我正在尝试使用D，P和Q（E，Dp和Dq <找到(p sup> -1 mod q)也可用。）
根据this answer和this answer并使用以下方法更新this question，我应该D。
为了测试这个，我生成了密钥对，并尝试从现有组件中计算组件，并将结果与原件进行比较。除了D之外，所有结果都很好。我从上面的答案中复制了我的计算错误。
如果有人可以告诉我我做错了什么，那就太好了。
测试代码
using System;
using System.Numerics;
using System.Security.Cryptography;
using System.Text;
class Program {
static RSAParameters key = new RSAParameters() {
P = new byte[]{
0xDE, 0xA6, 0x35, 0x0B, 0x0A, 0xA5, 0xD7, 0xA0, 0x5C, 0x49, 0xEA, 0xD1, 0x3F, 0xA6, 0xF5, 0x12,
0x19, 0x06, 0x25, 0x8A, 0xD9, 0xA7, 0x07, 0xE7, 0x0D, 0x8A, 0x7C, 0xB1, 0xD4, 0x81, 0x64, 0xFD,
0x04, 0xEC, 0x47, 0x33, 0x42, 0x0B, 0x22, 0xF2, 0x60, 0xBB, 0x75, 0x62, 0x53, 0x3E, 0x1A, 0x97,
0x9D, 0xEF, 0x25, 0xA7, 0xE5, 0x24, 0x3A, 0x30, 0x36, 0xA5, 0xF9, 0x8A, 0xF5, 0xFF, 0x1D, 0x1B
},
Q = new byte[]{
0xBE, 0xB9, 0x60, 0x12, 0x05, 0xB1, 0x61, 0xD9, 0x22, 0xD8, 0x84, 0x6E, 0x9A, 0x7B, 0xD1, 0x9B,
0x17, 0xA5, 0xDD, 0x02, 0x5E, 0x9D, 0xD8, 0x24, 0x06, 0x1B, 0xF3, 0xD8, 0x2F, 0x79, 0xFE, 0x78,
0x74, 0x3D, 0xC4, 0xE6, 0x17, 0xD2, 0xB7, 0x68, 0x78, 0x6F, 0x53, 0xE0, 0x38, 0x00, 0x86, 0xFB,
0x20, 0x2A, 0x1B, 0xBD, 0x91, 0x76, 0x3E, 0x33, 0x85, 0x9A, 0x31, 0xE6, 0x88, 0x60, 0x91, 0x81
},
DP = new byte[]{
0xAC, 0x28, 0x92, 0x6D, 0x46, 0x3F, 0x74, 0x1A, 0xA0, 0x21, 0xDB, 0xBB, 0x0E, 0xDF, 0xD7, 0x31,
0xB6, 0x3D, 0xC5, 0x7B, 0xB6, 0xCE, 0x6B, 0xD2, 0xE1, 0xEA, 0x8A, 0x7E, 0xAA, 0xD5, 0x9E, 0xB3,
0xF2, 0x41, 0x8C, 0xD0, 0x7A, 0xA9, 0xC7, 0xCC, 0xE8, 0xB5, 0x2A, 0x8F, 0xEB, 0xD3, 0xE2, 0x96,
0x07, 0xDD, 0xEA, 0x1D, 0x07, 0x96, 0x5A, 0x93, 0xFB, 0x3D, 0x9D, 0x56, 0x30, 0xDE, 0xA1, 0xAF
},
DQ = new byte[]{
0xA6, 0x9C, 0x44, 0x1B, 0x9A, 0x53, 0x89, 0xD9, 0xE8, 0xC1, 0xE2, 0x76, 0xC8, 0x87, 0x6F, 0xE5,
0x1F, 0x74, 0x6A, 0xAC, 0x5E, 0x41, 0x5F, 0x86, 0xA0, 0xBB, 0x9C, 0x79, 0xF7, 0x87, 0x87, 0xD0,
0x6C, 0x23, 0x65, 0xB5, 0x67, 0x8C, 0x51, 0x62, 0x77, 0x0B, 0x31, 0xE7, 0x86, 0xA4, 0x97, 0x46,
0x1B, 0xA4, 0x0D, 0x55, 0xBE, 0x13, 0xE0, 0x64, 0x9B, 0xCA, 0xC6, 0xDA, 0xCF, 0xBA, 0x24, 0x81
},
InverseQ = new byte[]{
0x02, 0x42, 0x90, 0xAE, 0xFF, 0xFE, 0xB6, 0xCB, 0x53, 0xFF, 0x96, 0x17, 0xC6, 0xE4, 0x3F, 0xE6,
