Bignum Bignum::operator*(const Bignum & bn) const {
Bignum tmp; short symbol = 0, bit;
const Bignum & self = *this;
Bignum::Info info = Bignum::Info(self, bn);
if (self[info.a_index] < 0) { symbol ^= 1; }
if (bn[info.b_index] < 0) { symbol ^= 1; }
self[info.a_index] = llabs(self[info.a_index]);
bn[info.b_index] = llabs(bn[info.b_index]);
for (int i = bignum_len - 1; i >= info.a_index; i--) {
for (int j = bignum_len - 1; j >= info.b_index; j--) {
bit = i - ( (bignum_len - 1) - j);
tmp[bit] += (self[i] * bn[j]);
tmp.carry(bit);
}
}
if (symbol) {
tmp[-tmp.bignum_used_len()] *= -1;
}
return tmp;
}
bool CommitmentProofOfKnowledge::Verify(const Bignum& A, const Bignum& B) const
{
// TODO: First verify that the values
// S1, S2 and S3 and "challenge" are in the correct ranges
if((this->challenge < Bignum(0)) || (this->challenge > (Bignum(2).pow(256) - Bignum(1)))){
return false;
}
// Compute T1 = g1^S1 * h1^S2 * inverse(A^{challenge}) mod p1
Bignum T1 = A.pow_mod(this->challenge, ap->modulus).inverse(ap->modulus).mul_mod(
(ap->g.pow_mod(S1, ap->modulus).mul_mod(ap->h.pow_mod(S2, ap->modulus), ap->modulus)),
ap->modulus);
// Compute T2 = g2^S1 * h2^S3 * inverse(B^{challenge}) mod p2
Bignum T2 = B.pow_mod(this->challenge, bp->modulus).inverse(bp->modulus).mul_mod(
(bp->g.pow_mod(S1, bp->modulus).mul_mod(bp->h.pow_mod(S3, bp->modulus), bp->modulus)),
bp->modulus);
// Hash T1 and T2 along with all of the public parameters
Bignum computedChallenge = calculateChallenge(A, B, T1, T2);
// Return success if the computed challenge matches the incoming challenge
if(computedChallenge == this->challenge){
return true;
}
// Otherwise return failure
return false;
}
int rb_big_sign(VALUE obj) {
NativeMethodEnvironment* env = NativeMethodEnvironment::get();
Bignum* big = c_as<Bignum>(env->get_object(obj));
return big->mp_val()->sign != MP_NEG;
}
void test_coerce_bignum() {
Fixnum* one = Fixnum::from(1);
Bignum* e = Bignum::create(state, one);
Array* ary = one->coerce(state, e);
Fixnum* a = try_as<Fixnum>(ary->get(state, 0));
Fixnum* b = try_as<Fixnum>(ary->get(state, 1));
TS_ASSERT_EQUALS(2, ary->size());
TS_ASSERT(a);
TS_ASSERT(b);
TS_ASSERT_EQUALS(one, a);
TS_ASSERT_EQUALS(one, b);
Bignum* f = Bignum::from(state, 9223372036854775807LL);
ary = one->coerce(state, f);
Bignum* c = try_as<Bignum>(ary->get(state, 0));
Bignum* d = try_as<Bignum>(ary->get(state, 1));
TS_ASSERT_EQUALS(2, ary->size());
TS_ASSERT(c);
TS_ASSERT(d);
TS_ASSERT_EQUALS(cTrue, c->equal(state, f));
TS_ASSERT_EQUALS(cTrue, d->equal(state, e));
}
int rb_big_bytes_used(VALUE obj) {
NativeMethodEnvironment* env = NativeMethodEnvironment::get();
Bignum* big = c_as<Bignum>(env->get_object(obj));
return big->size(env->state())->to_native();
}
bool
Test_GenerateGroupParams()
{
uint32_t pLen = 1024, qLen = 256, count;
IntegerGroupParams group;
for (count = 0; count < 1; count++) {
try {
group = deriveIntegerGroupParams(calculateSeed(GetTestModulus(), "test", ZEROCOIN_DEFAULT_SECURITYLEVEL, "TEST GROUP"), pLen, qLen);
} catch (std::runtime_error e) {
cout << "Caught exception " << e.what() << endl;
return false;
}
// Now perform some simple tests on the resulting parameters
if (group.groupOrder.bitSize() < qLen || group.modulus.bitSize() < pLen) {
return false;
}
Bignum c = group.g.pow_mod(group.groupOrder, group.modulus);
//cout << "g^q mod p = " << c << endl;
if (!(c.isOne())) return false;
// Try at multiple parameter sizes
pLen = pLen * 1.5;
qLen = qLen * 1.5;
}
return true;
}
void test_mul_with_bignum() {
Fixnum* one = as<Fixnum>(Fixnum::from(2));
Bignum* two = Bignum::from(state, (native_int)FIXNUM_MAX + 10);
Integer* three = one->mul(state, two);
TS_ASSERT_EQUALS(three->class_object(state), G(bignum));
