// TODO: get parameters from the commitment group
CommitmentProofOfKnowledge::CommitmentProofOfKnowledge(const IntegerGroupParams* aParams,
const IntegerGroupParams* bParams, const Commitment& a, const Commitment& b):
ap(aParams),bp(bParams)
{
CBigNum r1, r2, r3;
// First: make sure that the two commitments have the
// same contents.
if (a.getContents() != b.getContents()) {
throw std::runtime_error("Both commitments must contain the same value");
}
// Select three random values "r1, r2, r3" in the range 0 to (2^l)-1 where l is:
// length of challenge value + max(modulus 1, modulus 2, order 1, order 2) + margin.
// We set "margin" to be a relatively generous security parameter.
//
// We choose these large values to ensure statistical zero knowledge.
uint32_t randomSize = COMMITMENT_EQUALITY_CHALLENGE_SIZE + COMMITMENT_EQUALITY_SECMARGIN +
std::max(std::max(this->ap->modulus.bitSize(), this->bp->modulus.bitSize()),
std::max(this->ap->groupOrder.bitSize(), this->bp->groupOrder.bitSize()));
CBigNum maxRange = (CBigNum(2).pow(randomSize) - CBigNum(1));
r1 = CBigNum::randBignum(maxRange);
r2 = CBigNum::randBignum(maxRange);
r3 = CBigNum::randBignum(maxRange);
// Generate two random, ephemeral commitments "T1, T2"
// of the form:
// T1 = g1^r1 * h1^r2 mod p1
// T2 = g2^r1 * h2^r3 mod p2
//
// Where (g1, h1, p1) are from "aParams" and (g2, h2, p2) are from "bParams".
CBigNum T1 = this->ap->g.pow_mod(r1, this->ap->modulus).mul_mod((this->ap->h.pow_mod(r2, this->ap->modulus)), this->ap->modulus);
CBigNum T2 = this->bp->g.pow_mod(r1, this->bp->modulus).mul_mod((this->bp->h.pow_mod(r3, this->bp->modulus)), this->bp->modulus);
// Now hash commitment "A" with commitment "B" as well as the
// parameters and the two ephemeral commitments "T1, T2" we just generated
this->challenge = calculateChallenge(a.getCommitmentValue(), b.getCommitmentValue(), T1, T2);
// Let "m" be the contents of the commitments "A, B". We have:
// A = g1^m * h1^x mod p1
// B = g2^m * h2^y mod p2
// T1 = g1^r1 * h1^r2 mod p1
// T2 = g2^r1 * h2^r3 mod p2
//
// Now compute:
// S1 = r1 + (m * challenge) -- note, not modular arithmetic
// S2 = r2 + (x * challenge) -- note, not modular arithmetic
// S3 = r3 + (y * challenge) -- note, not modular arithmetic
this->S1 = r1 + (a.getContents() * this->challenge);
this->S2 = r2 + (a.getRandomness() * this->challenge);
this->S3 = r3 + (b.getRandomness() * this->challenge);
// We're done. The proof is S1, S2, S3 and "challenge", all of which
// are stored in member variables.
}
// TODO: get parameters from the commitment group
CommitmentProofOfKnowledge::CommitmentProofOfKnowledge(const IntegerGroupParams* aParams,
const IntegerGroupParams* bParams, const Commitment& a, const Commitment& b):
ap(aParams),bp(bParams) {
Bignum r1;
// First: make sure that the two commitments have the
// same contents.
if(a.getContents() != b.getContents()){
throw std::invalid_argument("Both commitments must contain the same value");
}
// In order to ensure statistical zero knowledge, we pick "r1" out of the
// largest possible range. In this case, the smaller of the two group orders.
if(this->ap->groupOrder < this->bp->groupOrder){
r1 = Bignum::randBignum(ap->groupOrder);
}else{
r1 = Bignum::randBignum(bp->groupOrder);
}
// Generate two random, ephemeral commitments "T1, T2" to "r1" under the two different
// sets of commitment parameters.
Commitment t1(aParams, r1);
Commitment t2(bParams, r1);
Bignum T1 = t1.getCommitmentValue();
Bignum T2 = t2.getCommitmentValue();
// Now hash commitment "A" with commitment "B" as well as the
// parameters and the two ephemeral commitments "T1, T2" we just generated
this->challenge = calculateChallenge(a.getCommitmentValue(), b.getCommitmentValue(), T1, T2);
// Let "m" be the contents of the commitments. We'll implicitly define
// A = g1^m * h1^x mod p1
// B = g2^m * h2^y mod p2
// T1 = g1^r1 * h1^r2 mod p1
// T2 = g2^r1 * h2^r3 mod p2
//
// Now compute:
// S1 = r1 + (m * challenge)
// S2 = r2 + (x * challenge)
// S3 = r3 + (y * challenge)
S1 = t1.getContents() + (a.getContents() * challenge);
S2 = t1.getRandomness() + (a.getRandomness() * challenge);
S3 = t2.getRandomness() + (b.getRandomness() * challenge);
// We're done. The proof is S1, S2, S3 and "challenge".
