public static GroupElement MultiModPow(Group group, GroupElement[] bases, BigInteger[] exponents)
{
if (bases == null || exponents == null || bases.Length != exponents.Length)
{
throw new ArgumentException();
}
GroupElement ge = group.Identity;
for (int i = 0; i < bases.Length; i++)
{
ge = ge.Multiply(bases[i].Exponentiate(exponents[i]));
}
return(ge);
}
internal static BigInteger GenerateSetMembershipChallenge(HashAlgorithm hash, BigInteger q, Group Gq, GroupElement g, GroupElement g1, BigInteger[] S, GroupElement tildeCid, GroupElement[] a)
{
object[] protocolArray = { g, g1, S, tildeCid, a };
BigInteger c = HashToZq(hash, ConcatArrays(new object[][] { GetGroupDescription(Gq), protocolArray }), q);
return(c);
}
public static BigInteger GenerateRevocationChallenge(HashAlgorithm hash, BigInteger q, GroupElement g, GroupElement g1, GroupElement gt, GroupElement K, GroupElement tildeCid, GroupElement X, GroupElement Y, GroupElement Cd, GroupElement T1, GroupElement T2, GroupElement T3)
{
object[] hashArray = new object[] { g, g1, gt, K, tildeCid, X, Y, Cd, T1, T2, T3 };
BigInteger c = HashToZq(hash, hashArray, q);
return(c);
}
// H(UID_p, UID_t, H, Cxb, E1, E2, Cxb’, E1’, E2’, additionalInfo)
public static BigInteger GenerateIdEscrowChallenge(HashAlgorithm hash, BigInteger q, byte[] UIDp, byte[] UIDt, GroupElement H, GroupElement Cxb, GroupElement E1, GroupElement E2, GroupElement CxbPrime, GroupElement E1Prime, GroupElement E2Prime, byte[] additionalInfo)
{
object[] hashArray = new object[] { UIDp, UIDt, H, Cxb, E1, E2, CxbPrime, E1Prime, E2Prime, additionalInfo };
BigInteger c = HashToZq(hash, hashArray, q);
return(c);
}
public static void GenerateChallenge(int d, HashAlgorithm hash, BigInteger q, byte[] UIDt, byte[] a, int p, byte[] ap, GroupElement Ps, byte[] m, byte[] md, int[] D, BigInteger[] x, int[] C, GroupElement[] tildeC, byte[][] tildeA, out BigInteger c, out byte[] cp)
{
int pPrime = (p == d) ? 0 : p;
object[] hashArray = new object[] { UIDt, a, D, x, C, tildeC, tildeA, pPrime, ap, Ps, m };
cp = HashToBytes(hash, hashArray);
c = HashToZq(hash, new object[] { new byte[][] { cp, md } }, q);
}
public static byte[] ComputeTokenID(HashAlgorithm hash, GroupElement h, GroupElement sigmaZPrime, BigInteger sigmaCPrime, BigInteger sigmaRPrime)
{
return(HashToBytes(hash, new object[] { h, sigmaZPrime, sigmaCPrime, sigmaRPrime }));
}
public static BigInteger ComputeXt(HashAlgorithm hash, byte[] UIDp, Group Gq, GroupElement[] g, GroupElement g_d, byte[] e, byte[] S, byte[] TI, out byte[] P)
{
bool deviceSupported = (g_d != null);
GroupElement[] gToHash = new GroupElement[g.Length + (deviceSupported ? 1 : 0)];
for (int i = 0; i < g.Length; i++)
{
gToHash[i] = g[i];
}
if (deviceSupported)
{
gToHash[g.Length] = g_d;
}
object[] preGqArray = new object[] { UIDp };
object[] postGqArray = new object[] { gToHash, e, S };
object[] GqArray = null;
if (Gq is SubGroup)
{
SubGroup subgroup = (SubGroup)Gq;
GqArray = new object[] { subgroup.P, subgroup.Order, subgroup.Generator.ToByteArray() };
}
else if (Gq is P256ECGroup)
{
FpCurve curve = (FpCurve)RecommendedParameters.P256.parameters.Curve;
GqArray = new object[] { curve.Q, curve.A.ToBigInteger(), curve.B.ToBigInteger(), RecommendedParameters.P256.g, RecommendedParameters.P256.parameters.N, BigInteger.One };
}
P = HashToBytes(hash, ConcatArrays(new object[][] { preGqArray, GqArray, postGqArray }));
BigInteger xt = HashToZq(hash, new object[] { (byte)1, P, TI }, Gq.Order);
return(xt);
}
public abstract GroupElement Multiply(GroupElement other);
public override GroupElement Multiply(GroupElement other)
{
return(new SubgroupElement(i.Multiply((other as SubgroupElement).i).Mod(subgroup.P), subgroup));