Bignum* expected = as<Bignum>(two->mul(state, Fixnum::from(2)));
TS_ASSERT_EQUALS(cTrue, as<Bignum>(three)->equal(state, expected));
}
void test_get_type() {
TS_ASSERT_EQUALS(Qnil->get_type(), NilType);
TS_ASSERT_EQUALS(Qtrue->get_type(), TrueType);
TS_ASSERT_EQUALS(Qfalse->get_type(), FalseType);
TS_ASSERT_EQUALS(state->symbol("blah")->get_type(), SymbolType);
Object* obj = util_new_object();
Bignum* big = Bignum::from(state, (native_int)13);
TS_ASSERT_EQUALS(obj->get_type(), ObjectType);
TS_ASSERT_EQUALS(big->get_type(), BignumType);
}
double rb_big2dbl(VALUE obj) {
NativeMethodEnvironment* env = NativeMethodEnvironment::get();
Bignum* big = c_as<Bignum>(env->get_object(obj));
double d = big->to_double(env->state());
if(std::isinf(d)) {
rb_warn("Bignum out of Float range");
d = HUGE_VAL;
}
return d;
}
bool
Test_MintAndSpend()
{
try {
// This test assumes a list of coins were generated in Test_MintCoin()
if (gCoins[0] == NULL)
{
// No coins: mint some.
Test_MintCoin();
if (gCoins[0] == NULL) {
return false;
}
}
// Accumulate the list of generated coins into a fresh accumulator.
// The first one gets marked as accumulated for a witness, the
// others just get accumulated normally.
Accumulator acc(&g_Params->accumulatorParams);
AccumulatorWitness wAcc(g_Params, acc, gCoins[0]->getPublicCoin());
for (uint32_t i = 0; i < TESTS_COINS_TO_ACCUMULATE; i++) {
acc += gCoins[i]->getPublicCoin();
wAcc +=gCoins[i]->getPublicCoin();
}
// Now spend the coin
SpendMetaData m(1,1);
CDataStream cc(SER_NETWORK, PROTOCOL_VERSION);
cc << *gCoins[0];
PrivateCoin myCoin(g_Params,cc);
CoinSpend spend(g_Params, myCoin, acc, wAcc, m);
// Serialize the proof and deserialize into newSpend
CDataStream ss(SER_NETWORK, PROTOCOL_VERSION);
ss << spend;
gProofSize = ss.size();
CoinSpend newSpend(g_Params, ss);
// See if we can verify the deserialized proof (return our result)
bool ret = newSpend.Verify(acc, m);
// Extract the serial number
Bignum serialNumber = newSpend.getCoinSerialNumber();
gSerialNumberSize = ceil((double)serialNumber.bitSize() / 8.0);
return ret;
} catch (runtime_error &e) {
cout << e.what() << endl;
return false;
}
return false;
}
double rb_big2dbl(VALUE obj) {
NativeMethodEnvironment* env = NativeMethodEnvironment::get();
Bignum* big = c_as<Bignum>(env->get_object(obj));
double d = big->to_double(env->state());
if(isinf(d)) {
rb_warn("Bignum out of Float range");
if(big->mp_val()->sign == MP_NEG) {
d = -HUGE_VAL;
} else {
d = HUGE_VAL;
}
}
return d;
}
bool Test_InvalidCoin()
{
Bignum coinValue;
try {
// Pick a random non-prime Bignum
for (uint32_t i = 0; i < NON_PRIME_TESTS; i++) {
coinValue = Bignum::randBignum(g_Params->coinCommitmentGroup.modulus);
coinValue = coinValue * 2;
if (!coinValue.isPrime()) break;
}
PublicCoin pubCoin(g_Params);
if (pubCoin.validate()) {
// A blank coin should not be valid!
return false;
}
PublicCoin pubCoin2(g_Params, coinValue, ZQ_LOVELACE);
if (pubCoin2.validate()) {
// A non-prime coin should not be valid!
return false;
}
PublicCoin pubCoin3 = pubCoin2;
if (pubCoin2.validate()) {
// A copy of a non-prime coin should not be valid!
return false;
}
// Serialize and deserialize the coin
CDataStream ss(SER_NETWORK, PROTOCOL_VERSION);
ss << pubCoin;
PublicCoin pubCoin4(g_Params, ss);
if (pubCoin4.validate()) {
// A deserialized copy of a non-prime coin should not be valid!
return false;
}
} catch (runtime_error &e) {
cout << "Caught exception: " << e.what() << endl;
return false;
}
return true;
}
void PrivateCoin::mintCoinFast(const CoinDenomination denomination) {
// Generate a random serial number in the range 0...{q-1} where
// "q" is the order of the commitment group.