}
AccumulatorProofOfKnowledge::AccumulatorProofOfKnowledge(const AccumulatorAndProofParams* p,
const Commitment& commitmentToCoin, const AccumulatorWitness& witness,
Accumulator& a): params(p) {
Bignum sg = params->accumulatorPoKCommitmentGroup.g;
Bignum sh = params->accumulatorPoKCommitmentGroup.h;
Bignum g_n = params->accumulatorQRNCommitmentGroup.g;
Bignum h_n = params->accumulatorQRNCommitmentGroup.h;
Bignum e = commitmentToCoin.getContents();
Bignum r = commitmentToCoin.getRandomness();
Bignum r_1 = Bignum::randBignum(params->accumulatorModulus/4);
Bignum r_2 = Bignum::randBignum(params->accumulatorModulus/4);
Bignum r_3 = Bignum::randBignum(params->accumulatorModulus/4);
this->C_e = g_n.pow_mod(e, params->accumulatorModulus) * h_n.pow_mod(r_1, params->accumulatorModulus);
this->C_u = witness.getValue() * h_n.pow_mod(r_2, params->accumulatorModulus);
this->C_r = g_n.pow_mod(r_2, params->accumulatorModulus) * h_n.pow_mod(r_3, params->accumulatorModulus);
Bignum r_alpha = Bignum::randBignum(params->maxCoinValue * Bignum(2).pow(params->k_prime + params->k_dprime));
if(!(Bignum::randBignum(Bignum(3)) % 2)) {
r_alpha = 0-r_alpha;
}
Bignum r_gamma = Bignum::randBignum(params->accumulatorPoKCommitmentGroup.modulus);
Bignum r_phi = Bignum::randBignum(params->accumulatorPoKCommitmentGroup.modulus);
Bignum r_psi = Bignum::randBignum(params->accumulatorPoKCommitmentGroup.modulus);
Bignum r_sigma = Bignum::randBignum(params->accumulatorPoKCommitmentGroup.modulus);
Bignum r_xi = Bignum::randBignum(params->accumulatorPoKCommitmentGroup.modulus);
Bignum r_epsilon = Bignum::randBignum((params->accumulatorModulus/4) * Bignum(2).pow(params->k_prime + params->k_dprime));
if(!(Bignum::randBignum(Bignum(3)) % 2)) {
r_epsilon = 0-r_epsilon;
}
Bignum r_eta = Bignum::randBignum((params->accumulatorModulus/4) * Bignum(2).pow(params->k_prime + params->k_dprime));
if(!(Bignum::randBignum(Bignum(3)) % 2)) {
r_eta = 0-r_eta;
}
Bignum r_zeta = Bignum::randBignum((params->accumulatorModulus/4) * Bignum(2).pow(params->k_prime + params->k_dprime));
if(!(Bignum::randBignum(Bignum(3)) % 2)) {
r_zeta = 0-r_zeta;
}
Bignum r_beta = Bignum::randBignum((params->accumulatorModulus/4) * params->accumulatorPoKCommitmentGroup.modulus * Bignum(2).pow(params->k_prime + params->k_dprime));
if(!(Bignum::randBignum(Bignum(3)) % 2)) {
r_beta = 0-r_beta;
}
Bignum r_delta = Bignum::randBignum((params->accumulatorModulus/4) * params->accumulatorPoKCommitmentGroup.modulus * Bignum(2).pow(params->k_prime + params->k_dprime));
if(!(Bignum::randBignum(Bignum(3)) % 2)) {
r_delta = 0-r_delta;
}
this->st_1 = (sg.pow_mod(r_alpha, params->accumulatorPoKCommitmentGroup.modulus) * sh.pow_mod(r_phi, params->accumulatorPoKCommitmentGroup.modulus)) % params->accumulatorPoKCommitmentGroup.modulus;
this->st_2 = (((commitmentToCoin.getCommitmentValue() * sg.inverse(params->accumulatorPoKCommitmentGroup.modulus)).pow_mod(r_gamma, params->accumulatorPoKCommitmentGroup.modulus)) * sh.pow_mod(r_psi, params->accumulatorPoKCommitmentGroup.modulus)) % params->accumulatorPoKCommitmentGroup.modulus;
this->st_3 = ((sg * commitmentToCoin.getCommitmentValue()).pow_mod(r_sigma, params->accumulatorPoKCommitmentGroup.modulus) * sh.pow_mod(r_xi, params->accumulatorPoKCommitmentGroup.modulus)) % params->accumulatorPoKCommitmentGroup.modulus;
this->t_1 = (h_n.pow_mod(r_zeta, params->accumulatorModulus) * g_n.pow_mod(r_epsilon, params->accumulatorModulus)) % params->accumulatorModulus;
this->t_2 = (h_n.pow_mod(r_eta, params->accumulatorModulus) * g_n.pow_mod(r_alpha, params->accumulatorModulus)) % params->accumulatorModulus;
this->t_3 = (C_u.pow_mod(r_alpha, params->accumulatorModulus) * ((h_n.inverse(params->accumulatorModulus)).pow_mod(r_beta, params->accumulatorModulus))) % params->accumulatorModulus;
this->t_4 = (C_r.pow_mod(r_alpha, params->accumulatorModulus) * ((h_n.inverse(params->accumulatorModulus)).pow_mod(r_delta, params->accumulatorModulus)) * ((g_n.inverse(params->accumulatorModulus)).pow_mod(r_beta, params->accumulatorModulus))) % params->accumulatorModulus;
CHashWriter hasher(0,0);
hasher << *params << sg << sh << g_n << h_n << commitmentToCoin.getCommitmentValue() << C_e << C_u << C_r << st_1 << st_2 << st_3 << t_1 << t_2 << t_3 << t_4;
//According to the proof, this hash should be of length k_prime bits. It is currently greater than that, which should not be a problem, but we should check this.