}
public override GroupElement Multiply(GroupElement other)
{
return(new ECElement(point.Add((other as ECElement).point) as FpPoint));
}
internal static void GenerateTestVectors(string uidp, Formatter formater, bool subgroup, bool supportDevice, bool lite, int maxAttribIndex, int[] D)
{
if (formater.type != Formatter.Type.doc)
{
throw new ArgumentException("Test vectors can only be printed with a doc formatter");
}
int n = maxAttribIndex;
int t = n + 1; // the token information field index
int d = n + 2; // the device field index
//
// hash input formatting
//
string UIDh = "SHA-256";
formater.PrintText("UIDh", UIDh);
HashAlgorithm hash = HashAlgorithm.Create(UIDh);
//
// Setup
//
Group Gq;
if (subgroup)
{
Gq = new SubGroup(RecommendedParameters.L2048N256.p, RecommendedParameters.L2048N256.q, RecommendedParameters.L2048N256.g, RecommendedParameters.L2048N256.Oid);
}
else
{
Gq = new P256ECGroup();
}
BigInteger q = Gq.Order;
// issuer setup
byte[] UIDp = Encoding.UTF8.GetBytes(uidp);
formater.PrintHex("UIDp", UIDp);
formater.PrintText("GroupName", Gq.OID);
BigInteger y0 = ProtocolHelper.GetRandom(q, true);
formater.PrintBigInteger("y0", y0);
GroupElement[] generators = new GroupElement[n + 2];
generators[0] = Gq.Generator.Exponentiate(y0);
generators[0].Print("g0", formater);
GroupElement deviceGenerator;
if (subgroup)
{
SubGroup sgGq = (SubGroup)Gq;
for (int i = 1; i <= n; i++)
{
// we use the recommended parameters for the g_i
generators[i] = new SubgroupElement(RecommendedParameters.L2048N256.g_i[i - 1], sgGq);
}
generators[t] = new SubgroupElement(RecommendedParameters.L2048N256.g_t, sgGq);
deviceGenerator = new SubgroupElement(RecommendedParameters.L2048N256.g_d, sgGq);
}
else
{
for (int i = 1; i <= n; i++)
{
// we use the recommended parameters for the g_i
generators[i] = new ECElement(RecommendedParameters.P256.g_i[i - 1]); // g_i is zero-based
}
generators[t] = new ECElement(RecommendedParameters.P256.g_t);
deviceGenerator = new ECElement(RecommendedParameters.P256.g_d);
}
byte[] e = { 0x00, 0x01, 0x01, 0x00, 0x00 };
for (int i = 0; i < e.Length; i++)
{
formater.PrintHex("e" + (i + 1), new byte[] { e[i] });
}
string SValue = "Issuer parameters specification";
byte[] S = Encoding.UTF8.GetBytes(SValue);
formater.PrintHex("S", S);
//
// ID Escrow - Auditor setup
//
BigInteger ie_x = null;
GroupElement ie_H = null;
byte[] ie_additionalInfo = null;
if (!lite) // TODO: only print if hasCommitment is true
{
ie_x = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("ie_x", ie_x);
ie_H = Gq.Generator.Exponentiate(ie_x);
ie_H.Print("ie_H", formater);
String ie_additionalInfoValue = "ID Escrow policy";
ie_additionalInfo = Encoding.UTF8.GetBytes(ie_additionalInfoValue);
formater.PrintHex("ie_additionalInfo", ie_additionalInfo);
}
//
// Revocation Authority setup
//
BigInteger r_delta = null;
GroupElement r_K = null;
BigInteger[] r_R = null;
GroupElement r_V = null;
if (!lite) // TODO: only print if hasCommitment is true
{
r_delta = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("r_delta", r_delta);
// K = g^delta
r_K = Gq.Generator.Exponentiate(r_delta);
r_K.Print("r_K", formater);
// revoked values
r_R = new BigInteger[4];
BigInteger vExponentAccumulator = BigInteger.One;
for (int i = 0; i < r_R.Length; i++)
{
r_R[i] = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("r_R" + (i + 1), r_R[i]);
// V = gt ^ Prod_{x \in R} (delta + x)
vExponentAccumulator = vExponentAccumulator.Multiply(r_delta.Add(r_R[i]).Mod(q)).Mod(q);
}
r_V = generators[t].Exponentiate(vExponentAccumulator);
r_V.Print("r_V", formater);
}
//
// Issuance
//
// issuance protocol input
byte[][] A =
{