Bignum s = Bignum::randBignum(this->params->coinCommitmentGroup.groupOrder);
// Generate a random number "r" in the range 0...{q-1}
Bignum r = Bignum::randBignum(this->params->coinCommitmentGroup.groupOrder);
// Manually compute a Pedersen commitment to the serial number "s" under randomness "r"
// C = g^s * h^r mod p
Bignum commitmentValue = this->params->coinCommitmentGroup.g.pow_mod(s, this->params->coinCommitmentGroup.modulus).mul_mod(this->params->coinCommitmentGroup.h.pow_mod(r, this->params->coinCommitmentGroup.modulus), this->params->coinCommitmentGroup.modulus);
// Repeat this process up to MAX_COINMINT_ATTEMPTS times until
// we obtain a prime number
for (uint32_t attempt = 0; attempt < MAX_COINMINT_ATTEMPTS; attempt++) {
// First verify that the commitment is a prime number
// in the appropriate range. If not, we'll throw this coin
// away and generate a new one.
if (commitmentValue.isPrime(ZEROCOIN_MINT_PRIME_PARAM) &&
commitmentValue >= params->accumulatorParams.minCoinValue &&
commitmentValue <= params->accumulatorParams.maxCoinValue) {
// Found a valid coin. Store it.
this->serialNumber = s;
this->randomness = r;
this->publicCoin = PublicCoin(params, commitmentValue, denomination);
// Success! We're done.
return;
}
// Generate a new random "r_delta" in 0...{q-1}
Bignum r_delta = Bignum::randBignum(this->params->coinCommitmentGroup.groupOrder);
// The commitment was not prime. Increment "r" and recalculate "C":
// r = r + r_delta mod q
// C = C * h mod p
r = (r + r_delta) % this->params->coinCommitmentGroup.groupOrder;
commitmentValue = commitmentValue.mul_mod(this->params->coinCommitmentGroup.h.pow_mod(r_delta, this->params->coinCommitmentGroup.modulus), this->params->coinCommitmentGroup.modulus);
}
// We only get here if we did not find a coin within
// MAX_COINMINT_ATTEMPTS. Throw an exception.
throw ZerocoinException("Unable to mint a new Zerocoin (too many attempts)");
}
Bignum::Bignum(const Bignum & bn) {
init();
decimal_point = bn.decimal_point;
bignum_len = bn.bignum_len;
for (int i = bignum_len - bn.bignum_used_len(); i < bignum_len; i++) {
bignum[i] = bn[i];
}
}
/** Verifies that a commitment c is accumulated in accumulator a
*/
bool AccumulatorProofOfKnowledge:: Verify(const Accumulator& a, const Bignum& valueOfCommitmentToCoin) const {
Bignum sg = params->accumulatorPoKCommitmentGroup.g;
Bignum sh = params->accumulatorPoKCommitmentGroup.h;
Bignum g_n = params->accumulatorQRNCommitmentGroup.g;
Bignum h_n = params->accumulatorQRNCommitmentGroup.h;
//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.
CHashWriter hasher(0,0);
hasher << *params << sg << sh << g_n << h_n << valueOfCommitmentToCoin << C_e << C_u << C_r << st_1 << st_2 << st_3 << t_1 << t_2 << t_3 << t_4;
Bignum c = Bignum(hasher.GetHash()); //this hash should be of length k_prime bits
Bignum st_1_prime = (valueOfCommitmentToCoin.pow_mod(c, params->accumulatorPoKCommitmentGroup.modulus) * sg.pow_mod(s_alpha, params->accumulatorPoKCommitmentGroup.modulus) * sh.pow_mod(s_phi, params->accumulatorPoKCommitmentGroup.modulus)) % params->accumulatorPoKCommitmentGroup.modulus;
Bignum st_2_prime = (sg.pow_mod(c, params->accumulatorPoKCommitmentGroup.modulus) * ((valueOfCommitmentToCoin * sg.inverse(params->accumulatorPoKCommitmentGroup.modulus)).pow_mod(s_gamma, params->accumulatorPoKCommitmentGroup.modulus)) * sh.pow_mod(s_psi, params->accumulatorPoKCommitmentGroup.modulus)) % params->accumulatorPoKCommitmentGroup.modulus;
Bignum st_3_prime = (sg.pow_mod(c, params->accumulatorPoKCommitmentGroup.modulus) * (sg * valueOfCommitmentToCoin).pow_mod(s_sigma, params->accumulatorPoKCommitmentGroup.modulus) * sh.pow_mod(s_xi, params->accumulatorPoKCommitmentGroup.modulus)) % params->accumulatorPoKCommitmentGroup.modulus;
Bignum t_1_prime = (C_r.pow_mod(c, params->accumulatorModulus) * h_n.pow_mod(s_zeta, params->accumulatorModulus) * g_n.pow_mod(s_epsilon, params->accumulatorModulus)) % params->accumulatorModulus;
