a % b); 54 } 55 56 //分数模运算 57 int calculateMod(int numerator, int denominator, int p) { 58 int numeratorModP = (numerator % p + p) % p; // 处理负数情况 59 int denominatorModP = (denominator % p + p) % p; 60 61 // 计算逆元 62 int denominatorInverse = 1; 63 for (int i = 1; i < p; ++i) 64 {//判断i是否为分母的逆元 65 if ((denominatorModP * i) % p == 1) { 66 denominatorInverse = i; 67 break; 68 } 69 } 70 71 int result = (numeratorModP * denominatorInverse) % p; 72 return result; 73 } 74 75 // 计算P+Q函数 76 vector<int> calculate_p_q(int x1, int y1, int x2, int y2, int a, int b, int p) { 77 int flag = 1; //标记PQ是否同号,1为同号,-1为异号 78 int member; //PQ的分子 79 int denominator; //PQ的分母 80 if (x1 == x2 && y1 == y2) { //P、Q重合,则P+Q=3P 81 member = 3 * (x1 * x1) + a; 82 denominator = 2 * y1; 83 } 84 else { //P、Q不重合 85 member = y2 - y1; 86 denominator = x2 - x1; 87 88 } 89 90 //将分子和分母化为最简 91 int gcd_value = get_gcd(member, denominator); 92 member = member / gcd_value; 93 denominator = denominator / gcd_value; 94 95 // 计算x3,y3 96 int x3 = calculateMod(member * member - x1 * denominator * denominator - x2 * denominator * denominator, denominator * denominator, p); 97 int y3 = calculateMod(member * (x1 - x3) - denominator * y1, denominator, p); 98 vector<int> result; //定义一个向量用于返回值 99 result.push_back(x3); //将x3的值添加到result中 100 result.push_back(y3); //将y3的值添加到result中 101 return result; //返回result 102 } 103 104 // 计算nP函数 105 vector<int> calculate_np(int x, int y, int n, int a, int b, int p) { 106 // 记录P的坐标 107 int x_old = x; 108 int y_old = y; 109 vector<int> result; // 用于返回P的值 110 for (int i = 0; i < n - 1; i++) { 111 // 调用calculate_p_q()函数计算P+P的值 112 vector<int> P_add_P = calculate_p_q(x_old, y_old, x_old, y_old, a, b, p); 113 // 更新P的坐标值为P+P 114 x_old = P_add_P[0]; 115 y_old = P_add_P[1]; 116 } 117 // 将P的新坐标添加到result中 118 result.push_back(x_old); 119 result.push_back(y_old); 120 return result; 121 } 122 vector<int> encrypt(int g1,int g2,int d,int a,int b,int p) 123 { 124 //加密过程 125 vector<int> publickey = calculate_np(g1, g2, d, a, b, p);//计算公钥 126 cout << "公钥为：" << '(' << publickey[0] << ',' << publickey[1] << ')' << endl; 127 int k; 128 cout << "请输入随机数k："; 129 cin >> k; 130 vector<int> kY = calculate_np(publickey[0], publickey[1], k, a, b, p); 131 vector<int> c1, c2;//密文 132 c1 = calculate_np(g1, g2, k, a, b, p);//计算c1 133 int x, y; 134 cout << "请输入明文x, y:"; 135 cin >> x >> y; 136 c2 = calculate_p_q(x, y, kY[0], kY[1], a, b, p);//计算c2 137 //储存密文 138 vector<int> result; 139 result.push_back(c1[0]); result.push_back(c1[1]); result.push_back(c2[0]); result.push_back(c2[1]); 140 return result; 141 } 142 143 //解密过程 144 vector<int> decrypt(vector<int> c,int d,int a,int b,int p) 145 { 146 vector<int> dc1 = calculate_np(c[0], c[1], d, a, b, p); 147 dc1[1] = -dc1[1]; // -dc1 148 vector<int> result; 149 result = calculate_p_q(c[2],c[3],dc1[0],dc1[1], a, b, p); 150 return result; 151 } 152 153 154 int main() 155 { 156 int a, b, p; 157 cout << "请输入a,b,p:";//生成元 158 cin >> a >> b >> p; 159 ellipse A(a, b, p); 160 cout << "椭圆曲线上的点集为："; 161 A.showPoint(); 162 int g1, g2; 163 cout << endl << "请输入生成元："; 164 cin >> g1 >> g2; 165 int d;//私钥 166 cout << "请输入私钥d："; 167 cin >> d; 168 //加密过程 169 vector<int> c= encrypt(g1, g2, d, a, b, p); 170 cout << "密文为：" << endl; 171 cout << '(' << c[0] << ',' << c[1] << ')'; 172 cout << '(' << c[2] << ',' << c[3] << ')'; 173 174 //解密过程 175 vector<int>m= decrypt(c,d, a, b, p); 176 cout <<endl <<"明文为：" << endl; 177 cout << '(' << m[0] << ',' << m[1] << ')'; 178 }
下图是运行及结果
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