Bignum c = Bignum(hasher.GetHash());
this->s_alpha = r_alpha - c*e;
this->s_beta = r_beta - c*r_2*e;
this->s_zeta = r_zeta - c*r_3;
this->s_sigma = r_sigma - c*((e+1).inverse(params->accumulatorPoKCommitmentGroup.groupOrder));
this->s_eta = r_eta - c*r_1;
this->s_epsilon = r_epsilon - c*r_2;
this->s_delta = r_delta - c*r_3*e;
this->s_xi = r_xi + c*r*((e+1).inverse(params->accumulatorPoKCommitmentGroup.groupOrder));
this->s_phi = (r_phi - c*r) % params->accumulatorPoKCommitmentGroup.groupOrder;
this->s_gamma = r_gamma - c*((e-1).inverse(params->accumulatorPoKCommitmentGroup.groupOrder));
this->s_psi = r_psi + c*r*((e-1).inverse(params->accumulatorPoKCommitmentGroup.groupOrder));
}
SerialNumberSignatureOfKnowledge::SerialNumberSignatureOfKnowledge(const
ZerocoinParams* p, const PrivateCoin& coin, const Commitment& commitmentToCoin,
uint256 msghash):params(p),
s_notprime(p->zkp_iterations),
sprime(p->zkp_iterations) {
// Sanity check: verify that the order of the "accumulatedValueCommitmentGroup" is
// equal to the modulus of "coinCommitmentGroup". Otherwise we will produce invalid
// proofs.
if (params->coinCommitmentGroup.modulus != params->serialNumberSoKCommitmentGroup.groupOrder) {
throw std::runtime_error("Groups are not structured correctly.");
}
CBigNum a = params->coinCommitmentGroup.g;
CBigNum b = params->coinCommitmentGroup.h;
CBigNum g = params->serialNumberSoKCommitmentGroup.g;
CBigNum h = params->serialNumberSoKCommitmentGroup.h;
CHashWriter hasher(0,0);
hasher << *params << commitmentToCoin.getCommitmentValue() << coin.getSerialNumber() << msghash;
vector<CBigNum> r(params->zkp_iterations);
vector<CBigNum> v_seed(params->zkp_iterations);
vector<CBigNum> v_expanded(params->zkp_iterations);
vector<CBigNum> c(params->zkp_iterations);
for(uint32_t i=0; i < params->zkp_iterations; i++) {
r[i] = CBigNum::randBignum(params->coinCommitmentGroup.groupOrder);
//use a random 256 bit seed that expands to 1024 bit for v[i]
while (true) {
uint256 hashRand = CBigNum::randBignum(CBigNum(~uint256(0))).getuint256();
CBigNum bnExpanded = SeedTo1024(hashRand);
if(bnExpanded > params->serialNumberSoKCommitmentGroup.groupOrder)
continue;
v_seed[i] = CBigNum(hashRand);
v_expanded[i] = bnExpanded;
break;
}
}
for(uint32_t i=0; i < params->zkp_iterations; i++) {
// compute g^{ {a^x b^r} h^v} mod p2
c[i] = challengeCalculation(coin.getSerialNumber(), r[i], v_expanded[i]);
}
// We can't hash data in parallel either
// because OPENMP cannot not guarantee loops
// execute in order.
for(uint32_t i=0; i < params->zkp_iterations; i++) {
hasher << c[i];
}
this->hash = hasher.GetHash();
unsigned char *hashbytes = (unsigned char*) &hash;
for(uint32_t i = 0; i < params->zkp_iterations; i++) {
int bit = i % 8;
int byte = i / 8;
bool challenge_bit = ((hashbytes[byte] >> bit) & 0x01);
if (challenge_bit) {
s_notprime[i] = r[i];
sprime[i] = v_seed[i];
} else {
s_notprime[i] = r[i] - coin.getRandomness();
sprime[i] = v_expanded[i] - (commitmentToCoin.getRandomness() *
b.pow_mod(r[i] - coin.getRandomness(), params->serialNumberSoKCommitmentGroup.groupOrder));
}
}
}