Bignum t_2_prime = (C_e.pow_mod(c, params->accumulatorModulus) * h_n.pow_mod(s_eta, params->accumulatorModulus) * g_n.pow_mod(s_alpha, params->accumulatorModulus)) % params->accumulatorModulus;
Bignum t_3_prime = ((a.getValue()).pow_mod(c, params->accumulatorModulus) * C_u.pow_mod(s_alpha, params->accumulatorModulus) * ((h_n.inverse(params->accumulatorModulus)).pow_mod(s_beta, params->accumulatorModulus))) % params->accumulatorModulus;
Bignum t_4_prime = (C_r.pow_mod(s_alpha, params->accumulatorModulus) * ((h_n.inverse(params->accumulatorModulus)).pow_mod(s_delta, params->accumulatorModulus)) * ((g_n.inverse(params->accumulatorModulus)).pow_mod(s_beta, params->accumulatorModulus))) % params->accumulatorModulus;
bool result = false;
bool result_st1 = (st_1 == st_1_prime);
bool result_st2 = (st_2 == st_2_prime);
bool result_st3 = (st_3 == st_3_prime);
bool result_t1 = (t_1 == t_1_prime);
bool result_t2 = (t_2 == t_2_prime);
bool result_t3 = (t_3 == t_3_prime);
bool result_t4 = (t_4 == t_4_prime);
bool result_range = ((s_alpha >= -(params->maxCoinValue * Bignum(2).pow(params->k_prime + params->k_dprime + 1))) && (s_alpha <= (params->maxCoinValue * Bignum(2).pow(params->k_prime + params->k_dprime + 1))));
result = result_st1 && result_st2 && result_st3 && result_t1 && result_t2 && result_t3 && result_t4 && result_range;
return result;
}
/////////////////////////////////////
//
// Operator Computing
//
////////////////////////////////////
Bignum Bignum::operator+(const Bignum & bn) const {
Bignum tmp;
const Bignum & self = *this;
Bignum::Info info = Bignum::Info(self, bn);
if (bn[info.b_index] < 0) {
tmp = bn;
tmp[info.b_index] = llabs(tmp[info.b_index]);
return self - bn;
} else if (self[info.a_index] < 0) {
tmp = self;
tmp[info.a_index] = llabs(tmp[info.a_index]);
return bn - tmp;
}
for (int i = bignum_len - 1; i >= 0; i--) {
tmp[i] += self[i] + bn[i];
tmp.carry(i);
}
return tmp;
}
void
CalculateParams(Params ¶ms, Bignum N, string aux, uint32_t securityLevel)
{
cout << "GNOSIS DEBUG: CalculateParams in ParamGeneration.cpp" << endl;
params.initialized = false;
params.accumulatorParams.initialized = false;
cout << "GNOSIS DEBUG: aux is " << aux << endl;
// Verify that |N| is > 1023 bits.
uint32_t NLen = N.bitSize();
cout << "GNOSIS DEBUG: NLen is " << NLen << endl;
if (NLen < 1023) {
throw ZerocoinException("Modulus must be at least 1023 bits");
}
// Verify that "securityLevel" is at least 80 bits (minimum).
if (securityLevel < 80) {
throw ZerocoinException("Security level must be at least 80 bits.");
}
cout << "GNOSIS DEBUG: securityLevel is " << securityLevel << endl;
// Set the accumulator modulus to "N".
params.accumulatorParams.accumulatorModulus = N;
// Calculate the required size of the field "F_p" into which
// we're embedding the coin commitment group. This may throw an
// exception if the securityLevel is too large to be supported
// by the current modulus.
uint32_t pLen = 0;
uint32_t qLen = 0;
calculateGroupParamLengths(NLen - 2, securityLevel, &pLen, &qLen);
// Calculate candidate parameters ("p", "q") for the coin commitment group
// using a deterministic process based on "N", the "aux" string, and
// the dedicated string "COMMITMENTGROUP".
params.coinCommitmentGroup = deriveIntegerGroupParams(calculateSeed(N, aux, securityLevel, STRING_COMMIT_GROUP),
pLen, qLen);
// g and h are invalid, since they are now different for each coin; see
// "Rational Zero" by Garman et al., section 4.4.
params.coinCommitmentGroup.invalidateGenerators();
PRINT_BIGNUM("params.coinCommitmentGroup.groupOrder", params.coinCommitmentGroup.groupOrder);
PRINT_BIGNUM("params.coinCommitmentGroup.modulus", params.coinCommitmentGroup.modulus);
// Next, we derive parameters for a second Accumulated Value commitment group.