new BigInteger("1234567890", 10).ToByteArray(), // user unique ID, direct-encoding
Encoding.UTF8.GetBytes("Alice Smith"), // name, hashed
Encoding.UTF8.GetBytes("USA"), // country of residence, hashed
new byte[] { 0x02 }, // level, direct-encoding
new BigInteger("25", 10).ToByteArray() // age, direct-encoding
};
for (int i = 0; i < A.Length; i++)
{
formater.PrintHex("A" + (i + 1), A[i]);
}
byte[] TI = Encoding.UTF8.GetBytes("Token information field value");
formater.PrintHex("TI", TI);
byte[] PI = Encoding.UTF8.GetBytes("Prover information field value");
formater.PrintHex("PI", PI);
BigInteger[] x = new BigInteger[n + 2];
BigInteger xd = null;
x[0] = BigInteger.One;
for (int i = 0; i < n; i++)
{
x[i + 1] = ProtocolHelper.ComputeXi(hash, q, e[i], A[i]);
formater.PrintBigInteger("x" + (i + 1), x[i + 1]);
}
byte[] P;
x[t] = ProtocolHelper.ComputeXt(hash, UIDp, Gq, generators, supportDevice ? deviceGenerator : null, e, S, TI, out P);
formater.PrintHex("P", P);
formater.PrintBigInteger("xt", x[t]);
// device info
GroupElement hd = Gq.Identity;
if (supportDevice)
{
xd = ProtocolHelper.ComputeXi(hash, q, (byte)1, Encoding.UTF8.GetBytes("Device secret"));
formater.PrintBigInteger("xd", xd);
hd = deviceGenerator.Exponentiate(xd);
hd.Print("hd", formater);
}
GroupElement gamma = ProtocolHelper.MultiModPow(Gq, generators, x);
if (supportDevice)
{
gamma = gamma.Multiply(hd);
}
gamma.Print("gamma", formater);
// issuer precomputations
GroupElement sigmaZ = gamma.Exponentiate(y0);
sigmaZ.Print("sigmaZ", formater);
BigInteger issuerW = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("w", issuerW);
GroupElement sigmaA = Gq.Generator.Exponentiate(issuerW);
sigmaA.Print("sigmaA", formater);
GroupElement sigmaB = gamma.Exponentiate(issuerW);
sigmaB.Print("sigmaB", formater);
// user precomputations
BigInteger alpha = ProtocolHelper.GetRandom(q, true);
formater.PrintBigInteger("alpha", alpha);
BigInteger beta1 = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("beta1", beta1);
BigInteger beta2 = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("beta2", beta2);
GroupElement h = gamma.Exponentiate(alpha);
h.Print("h", formater);
BigInteger alphaInverse = alpha.ModInverse(q);
formater.PrintBigInteger("alphaInverse", alphaInverse);
// user second message computations
GroupElement sigmaZPrime = sigmaZ.Exponentiate(alpha);
sigmaZPrime.Print("sigmaZPrime", formater);
GroupElement sigmaAPrime = generators[0].Exponentiate(beta1).Multiply(Gq.Generator.Exponentiate(beta2)).Multiply(sigmaA);
sigmaAPrime.Print("sigmaAPrime", formater);
GroupElement sigmaBPrime = sigmaZPrime.Exponentiate(beta1).Multiply(h.Exponentiate(beta2)).Multiply(sigmaB.Exponentiate(alpha));
sigmaBPrime.Print("sigmaBPrime", formater);
BigInteger sigmaCPrime = ProtocolHelper.HashToZq(hash, new object[] { h, PI, sigmaZPrime, sigmaAPrime, sigmaBPrime }, q);
formater.PrintBigInteger("sigmaCPrime", sigmaCPrime);
BigInteger sigmaC = sigmaCPrime.Add(beta1).Mod(q);
formater.PrintBigInteger("sigmaC", sigmaC);
// issuer third message computations
BigInteger sigmaR = sigmaC.Multiply(y0).Add(issuerW).Mod(q);
formater.PrintBigInteger("sigmaR", sigmaR);
// user token generation computations
BigInteger sigmaRPrime = sigmaR.Add(beta2).Mod(q);
formater.PrintBigInteger("sigmaRPrime", sigmaRPrime);
// token signature verification
if (!(sigmaAPrime.Multiply(sigmaBPrime)).Equals(
((Gq.Generator.Multiply(h)).Exponentiate(sigmaRPrime)).Multiply(
(generators[0].Multiply(sigmaZPrime)).Exponentiate(sigmaCPrime.Negate().Mod(q))))
)
{
throw new Exception("signature is invalid");
}
//
// Presentation
//
// parameters
formater.PrintSet("D", D);
int[] U = new int[n - D.Length];
int dIndex = 0, uIndex = 0;
