private BigInteger ExtractUsersPrivateKey(
ECPoint G, BigInteger N,
string message1,
string message2,
BigInteger r1, BigInteger s1,
BigInteger r2, BigInteger s2,
BigInteger attackersPrivate, ECPoint attackersPublic,
ECPoint usersPublic)
{
var m1 = new BigInteger(1, Hash(Encoding.UTF8.GetBytes(message1))); // hash of m1
var m2 = new BigInteger(1, Hash(Encoding.UTF8.GetBytes(message2))); // hash of m2
//calculate the result of verifying signature 1
var w = s1.ModInverse(N);
var u1 = m1.Multiply(w).Mod(N);
var u2 = r1.Multiply(w).Mod(N);
var verifyPoint = ECAlgorithms.SumOfTwoMultiplies(G, u1, usersPublic, u2).Normalize();
//reinit K calculator to reproduce a,b,h,e
var kCalc = new RandomDsaKCalculator();
kCalc.Init(N, new SecureRandom(new SeededGenerator(Hash(Encoding.UTF8.GetBytes(message2)))));
var a = kCalc.NextK();
var b = kCalc.NextK();
var h = kCalc.NextK();
var e = kCalc.NextK();
var Z1 = verifyPoint.Multiply(a).Add(verifyPoint.Multiply(attackersPrivate).Multiply(b));
//cycle through all possible j & u
for (int i = 0; i < 2; i++)
{
for (int l = 0; l < 2; l++)
{
var j = new BigInteger(i.ToString());
var u = new BigInteger(l.ToString());
var Z2 = Z1.Add(G.Multiply(j).Multiply(h)).Add(attackersPublic.Multiply(u).Multiply(e)).Normalize();
var zX = Z2.AffineXCoord.ToBigInteger().ToByteArray();
var hash = Hash(zX);
var kCandidate = new BigInteger(1, hash);
var verifyPointCandidate = G.Multiply(kCandidate).Normalize();
var rCandidate = verifyPointCandidate.AffineXCoord.ToBigInteger().Mod(N);
if (rCandidate.Equals(r2)) // Gotcha!
{
return(s2.Multiply(kCandidate).Subtract(m2).Multiply(r2.ModInverse(N)).Mod(N));
}
}
}
return(null);
}
public void Backdoor()
{
var random = new SecureRandom();
var curve = CustomNamedCurves.GetByName("secp521r1");
var gen = new ECKeyPairGenerator("ECDSA");
var G = curve.G;
var N = curve.N;
var paramz = new ECDomainParameters(curve.Curve, G, N);
gen.Init(new ECKeyGenerationParameters(paramz, random));
var kCalc = new RandomDsaKCalculator(); // kCalc generates random values [1, N-1]
kCalc.Init(N, random);
var attackersKeyPair = gen.GenerateKeyPair();
var v = ((ECPrivateKeyParameters)attackersKeyPair.Private).D; //attacker's private
var V = G.Multiply(v); //attackers public
var usersKeyPair = gen.GenerateKeyPair(); //user's public
var D = ((ECPrivateKeyParameters)usersKeyPair.Private).D; //user's private
var Q = ((ECPublicKeyParameters)usersKeyPair.Public).Q; //user's public
const string message1 = "First message to sign";
var m1 = new BigInteger(1, Hash(Encoding.UTF8.GetBytes(message1))); // hash of m1
//Generate signature 1
var k1 = kCalc.NextK(); // k1 is true random
var signaturePoint1 = G.Multiply(k1).Normalize();
//(r1, s1) - signature 1
var r1 = signaturePoint1.AffineXCoord.ToBigInteger().Mod(N);
var s1 = k1.ModInverse(N).Multiply(m1.Add(D.Multiply(r1)));
//verify signature 1
var w = s1.ModInverse(N);
var u1 = m1.Multiply(w).Mod(N);
var u2 = r1.Multiply(w).Mod(N);
var verifyPoint1 = ECAlgorithms.SumOfTwoMultiplies(G, u1, Q, u2).Normalize();
var valid1 = verifyPoint1.AffineXCoord.ToBigInteger().Mod(N).Equals(r1);
//Generate signature 2
const string message2 = "Second message to sign";
var m2 = new BigInteger(1, Hash(Encoding.UTF8.GetBytes(message2))); // hash of m2
//here we generate a,b,h,e < N using seed = hash(m2)
kCalc.Init(N, new SecureRandom(new SeededGenerator(Hash(Encoding.UTF8.GetBytes(message2)))));
var a = kCalc.NextK();
var b = kCalc.NextK();
var h = kCalc.NextK();
var e = kCalc.NextK();
//u,j - true random
var u = (random.Next() % 2) == 1 ? BigInteger.One : BigInteger.Zero;
var j = (random.Next() % 2) == 1 ? BigInteger.One : BigInteger.Zero;
//compute hidden field element
var Z = G.Multiply(k1).Multiply(a)
.Add(V.Multiply(k1).Multiply(b))
.Add(G.Multiply(h).Multiply(j))
.Add(V.Multiply(e).Multiply(u))
.Normalize();
var zX = Z.AffineXCoord.ToBigInteger().ToByteArray();
var hash = Hash(zX);
var k2 = new BigInteger(1, hash);
var signaturePoint2 = G.Multiply(k2).Normalize();
//(r2, s2) = signature 2
var r2 = signaturePoint2.AffineXCoord.ToBigInteger().Mod(N);
var s2 = k2.ModInverse(N).Multiply(m2.Add(D.Multiply(r2)));
//verify signature 2
w = s2.ModInverse(N);
u1 = m2.Multiply(w).Mod(N);
u2 = r2.Multiply(w).Mod(N);
var verifyPoint2 = ECAlgorithms.SumOfTwoMultiplies(G, u1, Q, u2).Normalize();
var valid2 = verifyPoint2.AffineXCoord.ToBigInteger().Mod(N).Equals(r2);
if (valid1 && valid2)
{
//compute user's private key
var d = ExtractUsersPrivateKey(G, N, message1, message2, r1, s1, r2, s2, v, V, Q);
Console.WriteLine("Ecdsa private key restored: {0}", d.Equals(D));
}
else
{
Console.WriteLine("Something's wrong");
}
}