// This is a Schnorr group with the specific property that the order of the group
// must be exactly equal to "q" from the commitment group. We set
// the modulus of the new group equal to "2q+1" and test to see if this is prime.
params.serialNumberSoKCommitmentGroup = deriveIntegerGroupFromOrder(params.coinCommitmentGroup.modulus);
PRINT_GROUP_PARAMS(params.serialNumberSoKCommitmentGroup);
// Calculate the parameters for the internal commitment
// using the same process.
params.accumulatorParams.accumulatorPoKCommitmentGroup = deriveIntegerGroupParams(calculateSeed(N, aux, securityLevel, STRING_AIC_GROUP),
qLen + 300, qLen + 1);
PRINT_GROUP_PARAMS(params.accumulatorParams.accumulatorPoKCommitmentGroup);
// Calculate the parameters for the accumulator QRN commitment generators. This isn't really
// a whole group, just a pair of random generators in QR_N.
uint32_t resultCtr;
params.accumulatorParams.accumulatorQRNCommitmentGroup.g(generateIntegerFromSeed(NLen - 1,
calculateSeed(N, aux, securityLevel, STRING_QRNCOMMIT_GROUPG),
&resultCtr).pow_mod(Bignum(2), N));
params.accumulatorParams.accumulatorQRNCommitmentGroup.h(generateIntegerFromSeed(NLen - 1,
calculateSeed(N, aux, securityLevel, STRING_QRNCOMMIT_GROUPH),
&resultCtr).pow_mod(Bignum(2), N));
PRINT_BIGNUM("params.accumulatorParams.accumulatorQRNCommitmentGroup.g", params.accumulatorParams.accumulatorQRNCommitmentGroup.g());
PRINT_BIGNUM("params.accumulatorParams.accumulatorQRNCommitmentGroup.h", params.accumulatorParams.accumulatorQRNCommitmentGroup.h());
// Calculate the accumulator base, which we calculate as "u = C**2 mod N"
// where C is an arbitrary value. In the unlikely case that "u = 1" we increment
// "C" and repeat.
Bignum constant(ACCUMULATOR_BASE_CONSTANT);
params.accumulatorParams.accumulatorBase = Bignum(1);
for (uint32_t count = 0; count < MAX_ACCUMGEN_ATTEMPTS && params.accumulatorParams.accumulatorBase.isOne(); count++) {
params.accumulatorParams.accumulatorBase = constant.pow_mod(Bignum(2), params.accumulatorParams.accumulatorModulus);
}
// Compute the accumulator range. The upper range is the largest possible coin commitment value.
// The lower range is sqrt(upper range) + 1. Since OpenSSL doesn't have
// a square root function we use a slightly higher approximation.
params.accumulatorParams.maxCoinValue = params.coinCommitmentGroup.modulus;
params.accumulatorParams.minCoinValue = Bignum(2).pow((params.coinCommitmentGroup.modulus.bitSize() / 2) + 3);
// If all went well, mark params as successfully initialized.
params.accumulatorParams.initialized = true;
// If all went well, mark params as successfully initialized.
params.initialized = true;
}
int main(int argc, char **argv)
{
static Bignum resultModulus(0);
uint32_t numBits = DEFAULT_MODULUS_SIZE;
ofstream outfile;
char* outfileName;
bool writeToFile = false;
while ((argc > 1) && (argv[1][0] == '-'))
{
switch (argv[1][1])
{
case 'b':
numBits = atoi(argv[2]);
++argv;
--argc;
break;
case 'o':
outfileName = argv[2];
writeToFile = true;
break;
case 'h':
usage();
break;
default:
printf("Wrong Argument: %s\n", argv[1]);
usage();
break;
}
++argv;
--argc;
}
if (numBits < MIN_MODULUS_SIZE) {
cout << "Modulus is below minimum length (" << MIN_MODULUS_SIZE << ") bits" << endl;
return(0);
}
PrintWarning();
cout << "Modulus size set to " << numBits << " bits." << endl;
cout << "Generating parameters. This may take a few minutes..." << endl;
// Generate two safe primes "p" and "q"
Bignum *p, *q;
p = new Bignum(0);
q = new Bignum(0);
*p = Bignum::generatePrime(numBits / 2, true);
*q = Bignum::generatePrime(numBits / 2, true);
// Multiply to compute N
resultModulus = (*p) * (*q);
// Wipe out the factors
delete p;
delete q;
// Convert to a hexidecimal string
std::string resultHex = resultModulus.ToString(16);
cout << endl << "N = " << endl << resultHex << endl;
if (writeToFile) {
try {
outfile.open (outfileName);
outfile << resultHex;