for (int i = 1; i <= n; i++)
{
if (D.Length > 0 && D[dIndex] == i)
{
// i is disclosed, skip
dIndex++;
}
else
{
// i is undisclosed, add to U
U[uIndex++] = i;
}
}
formater.PrintSet("U", U);
// we commit to the first undisclosed index, if any
bool hasCommitment = !lite && U.Length > 0;
if (hasCommitment && (U.Length < 2 || !U.Contains <int>(1) || !U.Contains <int>(4)))
{
// we can't commit to attributes 1 and 4
throw new ArgumentException("U must have at least two elements");
}
int[] C = hasCommitment ? new int[] { 1, 4 } : new int[] { };
if (hasCommitment)
{
formater.PrintSet("C", C);
}
// p is the first committed attribute (if any), or the device attribute if device-protected
int p = hasCommitment ? (supportDevice ? d : C[0]) : 0;
if (hasCommitment)
{
formater.PrintText("p", supportDevice ? "d" : p.ToString());
}
byte[] s = Encoding.UTF8.GetBytes("VerifierUID");
if (hasCommitment)
{
formater.PrintHex("s", s);
}
byte[] m = Encoding.UTF8.GetBytes("VerifierUID+random data");
formater.PrintHex("m", m);
byte[] md = Encoding.UTF8.GetBytes("Direct message");
formater.PrintHex("md", md);
// Prover's precomputation
BigInteger[] w = new BigInteger[n + 2 + (supportDevice ? 1 : 0)];
BigInteger[] tildeO = new BigInteger[C.Length];
BigInteger[] tildeW = new BigInteger[C.Length];
GroupElement[] tildeC = new GroupElement[C.Length];
byte[][] tildeA = new byte[C.Length][];
GroupElement[] bases = new GroupElement[U.Length + 1 + (supportDevice ? 1 : 0)];
BigInteger[] exponents = new BigInteger[U.Length + 1 + (supportDevice ? 1 : 0)];
w[0] = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("w0", w[0]);
GroupElement ad = null;
BigInteger wdPrime = null;
GroupElement Ps = null;
GroupElement gs = null;
GroupElement apPrime = null;
if (p > 0 && s != null && hasCommitment)
{
gs = ProtocolHelper.GenerateScopeElement(Gq, s, formater);
gs.Print("gs", formater);
}
if (supportDevice)
{
w[d] = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("wd", w[d]);
// Device's commitment
wdPrime = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("wdPrime", wdPrime);
ad = deviceGenerator.Exponentiate(wdPrime);
ad.Print("ad", formater);
if (hasCommitment)
{
// Device's pseudonym
apPrime = gs.Exponentiate(wdPrime);
apPrime.Print("apPrime", formater);
Ps = gs.Exponentiate(xd);
Ps.Print("Ps", formater);
}
}
// Presentation proof generation
bases[0] = h;
exponents[0] = w[0];
if (supportDevice)
{
bases[1] = deviceGenerator;
exponents[1] = w[d];
}
for (int i = 0; i < U.Length; i++)
{
int index = U[i];
w[index] = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("w" + index, w[index]);
bases[i + 1 + (supportDevice ? 1 : 0)] = generators[index];
exponents[i + 1 + (supportDevice ? 1 : 0)] = w[index];
}
GroupElement aInput = ProtocolHelper.MultiModPow(Gq, bases, exponents);
if (supportDevice)
{
aInput = aInput.Multiply(ad);
}
byte[] a = ProtocolHelper.HashToBytes(hash, new Object[] { aInput });
formater.PrintHex("a", a);
byte[] ap = null;
// Generate commitments
for (int i = 0; i < C.Length; i++)
{
int index = C[i];
tildeO[i] = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("tildeO" + index, tildeO[i]);
tildeW[i] = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("tildeW" + index, tildeW[i]);
bases = new GroupElement[2] {
Gq.Generator, generators[1]
};
exponents = new BigInteger[2] {
x[index], tildeO[i]
};
tildeC[i] = ProtocolHelper.MultiModPow(Gq, bases, exponents);
tildeC[i].Print("tildeC" + index, formater);
exponents = new BigInteger[2] {
w[index], tildeW[i]
};
tildeA[i] = ProtocolHelper.HashToBytes(hash, new object[] { ProtocolHelper.MultiModPow(Gq, bases, exponents) });
formater.PrintHex("tildeA" + index, tildeA[i]);
}
// compute pseudonym
