如何从P和Q计算RSA的附加私钥参数？

Question

当读取的RSA私人密钥团,其中缺少一些RSA参数(DP,DQ,InverseQ,d),我怎么能计算从那些缺少这些参数都供应吗？我读过,有可能从P和Q,其计算这些被提供,但我不知道如何计算它们.

导入此密钥时,我在尝试使用私钥时出错,声称数据不存在(当然,虽然提供了P和Q).

我需要能够在多个平台上执行此操作,因此我担心这会使我需要拥有实际代码来计算它们.

在C#中

Answer 1

这是可以构建RSAParametersP,Q和公钥数据的完整集合的C#代码.我们假设这个方法的参数是big-endian(和RSAParameters一样).

private static RSAParameters Create(byte[] p, byte[] q, byte[] exponent, byte[] modulus)
{
    var addlParameters = GetFullPrivateParameters(
        p: new BigInteger(CopyAndReverse(p)),
        q: new BigInteger(CopyAndReverse(q)),
        e: new BigInteger(CopyAndReverse(exponent)),
        modulus: new BigInteger(CopyAndReverse(modulus)));

    return new RSAParameters
    {
        P = p,
        Q = q,
        Exponent = exponent,
        Modulus = modulus,
        D = addlParameters.D,
        DP = addlParameters.DP,
        DQ = addlParameters.DQ,
        InverseQ = addlParameters.InverseQ,
    };
}

private static RSAParameters GetFullPrivateParameters(BigInteger p, BigInteger q, BigInteger e, BigInteger modulus)
{
    var n = p * q;
    var phiOfN = n - p - q + 1; // OR: (p - 1) * (q - 1);

    var d = ModInverse(e, phiOfN);
    Assert.Equal(1, (d * e) % phiOfN);

    var dp = d % (p - 1);
    var dq = d % (q - 1);

    var qInv = ModInverse(q, p);
    //Assert.Equal(1, (qInv * q) % p);

    return new RSAParameters
    {
        D = CopyAndReverse(d.ToByteArray()),
        DP = CopyAndReverse(dp.ToByteArray()),
        DQ = CopyAndReverse(dq.ToByteArray()),
        InverseQ = CopyAndReverse(qInv.ToByteArray()),
    };
}


/// <summary>
/// Calculates the modular multiplicative inverse of <paramref name="a"/> modulo <paramref name="m"/>
/// using the extended Euclidean algorithm.
/// </summary>
/// <remarks>
/// This implementation comes from the pseudocode defining the inverse(a, n) function at
/// https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
/// </remarks>
public static BigInteger ModInverse(BigInteger a, BigInteger n)
{
    BigInteger t = 0, nt = 1, r = n, nr = a;

    if (n < 0)
    {
        n = -n;
    }

    if (a < 0)
    {
        a = n - (-a % n);
    }

    while (nr != 0)
    {
        var quot = r / nr;

        var tmp = nt; nt = t - quot * nt; t = tmp;
        tmp = nr; nr = r - quot * nr; r = tmp;
    }

    if (r > 1) throw new ArgumentException(nameof(a) + " is not convertible.");
    if (t < 0) t = t + n;
    return t;
}

private static byte[] CopyAndReverse(byte[] data)
{
    byte[] reversed = new byte[data.Length];
    Array.Copy(data, 0, reversed, 0, data.Length);
    Array.Reverse(reversed);
    return reversed;
}