outfile.close();
cout << endl << "Result has been written to file '" << outfileName << "'." << endl;
} catch (std::runtime_error &e) {
cout << "Unable to write to file:" << e.what() << endl;
}
}
}
Value RandomState::random(Value arg)
{
if (fixnump(arg))
{
long n = xlong(arg);
if (n > 0)
{
mpz_t limit;
mpz_init_set_si(limit, n);
mpz_t result;
mpz_init(result);
mpz_urandomm(result, _state, limit);
return normalize(result);
}
}
else if (bignump(arg))
{
Bignum * b = the_bignum(arg);
if (b->plusp())
{
mpz_t result;
mpz_init(result);
mpz_urandomm(result, _state, b->_z);
return normalize(result);
}
}
else if (single_float_p(arg))
{
float f = the_single_float(arg)->_f;
if (f > 0)
{
mpz_t fixnum_limit;
mpz_init_set_si(fixnum_limit, MOST_POSITIVE_FIXNUM);
mpz_t fixnum_result;
mpz_init(fixnum_result);
mpz_urandomm(fixnum_result, _state, fixnum_limit);
double double_result = mpz_get_si(fixnum_result);
double_result /= MOST_POSITIVE_FIXNUM;
return make_value(new SingleFloat(double_result * f));
}
}
else if (double_float_p(arg))
{
double d = the_double_float(arg)->_d;
if (d > 0)
{
mpz_t fixnum_limit;
mpz_init_set_si(fixnum_limit, MOST_POSITIVE_FIXNUM);
mpz_t fixnum_result;
mpz_init(fixnum_result);
mpz_urandomm(fixnum_result, _state, fixnum_limit);
double double_result = mpz_get_si(fixnum_result);
double_result /= MOST_POSITIVE_FIXNUM;
return make_value(new DoubleFloat(double_result * d));
}
}
return signal_type_error(arg,
list3(S_or,
list2(S_integer, list1(FIXNUM_ZERO)),
list2(S_float, list1(FIXNUM_ZERO))));
}
void
calculateGroupModulusAndOrder(uint256 seed, uint32_t pLen, uint32_t qLen,
Bignum *resultModulus, Bignum *resultGroupOrder,
uint256 *resultPseed, uint256 *resultQseed)
{
// Verify that the seed length is >= qLen
if (qLen > (sizeof(seed)) * 8) {
// TODO: The use of 256-bit seeds limits us to 256-bit group orders. We should probably change this.
// throw ZerocoinException("Seed is too short to support the required security level.");
}
#ifdef ZEROCOIN_DEBUG
cout << "calculateGroupModulusAndOrder: pLen = " << pLen << endl;
#endif
// Generate a random prime for the group order.
// This may throw an exception, which we'll pass upwards.
// Result is the value "resultGroupOrder", "qseed" and "qgen_counter".
uint256 qseed;
uint32_t qgen_counter;
*resultGroupOrder = generateRandomPrime(qLen, seed, &qseed, &qgen_counter);
// Using ⎡pLen / 2 + 1⎤ as the length and qseed as the input_seed, use the random prime
// routine to obtain p0 , pseed, and pgen_counter. We pass exceptions upward.
uint32_t p0len = ceil((pLen / 2.0) + 1);
uint256 pseed;
uint32_t pgen_counter;
Bignum p0 = generateRandomPrime(p0len, qseed, &pseed, &pgen_counter);
// Set x = 0, old_counter = pgen_counter
uint32_t old_counter = pgen_counter;
// Generate a random integer "x" of pLen bits
uint32_t iterations;
Bignum x = generateIntegerFromSeed(pLen, pseed, &iterations);
pseed += (iterations + 1);
// Set x = 2^{pLen−1} + (x mod 2^{pLen–1}).
Bignum powerOfTwo = Bignum(2).pow(pLen-1);
x = powerOfTwo + (x % powerOfTwo);
// t = ⎡x / (2 * resultGroupOrder * p0)⎤.
// TODO: we don't have a ceiling function
Bignum t = x / (Bignum(2) * (*resultGroupOrder) * p0);
// Now loop until we find a valid prime "p" or we fail due to
// pgen_counter exceeding ((4*pLen) + old_counter).
for ( ; pgen_counter <= ((4*pLen) + old_counter) ; pgen_counter++) {
// If (2 * t * resultGroupOrder * p0 + 1) > 2^{pLen}, then
// t = ⎡2^{pLen−1} / (2 * resultGroupOrder * p0)⎤.
powerOfTwo = Bignum(2).pow(pLen);
Bignum prod = (Bignum(2) * t * (*resultGroupOrder) * p0) + Bignum(1);
if (prod > powerOfTwo) {
// TODO: implement a ceil function
t = Bignum(2).pow(pLen-1) / (Bignum(2) * (*resultGroupOrder) * p0);
}
// Compute a candidate prime resultModulus = 2tqp0 + 1.
*resultModulus = (Bignum(2) * t * (*resultGroupOrder) * p0) + Bignum(1);
// Verify that resultModulus is prime. First generate a pseudorandom integer "a".