if (p > 0 && s != null)
{
GroupElement apInput = gs.Exponentiate(w[p]);
if (p == d)
{
apInput = apInput.Multiply(apPrime);
}
ap = ProtocolHelper.HashToBytes(hash, new Object[] { apInput });
formater.PrintHex("ap", ap);
if (p != d)
{
Ps = gs.Exponentiate(x[p]);
Ps.Print("Ps", formater);
}
}
BigInteger[] disclosedX = new BigInteger[D.Length];
for (int i = 0; i < D.Length; i++)
{
disclosedX[i] = x[D[i]];
}
byte[] UIDt = ProtocolHelper.ComputeTokenID(hash, h, sigmaZPrime, sigmaCPrime, sigmaRPrime);
formater.PrintHex("UIDt", UIDt);
BigInteger c;
byte[] cp;
ProtocolHelper.GenerateChallenge(d, hash, q, UIDt, a, p, ap, Ps, m, md, D, disclosedX, C, tildeC, tildeA, out c, out cp);
formater.PrintHex("cp", cp);
formater.PrintBigInteger("c", c);
BigInteger[] r = new BigInteger[n + 3];
r[0] = c.Multiply(alphaInverse).Add(w[0]).Mod(q);
formater.PrintBigInteger("r0", r[0]);
for (int i = 0; i < U.Length; i++)
{
int index = U[i];
r[index] = c.Negate().Mod(q).Multiply(x[index]).Add(w[index]).Mod(q);
formater.PrintBigInteger("r" + index, r[index]);
}
if (supportDevice)
{
// Device response computation
BigInteger rdPrime = c.Negate().Mod(q).Multiply(xd).Add(wdPrime).Mod(q);
formater.PrintBigInteger("rdPrime", rdPrime);
// a_d = g_d^r'_d h_d^c
GroupElement validation = deviceGenerator.Exponentiate(rdPrime).Multiply(hd.Exponentiate(c));
if (!ad.Equals(validation))
{
throw new Exception("invalid device response");
}
r[d] = rdPrime.Add(w[d]).Mod(q);
formater.PrintBigInteger("rd", r[d]);
}
BigInteger[] tildeR = new BigInteger[C.Length];
for (int i = 0; i < C.Length; i++)
{
int index = C[i];
tildeR[i] = c.Negate().Mod(q).Multiply(tildeO[i]).Add(tildeW[i]).Mod(q);
formater.PrintBigInteger("tildeR" + index, tildeR[i]);
}
//
// ID Escrow - Veriable encryption
//
int ie_bIndex = 0, commitmentIndex = 0;
BigInteger ie_r = null, ie_xbPrime = null, ie_rPrime = null, ie_obPrime = null, ie_c = null, ie_rxb = null, ie_rr = null, ie_rob = null;
GroupElement ie_E1 = null, ie_E2 = null, ie_CxbPrime = null, ie_E1Prime = null, ie_E2Prime = null;
if (hasCommitment)
{
ie_bIndex = 1; // x1 is used as ID escrow attribute
formater.PrintText("ie_b", ie_bIndex.ToString());
commitmentIndex = 0;
ie_r = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("ie_r", ie_r);
ie_E1 = Gq.Generator.Exponentiate(ie_r);
ie_E1.Print("ie_E1", formater);
ie_E2 = Gq.Generator.Exponentiate(x[1]).Multiply(ie_H.Exponentiate(ie_r));
ie_E2.Print("ie_E2", formater);
ie_xbPrime = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("ie_xbPrime", ie_xbPrime);
ie_rPrime = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("ie_rPrime", ie_rPrime);
ie_obPrime = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("ie_obPrime", ie_obPrime);
ie_CxbPrime = Gq.Generator.Exponentiate(ie_xbPrime).Multiply(generators[1].Exponentiate(ie_obPrime));
ie_CxbPrime.Print("ie_CxbPrime", formater);
ie_E1Prime = Gq.Generator.Exponentiate(ie_rPrime);
ie_E1Prime.Print("ie_E1Prime", formater);
ie_E2Prime = Gq.Generator.Exponentiate(ie_xbPrime).Multiply(ie_H.Exponentiate(ie_rPrime));
ie_E2Prime.Print("ie_E2Prime", formater);
ie_c = ProtocolHelper.GenerateIdEscrowChallenge(hash, q, UIDp, UIDt, ie_H, tildeC[commitmentIndex], ie_E1, ie_E2, ie_CxbPrime, ie_E1Prime, ie_E2Prime, ie_additionalInfo);
formater.PrintBigInteger("ie_c", ie_c);
ie_rxb = ie_c.Negate().Mod(q).Multiply(x[ie_bIndex]).Add(ie_xbPrime).Mod(q);
formater.PrintBigInteger("ie_rxb", ie_rxb);
ie_rr = ie_c.Negate().Mod(q).Multiply(ie_r).Add(ie_rPrime).Mod(q);
formater.PrintBigInteger("ie_rr", ie_rr);
ie_rob = ie_c.Negate().Mod(q).Multiply(tildeO[commitmentIndex]).Add(ie_obPrime).Mod(q);
formater.PrintBigInteger("ie_rob", ie_rob);
}
//