0xC7, 0xBC, 0xB2, 0xEB, 0x53, 0xA9, 0x47, 0xEE, 0x10, 0x36, 0x98, 0xEF, 0xA8, 0x3E, 0x9C, 0xF7,
0xF9, 0xCF, 0x24, 0xE5, 0xD7, 0x9A, 0xAF, 0x09, 0xCF, 0x28, 0xAA, 0x5D, 0x2A, 0xB7, 0x27, 0x73,
0x47, 0x2D, 0x54, 0x54, 0x61, 0xC5, 0xCE, 0x3E, 0xA4, 0x91, 0xF6, 0x9D, 0xF4, 0x65, 0x08, 0xDD
},
Exponent = new byte[]{
0x00, 0x01, 0x00, 0x01,
},
Modulus = new byte[]{
0xA5, 0xE0, 0x95, 0x08, 0x87, 0x69, 0x2B, 0xB4, 0x7F, 0x08, 0xFB, 0x4F, 0x66, 0x85, 0xD9, 0x95,
0x53, 0x0F, 0x7C, 0x99, 0x95, 0x16, 0xF4, 0x0D, 0xAD, 0x9E, 0x31, 0xD8, 0x20, 0xF4, 0x88, 0x63,
0xAE, 0x51, 0x04, 0xC2, 0xE9, 0x92, 0x3C, 0x1C, 0x90, 0xF8, 0xF4, 0x38, 0x6A, 0x86, 0xFD, 0x8F,
0xDE, 0x85, 0x22, 0xDD, 0xE8, 0x7E, 0x8D, 0xF2, 0xC5, 0xC9, 0x4E, 0x71, 0x2B, 0x56, 0x25, 0x1A,
0xEA, 0x66, 0x15, 0x19, 0x63, 0x70, 0x53, 0x79, 0xDF, 0x38, 0x49, 0x30, 0x74, 0x45, 0xBE, 0xA3,
0x28, 0x0D, 0x0E, 0x7A, 0x7D, 0xB6, 0x8B, 0xCA, 0x09, 0x56, 0x21, 0xE7, 0x98, 0x3E, 0x4B, 0x8B,
0xD0, 0x31, 0x27, 0x8E, 0x6F, 0x10, 0xA6, 0x6C, 0x1C, 0x48, 0xB5, 0x5E, 0x89, 0x7B, 0x74, 0x74,
0xB2, 0x57, 0x72, 0x6D, 0x18, 0xEB, 0xF3, 0xF5, 0x53, 0xCA, 0x8C, 0xBE, 0xB7, 0x29, 0xF5, 0x9B
},
D = new byte[]{
0x9F, 0x86, 0xE1, 0x4D, 0x96, 0x8C, 0xFA, 0xCF, 0x57, 0xED, 0x17, 0x64, 0x41, 0x41, 0x31, 0x04,
0x7F, 0x21, 0x41, 0xBF, 0xA2, 0xB6, 0xB4, 0x78, 0x03, 0x25, 0x44, 0xE2, 0x8A, 0xAF, 0x22, 0x0C,
0x5B, 0xB4, 0xE7, 0x53, 0x5C, 0xB6, 0x9A, 0xC1, 0x0E, 0x5B, 0x9E, 0xE4, 0x32, 0xEF, 0x28, 0x24,
0x98, 0xE8, 0x89, 0xA3, 0xC8, 0xD9, 0x0D, 0x43, 0x12, 0x1C, 0x8C, 0x28, 0x22, 0x79, 0x72, 0xAC,
0x66, 0x7B, 0x7D, 0xD2, 0xF9, 0x48, 0x06, 0xCD, 0x9D, 0x9A, 0xE6, 0x42, 0x92, 0xBA, 0x56, 0xA6,
0x63, 0x07, 0x1E, 0x25, 0x4E, 0xC8, 0x07, 0x58, 0x5B, 0x88, 0x60, 0x97, 0x92, 0xE2, 0xD5, 0xB9,
0xC6, 0x70, 0xBB, 0x63, 0x5A, 0xC3, 0xC3, 0xA6, 0x46, 0x5A, 0x1C, 0x9C, 0xBF, 0x61, 0x57, 0x9E,
0x9E, 0xFA, 0xC0, 0xC4, 0x8A, 0xC2, 0xBA, 0x88, 0x46, 0xA9, 0x7A, 0xF2, 0x7D, 0x4F, 0x6C, 0x01
}
};
public static BigInteger FromBigEndian(byte[] p) {
Array.Reverse(p);
if (p[p.Length - 1] > 127) {
Array.Resize(ref p, p.Length + 1);
p[p.Length - 1] = 0;
}
return new BigInteger(p);