Bignum a = generateIntegerFromSeed(pLen, pseed, &iterations);
pseed += iterations + 1;
// Set a = 2 + (a mod (resultModulus–3)).
a = Bignum(2) + (a % ((*resultModulus) - Bignum(3)));
// Set z = a^{2 * t * resultGroupOrder} mod resultModulus
Bignum z = a.pow_mod(Bignum(2) * t * (*resultGroupOrder), (*resultModulus));
// If GCD(z–1, resultModulus) == 1 AND (z^{p0} mod resultModulus == 1)
// then we have found our result. Return.
if ((resultModulus->gcd(z - Bignum(1))).isOne() &&
(z.pow_mod(p0, (*resultModulus))).isOne()) {
// Success! Return the seeds and primes.
*resultPseed = pseed;
*resultQseed = qseed;
return;
}
// This prime did not work out. Increment "t" and try again.
t = t + Bignum(1);
} // loop continues until pgen_counter exceeds a limit
// We reach this point only if we exceeded our maximum iteration count.
// Throw an exception.
throw ZerocoinException("Unable to generate a prime modulus for the group");
}
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));
}
Bignum
generateRandomPrime(uint32_t primeBitLen, uint256 in_seed, uint256 *out_seed,
uint32_t *prime_gen_counter)
{
// Verify that primeBitLen is not too small
if (primeBitLen < 2) {
throw ZerocoinException("Prime length is too short");
}
// If primeBitLen < 33 bits, perform the base case.
if (primeBitLen < 33) {
Bignum result(0);
// Set prime_seed = in_seed, prime_gen_counter = 0.
uint256 prime_seed = in_seed;
(*prime_gen_counter) = 0;
// Loop up to "4 * primeBitLen" iterations.
while ((*prime_gen_counter) < (4 * primeBitLen)) {
// Generate a pseudorandom integer "c" of length primeBitLength bits
uint32_t iteration_count;
Bignum c = generateIntegerFromSeed(primeBitLen, prime_seed, &iteration_count);
#ifdef ZEROCOIN_DEBUG
cout << "generateRandomPrime: primeBitLen = " << primeBitLen << endl;
cout << "Generated c = " << c << endl;
#endif
prime_seed += (iteration_count + 1);
(*prime_gen_counter)++;
// Set "intc" to be the least odd integer >= "c" we just generated
uint32_t intc = c.getulong();
intc = (2 * floor(intc / 2.0)) + 1;
#ifdef ZEROCOIN_DEBUG
cout << "Should be odd. c = " << intc << endl;
cout << "The big num is: c = " << c << endl;
#endif
// Perform trial division on this (relatively small) integer to determine if "intc"
// is prime. If so, return success.
if (primalityTestByTrialDivision(intc)) {
// Return "intc" converted back into a Bignum and "prime_seed". We also updated
// the variable "prime_gen_counter" in previous statements.
result = intc;
*out_seed = prime_seed;
// Success
return result;
}
} // while()
// If we reached this point there was an error finding a candidate prime
// so throw an exception.
throw ZerocoinException("Unable to find prime in Shawe-Taylor algorithm");
// END OF BASE CASE
}
// If primeBitLen >= 33 bits, perform the recursive case.
else {
// Recurse to find a new random prime of roughly half the size
uint32_t newLength = ceil((double)primeBitLen / 2.0) + 1;
Bignum c0 = generateRandomPrime(newLength, in_seed, out_seed, prime_gen_counter);
// Generate a random integer "x" of primeBitLen bits using the output
// of the previous call.
uint32_t numIterations;
Bignum x = generateIntegerFromSeed(primeBitLen, *out_seed, &numIterations);
(*out_seed) += numIterations + 1;
// Compute "t" = ⎡x / (2 * c0⎤
// TODO no Ceiling call
Bignum t = x / (Bignum(2) * c0);
// Repeat the following procedure until we find a prime (or time out)
for (uint32_t testNum = 0; testNum < MAX_PRIMEGEN_ATTEMPTS; testNum++) {
// If ((2 * t * c0) + 1 > 2^{primeBitLen}),
// then t = ⎡2^{primeBitLen} – 1 / (2 * c0)⎤.
if ((Bignum(2) * t * c0) > (Bignum(2).pow(Bignum(primeBitLen)))) {
t = ((Bignum(2).pow(Bignum(primeBitLen))) - Bignum(1)) / (Bignum(2) * c0);
}
// Set c = (2 * t * c0) + 1
Bignum c = (Bignum(2) * t * c0) + Bignum(1);
// Increment prime_gen_counter
(*prime_gen_counter)++;
// Test "c" for primality as follows:
// 1. First pick an integer "a" in between 2 and (c - 2)
Bignum a = generateIntegerFromSeed(c.bitSize(), (*out_seed), &numIterations);
a = Bignum(2) + (a % (c - Bignum(3)));
(*out_seed) += (numIterations + 1);
// 2. Compute "z" = a^{2*t} mod c
Bignum z = a.pow_mod(Bignum(2) * t, c);
// 3. Check if "c" is prime.