// Revocation
//
int r_id = 0;
BigInteger r_d = null, r_cPrime = null;
BigInteger[] r_t = new BigInteger[2 + 1], r_k = new BigInteger[6 + 1], r_s = new BigInteger[6 + 1]; // we waste index 0 for readability
GroupElement r_W = null, r_Q = null, r_X = null, r_Y = null, r_Cd = null, r_T1 = null, r_T2 = null, r_T3 = null;
if (hasCommitment)
{
r_id = 1; // x1 is used as revocation attribute
formater.PrintText("r_id", r_id.ToString());
// compute revocation witness
r_d = BigInteger.One;
BigInteger deltaPlusXid = r_delta.Add(x[r_id]); // delta + x_id
BigInteger wExponentAccumulator = BigInteger.One;
for (int i = 0; i < r_R.Length; i++)
{
// rd = Prod_{x \in R}(x - xid)
// r_d = r_d.Multiply(r_delta.Add(r_R[i])).Mod(deltaPlusXid); // TODO (FIXME): delete, old calculation
r_d = r_d.Multiply(r_R[i].Subtract(x[r_id]).Mod(q)).Mod(q);
wExponentAccumulator = wExponentAccumulator.Multiply(r_delta.Add(r_R[i])).Mod(q);
}
formater.PrintBigInteger("r_d", r_d);
BigInteger wExponent = wExponentAccumulator.Subtract(r_d).Mod(q).Multiply(
r_delta.Add(x[r_id]).ModInverse(q)
).Mod(q);
// W = gt ^ [( Prod_{x \in R}(delta + x)-d ) / (delta + x_id)]
r_W = generators[t].Exponentiate(wExponent);
r_W.Print("r_W", formater);
// Q = V W^(-x_id) gt^(-d)
r_Q = r_V.Multiply(r_W.Exponentiate(x[r_id].Negate())).Multiply(generators[t].Exponentiate(r_d.Negate()));
r_Q.Print("r_Q", formater);
// compute non-revocation proof
for (int i = 1; i <= 2; i++)
{
r_t[i] = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("r_t" + i, r_t[i]);
}
for (int i = 1; i <= 6; i++)
{
r_k[i] = ProtocolHelper.GetRandom(q);
formater.PrintBigInteger("r_k" + i, r_k[i]);
}
// X = W g^t1
r_X = r_W.Multiply(Gq.Generator.Exponentiate(r_t[1]));
r_X.Print("r_X", formater);
// Y = Q K^t1
r_Y = r_Q.Multiply(r_K.Exponentiate(r_t[1]));
r_Y.Print("r_Y", formater);
// Cd = gt^d g1^t2
r_Cd = generators[t].Exponentiate(r_d).Multiply(generators[1].Exponentiate(r_t[2]));
r_Cd.Print("r_Cd", formater);
// w = d^(-1) mod q
BigInteger r_w = r_d.ModInverse(q);
formater.PrintBigInteger("r_w", r_w);
// z = t1 tildaO_id - t2 mod q
BigInteger r_z = r_t[1].Multiply(tildeO[commitmentIndex]).Mod(q).Subtract(r_t[2]).Mod(q);
formater.PrintBigInteger("r_z", r_z);
// z' = -t2 w mod q
BigInteger r_zPrime = r_t[2].Negate().Multiply(r_w).Mod(q);
formater.PrintBigInteger("r_zPrime", r_zPrime);
// T1 = X^k1 (tildeC_id K)^-k2 g1^k3
r_T1 =
r_X.Exponentiate(r_k[1]).Multiply(
tildeC[commitmentIndex].Multiply(r_K).Exponentiate(r_k[2].Negate()).Multiply(
generators[1].Exponentiate(r_k[3])));
r_T1.Print("r_T1", formater);
// T2 = g^k1 g1^k4
r_T2 = Gq.Generator.Exponentiate(r_k[1]).Multiply(generators[1].Exponentiate(r_k[4]));
r_T2.Print("r_T2", formater);
// T3 = Cd^k5 g1^k6
r_T3 = r_Cd.Exponentiate(r_k[5]).Multiply(generators[1].Exponentiate(r_k[6]));
r_T3.Print("r_T3", formater);
// c' = H(g, g1, gt, K, tildeC_id, X, Y, Cd, T1, T2, T3)
r_cPrime = ProtocolHelper.GenerateRevocationChallenge(hash, q, Gq.Generator, generators[1], generators[t], r_K, tildeC[commitmentIndex], r_X, r_Y, r_Cd, r_T1, r_T2, r_T3);
formater.PrintBigInteger("r_cPrime", r_cPrime);
BigInteger r_cPrimeNegate = r_cPrime.Negate().Mod(q);
// s1 = -c x_id + k1 mod q
// s2 = -c t1 + k2 mod q
// s3 = -c z + k3 mod q
// s4 = -c tildeO_id + k4 mod q
// s5 = -c w + k5 mod q
// s6 = -c z' + k6 mod q
BigInteger[] r_sVariable = { null, x[r_id], r_t[1], r_z, tildeO[commitmentIndex], r_w, r_zPrime };
for (int i = 1; i <= 6; i++)
{
r_s[i] = r_cPrimeNegate.Multiply(r_sVariable[i]).Add(r_k[i]).Mod(q);
formater.PrintBigInteger("r_s" + i, r_s[i]);
}
}
//
// Set membership
//
int sm_x_index = 0, sm_i = 0;
int smCommitmentIndex = 1; // x4 (2nd committed value) is used for set membership proof