}
static void Main(string[] args) {
using (RSACryptoServiceProvider rsa = new RSACryptoServiceProvider() { PersistKeyInCsp = false }) {
rsa.ImportParameters(key);
Console.Write("Testing Encrypt/Decrypt ... ");
string message = "Testing Some Data to Encrypt";
byte[] buffer = Encoding.ASCII.GetBytes(message);
byte[] encoded = rsa.Encrypt(buffer, true);
byte[] decoded = rsa.Decrypt(encoded, true);
string message1 = ASCIIEncoding.ASCII.GetString(decoded);
if (message == message1) {
Console.WriteLine("Ok :)");
} else {
Console.WriteLine("Bad Encryption :(");
Console.ReadKey();
return;
}
}
//Convert Key to BigIntegers
BigInteger P = FromBigEndian(key.P);
BigInteger Q = FromBigEndian(key.Q);
BigInteger DP = FromBigEndian(key.DP);
BigInteger DQ = FromBigEndian(key.DQ);
BigInteger InverseQ = FromBigEndian(key.InverseQ);
BigInteger E = FromBigEndian(key.Exponent);
BigInteger M = FromBigEndian(key.Modulus);
BigInteger D = FromBigEndian(key.D);
Console.WriteLine("Testing Numbers ... ");
BigInteger M1 = BigInteger.Multiply(P, Q); // M = P*Q
if (M1.CompareTo(M) == 0) {
Console.WriteLine(" M Ok :)");
} else {
Console.WriteLine(" Bad M:(");
Console.ReadKey();
return;
}
BigInteger PMinus1 = BigInteger.Subtract(P, BigInteger.One); // M = P*Q
BigInteger DP1 = BigInteger.Remainder(D, PMinus1); // M = P*Q
if (DP1.CompareTo(DP) == 0) {
Console.WriteLine(" DP Ok :)");
} else {
Console.WriteLine(" Bad DP :(");
Console.ReadKey();
return;
}
BigInteger QMinus1 = BigInteger.Subtract(Q, BigInteger.One); // M = P*Q
BigInteger DQ1 = BigInteger.Remainder(D, QMinus1); // M = P*Q
if (DQ1.CompareTo(DQ) == 0) {
Console.WriteLine(" DQ Ok :)");
} else {
Console.WriteLine(" Bad DQ :(");
Console.ReadKey();
return;
}
BigInteger Phi = BigInteger.Multiply(PMinus1, QMinus1);
BigInteger PhiMinus1 = BigInteger.Subtract(Phi, BigInteger.One);
BigInteger D1 = BigInteger.ModPow(E, PhiMinus1, Phi);
if (D1.CompareTo(D) == 0) {
Console.WriteLine(" D Ok :)");
} else {
Console.WriteLine(" Bad D :(");
Console.ReadKey();
return;
}
Console.ReadKey();
}
}
测试结果
Testing Encrypt/Decrypt ... Ok :)
Testing Numbers ...
