如何从P，Q和E计算用于RSA加密的D.
我试图用P ， Q和E找到D （ Dp ， Dq和(p -1 mod q)也可用）。
根据这个答案和这个问题的回答和更新使用以下方法我应该得到D
为了测试这一点，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D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。 如果有人能告诉我我做错了什么会很棒。
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 :(
首先，您需要validationGCD(e, φ) = 1因为d仅在该属性成立时才存在。 然后计算我在我对“C＃中的1 / BigInteger”的回答中描述的e modulo phi的模乘法逆 。
你的代码似乎假设e^(φ(n)-1) mod φ(n)是反向的，但这是不正确的。 我认为正确的公式是e^(φ(φ(n))-1) mod φ(n) ，但由于你只知道φ(n)而不是φ(φ(n)) ，所以使用起来不方便。
我建议通过将维基百科伪代码移植到C＃来使用扩展欧几里德算法。
作为旁注： d通常有多个等价值，因为你不需要e*d mod φ(n)=1但只是e*d mod λ(n)=1其中λ是Carmichael函数，见“为什么RSA加密密钥基于模数（phi（n））而不是模数n“on crypto.SE
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获取详细信息，我在C＃中测试了源代码，如下所示，结果是匹配的，
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; }
D可以通过以下公式计算：
var qq = BigInteger.Multiply(totient, n); var qw = BigInteger.Multiply(totient, qq); BigInteger d = BigInteger.ModPow(e, (qw - 1), totient);
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必须是有效的。