BigInteger sm_challenge = null;
BigInteger[] sm_S = null, sm_c = null, sm_r = null;
GroupElement[] sm_a = null;
if (hasCommitment)
{
// x4 is used as the committed attribute
sm_x_index = 4;
formater.PrintText("sm_x_index", sm_x_index.ToString());
sm_i = 2; // x4 is the 2nd value in the set
formater.PrintText("sm_i", sm_i.ToString());
int sm_n = 3; // size of the attribute set
formater.PrintText("sm_n", sm_n.ToString());
// attribute set
sm_S = new BigInteger[sm_n];
for (int i = 1; i <= sm_S.Length; i++)
{
sm_S[i - 1] = new BigInteger(i.ToString(), 10);
formater.PrintBigInteger("sm_s" + i, sm_S[i - 1]);
}
// SetMembershipProve
sm_a = new GroupElement[sm_S.Length];
sm_c = new BigInteger[sm_S.Length];
sm_r = new BigInteger[sm_S.Length];
BigInteger sm_sumC = BigInteger.Zero;
for (int i = 1; i <= sm_S.Length; i++)
{
if (i == sm_i)
{
continue;
}
sm_c[i - 1] = ProtocolHelper.GetRandom(q, true);
formater.PrintBigInteger("sm_c" + i, sm_c[i - 1]);
sm_sumC = sm_sumC.Add(sm_c[i - 1]).Mod(q);
sm_r[i - 1] = ProtocolHelper.GetRandom(q, true);
formater.PrintBigInteger("sm_r" + i, sm_r[i - 1]);
sm_a[i - 1] =
generators[1].Exponentiate(sm_r[i - 1]).Multiply(
Gq.Generator.Exponentiate(sm_S[i - 1].Multiply(sm_c[i - 1]).Mod(q)).Multiply(
tildeC[smCommitmentIndex].Exponentiate(sm_c[i - 1].Negate().Mod(q))));
sm_a[i - 1].Print("sm_a" + i, formater);
}
BigInteger sm_w = ProtocolHelper.GetRandom(q, true);
formater.PrintBigInteger("sm_w", sm_w);
sm_a[sm_i - 1] = generators[1].Exponentiate(sm_w);
sm_a[sm_i - 1].Print("sm_a" + sm_i, formater);
sm_challenge =
ProtocolHelper.GenerateSetMembershipChallenge(hash, q, Gq, Gq.Generator, generators[1], sm_S, tildeC[smCommitmentIndex], sm_a);
formater.PrintBigInteger("sm_c", sm_challenge);
sm_c[sm_i - 1] = sm_challenge.Subtract(sm_sumC).Mod(q);
formater.PrintBigInteger("sm_c" + sm_i, sm_c[sm_i - 1]);
sm_r[sm_i - 1] = sm_c[sm_i - 1].Multiply(tildeO[smCommitmentIndex]).Add(sm_w).Mod(q);
formater.PrintBigInteger("sm_r" + sm_i, sm_r[sm_i - 1]);
}
//
// Proof verification
//
// Verifier's U-Prove token signature verification
BigInteger digest = ProtocolHelper.HashToZq(hash, new object[] {
h,
PI,
sigmaZPrime,
Gq.Generator.Exponentiate(sigmaRPrime).Multiply(generators[0].Exponentiate(sigmaCPrime.Negate().Mod(q))),
h.Exponentiate(sigmaRPrime).Multiply(sigmaZPrime.Exponentiate(sigmaCPrime.Negate().Mod(q)))
}, q);
if (!sigmaCPrime.Equals(digest))
{
throw new Exception("invalid signature");
}
// Verifier's proof verification
bases = new GroupElement[D.Length + 2];
exponents = new BigInteger[D.Length + 2];
bases[0] = generators[0];
exponents[0] = BigInteger.One;
bases[1] = generators[t];
exponents[1] = x[t];
for (int i = 0; i < D.Length; i++)
{
int index = D[i];
bases[i + 2] = generators[index];
exponents[i + 2] = x[index];
}
GroupElement temp = ProtocolHelper.MultiModPow(Gq, bases, exponents);
bases = new GroupElement[U.Length + 2 + (supportDevice ? 1 : 0)];
exponents = new BigInteger[U.Length + 2 + (supportDevice ? 1 : 0)];
bases[0] = temp;
exponents[0] = c.Negate().Mod(q);
bases[1] = h;
exponents[1] = r[0];
if (supportDevice)
{
bases[2] = deviceGenerator;
exponents[2] = r[d];
}
for (int i = 0; i < U.Length; i++)
{
int index = U[i];
bases[i + 2 + (supportDevice ? 1 : 0)] = generators[index];
exponents[i + 2 + (supportDevice ? 1 : 0)] = r[index];
}
GroupElement digest2Input = ProtocolHelper.MultiModPow(Gq, bases, exponents);
byte[] digest2 = ProtocolHelper.HashToBytes(hash, new object[] { digest2Input });
if (!a.SequenceEqual(digest2))
{
throw new Exception("invalid proof");
}
if (hasCommitment)
{
// verify pseudonym
bases = new GroupElement[2] {
Ps, gs
};
exponents = new BigInteger[2] {