M Ok :)
DP Ok :)
DQ Ok :)
Bad D :(
答案 0 :(得分：5)
首先，您需要验证GCD(e, φ) = 1，因为d仅在该属性成立时才存在。然后计算我在modular multiplicative inverse of e modulo phi中描述的my answer to "1/BigInteger in C#"。
您的代码似乎假设e^(φ(n)-1) mod φ(n)是反向的，但这是不正确的。我认为正确的公式为e^(φ(φ(n))-1) mod φ(n)，但由于您只知道φ(n)而不是φ(φ(n))，因此使用起来不方便。
我建议使用扩展欧几里德算法将wikipedia伪代码移植到C＃。
作为旁注： d通常有多个等效值，因为您不需要e*d mod φ(n)=1，只需要e*d mod λ(n)=1，其中λ是Carmichael function请参阅"Why RSA encryption key is based on modulo(phi(n)) rather than modulo n" on crypto.SE
答案 1 :(得分：2)
扩展欧几里德算法可用于计算模逆，使用此链接：http://www.di-mgt.com.au/euclidean.html#extendedeuclidean来获取细节， 我在C＃中测试了源代码，如下所示，结果是匹配，
public static BigInteger modinv(BigInteger u, BigInteger v)
{
BigInteger inv, u1, u3, v1, v3, t1, t3, q;
BigInteger iter;
/* Step X1. Initialise */
u1 = 1;
u3 = u;
v1 = 0;
v3 = v;
/* Remember odd/even iterations */
iter = 1;
/* Step X2. Loop while v3 != 0 */
while (v3 != 0)
{
/* Step X3. Divide and "Subtract" */
q = u3 / v3;
t3 = u3 % v3;
t1 = u1 + q * v1;
/* Swap */
u1 = v1; v1 = t1; u3 = v3; v3 = t3;
iter = -iter;
}
/* Make sure u3 = gcd(u,v) == 1 */
if (u3 != 1)
return 0; /* Error: No inverse exists */
/* Ensure a positive result */
if (iter < 0)
inv = v - u1;
else
inv = u1;
return inv;
}
答案 2 :(得分：0)
D可以通过以下方式计算：
var qq = BigInteger.Multiply(totient, n);
var qw = BigInteger.Multiply(totient, qq);
BigInteger d = BigInteger.ModPow(e, (qw - 1), totient);
答案 3 :(得分：0)
Console.Write("Testing Encrypt/Decrypt using BigInteger ");
string message2 = "Testing Some Data to Encrypt";
byte[] buffer2 = Encoding.ASCII.GetBytes(message2);
BigInteger m = new BigInteger(buffer2);
BigInteger c = BigInteger.ModPow(m, E, M); //encrypt
BigInteger m2 = BigInteger.ModPow(c, D, M); //decrypt, m2 also equals m
byte[] decoded2 = m2.ToByteArray();
if (decoded2[0] == 0)
{
decoded2 = decoded2.Where(b => b != 0).ToArray();
}
string message3 = ASCIIEncoding.ASCII.GetString(decoded2);
if (message2 == message3)
{
Console.WriteLine("Ok :)");
}
else
{
Console.WriteLine("Bad Encryption :(");
Console.ReadKey();
return;
}
我用你的参数尝试了它并且它有效，所以E，D和M必须有效。