c, r[p]
};
digest2Input = ProtocolHelper.MultiModPow(Gq, bases, exponents);
digest2 = ProtocolHelper.HashToBytes(hash, new object[] { digest2Input });
if (!ap.SequenceEqual(digest2))
{
throw new Exception("invalid pseudonym");
}
// verify commitment
for (int i = 0; i < C.Length; i++)
{
int index = C[i];
bases = new GroupElement[3] {
tildeC[i], Gq.Generator, generators[1]
};
exponents = new BigInteger[3] {
c, r[index], tildeR[i]
};
digest2Input = ProtocolHelper.MultiModPow(Gq, bases, exponents);
digest2 = ProtocolHelper.HashToBytes(hash, new object[] { digest2Input });
if (!tildeA[i].SequenceEqual(digest2))
{
throw new Exception("invalid commitment: " + index);
}
}
//
// ID Escrow - Verifier validation
//
GroupElement ie_CxBDoublePrime = Gq.Generator.Exponentiate(ie_rxb).Multiply(generators[1].Exponentiate(ie_rob)).Multiply(tildeC[commitmentIndex].Exponentiate(ie_c));
GroupElement ie_E1DoublePrime = Gq.Generator.Exponentiate(ie_rr).Multiply(ie_E1.Exponentiate(ie_c));
GroupElement ie_E2DoublePrime = Gq.Generator.Exponentiate(ie_rxb).Multiply(ie_H.Exponentiate(ie_rr)).Multiply(ie_E2.Exponentiate(ie_c));
BigInteger ie_cPrime = ProtocolHelper.GenerateIdEscrowChallenge(hash, q, UIDp, UIDt, ie_H, tildeC[commitmentIndex], ie_E1, ie_E2, ie_CxBDoublePrime, ie_E1DoublePrime, ie_E2DoublePrime, ie_additionalInfo);
if (!ie_c.Equals(ie_cPrime))
{
throw new Exception("invalid ID escrow encryption");
}
//
// ID Escrow - Auditor decryption
//
GroupElement ie_PE = ie_E2.Multiply(ie_E1.Exponentiate(ie_x.Negate()));
if (!Gq.Generator.Exponentiate(x[ie_bIndex]).Equals(ie_PE))
{
throw new Exception("invalid ID escrow decryption");
}
//
// Revocation : verify non-revocation proof
//
// T1 = (V Y^-1 Cd^-1)^c' X^s1 (tildeC_id K)^-s2 g1^s3
GroupElement r_T1_2 =
(r_V.Multiply(r_Y.Exponentiate(BigInteger.One.Negate())).Multiply(r_Cd.Exponentiate(BigInteger.One.Negate()))).Exponentiate(r_cPrime).Multiply(
r_X.Exponentiate(r_s[1])).Multiply(
tildeC[commitmentIndex].Multiply(r_K).Exponentiate(r_s[2].Negate())).Multiply(
generators[1].Exponentiate(r_s[3]));
// T2 = tildeC_id^c' g^s1 g1^s4
GroupElement r_T2_2 =
tildeC[commitmentIndex].Exponentiate(r_cPrime).Multiply(Gq.Generator.Exponentiate(r_s[1])).Multiply(generators[1].Exponentiate(r_s[4]));
// T3 = gt^c' Cd^s5 g1^s6
GroupElement r_T3_2 =
generators[t].Exponentiate(r_cPrime).Multiply(r_Cd.Exponentiate(r_s[5])).Multiply(generators[1].Exponentiate(r_s[6]));
BigInteger r_cPrime_2 = ProtocolHelper.GenerateRevocationChallenge(hash, q, Gq.Generator, generators[1], generators[t], r_K, tildeC[commitmentIndex], r_X, r_Y, r_Cd, r_T1_2, r_T2_2, r_T3_2);
if (!r_cPrime.Equals(r_cPrime_2) || !r_Y.Equals(r_X.Exponentiate(r_delta)))
{
throw new Exception("invalid non-revocation proof");
}
//
// Set membership proof verification
//
BigInteger sm_challenge_2 = ProtocolHelper.GenerateSetMembershipChallenge(hash, q, Gq, Gq.Generator, generators[1], sm_S, tildeC[smCommitmentIndex], sm_a);
if (!sm_challenge_2.Equals(sm_challenge))
{
throw new Exception("invalid membership challenge");
}
BigInteger sm_sumC_2 = BigInteger.Zero;
// c_n is not serialized, so we recalculate it from the challenge and the other c values
for (int i = 0; i < sm_c.Length - 1; i++)
{
sm_sumC_2 = sm_sumC_2.Add(sm_c[i]);
}
// c_n = c - \sum_{1 <= j < n} (c_j) mod q
sm_c[sm_c.Length - 1] = sm_challenge_2.Subtract(sm_sumC_2).Mod(q); // replace c_n
for (int i = 0; i < sm_r.Length; i++)
{
if (!generators[1].Exponentiate(sm_r[i]).Equals(
tildeC[smCommitmentIndex].Exponentiate(sm_c[i]).Multiply(
Gq.Generator.Exponentiate(sm_c[i].Multiply(sm_S[i]).Negate().Mod(q)).Multiply(
sm_a[i]))
))
{
throw new Exception("invalid membership proof");
}
}
}
}