// Specifically, verify that gcd((z-1), c) == 1 AND (z^c0 mod c) == 1
// If so we return "c" as our result.
if (c.gcd(z - Bignum(1)).isOne() && z.pow_mod(c0, c).isOne()) {
// Return "c", out_seed and prime_gen_counter
// (the latter two of which were already updated)
return c;
}
// 4. If the test did not succeed, increment "t" and loop
t = t + Bignum(1);
} // end of test loop
}
// We only reach this point if the test loop has iterated MAX_PRIMEGEN_ATTEMPTS
// and failed to identify a valid prime. Throw an exception.
throw ZerocoinException("Unable to generate random prime (too many tests)");
}
bool
ZerocoinTutorial()
{
// The following simple code illustrates the call flow for Zerocoin
// applications. In a real currency network these operations would
// be split between individual payers/payees, network nodes and miners.
//
// For each call we specify the participant who would use it.
// Zerocoin uses exceptions (based on the runtime_error class)
// to indicate all sorts of problems. Always remember to catch them!
try {
/********************************************************************/
// What is it: Parameter loading
// Who does it: ALL ZEROCOIN PARTICIPANTS
// What it does: Loads a trusted Zerocoin modulus "N" and
// generates all associated parameters from it.
// We use a hardcoded "N" that we generated using
// the included 'paramgen' utility.
/********************************************************************/
// Load a test modulus from our hardcoded string (above)
Bignum testModulus;
testModulus.SetHex(std::string(TUTORIAL_TEST_MODULUS));
// Set up the Zerocoin Params object
libzerocoin::Params* params = new libzerocoin::Params(testModulus);
cout << "Successfully loaded parameters." << endl;
/********************************************************************/
// What is it: Coin generation
// Who does it: ZEROCOIN CLIENTS
// What it does: Generates a new 'zerocoin' coin using the
// public parameters. Once generated, the client
// will transmit the public portion of this coin
// in a ZEROCOIN_MINT transaction. The inputs
// to this transaction must add up to the zerocoin
// denomination plus any transaction fees.
/********************************************************************/
// The following constructor does all the work of minting a brand
// new zerocoin. It stores all the private values inside the
// PrivateCoin object. This includes the coin secrets, which must be
// stored in a secure location (wallet) at the client.
libzerocoin::PrivateCoin newCoin(params);
// Get a copy of the 'public' portion of the coin. You should
// embed this into a Zerocoin 'MINT' transaction along with a series
// of currency inputs totaling the assigned value of one zerocoin.
libzerocoin::PublicCoin pubCoin = newCoin.getPublicCoin();
cout << "Successfully minted a zerocoin." << endl;
// Serialize the public coin to a CDataStream object.
CDataStream serializedCoin(SER_NETWORK, PROTOCOL_VERSION);
serializedCoin << pubCoin;
/********************************************************************/
// What is it: Coin verification
// Who does it: TRANSACTION VERIFIERS
// What it does: Verifies the structure of a zerocoin obtained from
// a ZEROCOIN_MINT transaction. All coins must be
// verified before you operate on them.
// Note that this is only part of the transaction
// verification process! The client must also check
// that (1) the inputs to the transaction are valid
// and add up to the value of one zerocoin, (2) that
// this particular zerocoin has not been minted before.
/********************************************************************/
// Deserialize the public coin into a fresh object. This will
// automatically validate that the coin is correctly structured and
// will throw an exception if it isn't. (You need to handle those.)
libzerocoin::PublicCoin pubCoinNew(params, serializedCoin);
cout << "Deserialized and verified the coin." << endl;
/********************************************************************/
// What is it: Accumulator computation
// Who does it: ZEROCOIN CLIENTS & TRANSACTION VERIFIERS
// What it does: Collects a number of PublicCoin values drawn from
// the block chain and calculates an accumulator.
// This accumulator is incrementally computable;
// you can stop and serialize it at any point
// then continue accumulating new transactions.
// The accumulator is also order-independent, so
// the same coins can be accumulated in any order
// to give the same result.
// WARNING: do not accumulate the same coin twice!
/********************************************************************/
// Create an empty accumulator object
libzerocoin::Accumulator accumulator(params);
// Add several coins to it (we'll generate them here on the fly).
for (uint32_t i = 0; i < COINS_TO_ACCUMULATE; i++) {
libzerocoin::PrivateCoin testCoin(params);
accumulator += testCoin.getPublicCoin();
}
// Serialize the accumulator object.
//
// If you're using Accumulator Checkpoints, each miner would
// start by deserializing the accumulator checkpoint from the
// previous block (or creating a new Accumulator if no previous
// block exists). It will then add all the coins in the new block,
// then serialize the resulting Accumulator object to obtain the
// new checkpoint. All block verifiers should do the same thing
// to check their work.
CDataStream serializedAccumulator(SER_NETWORK, PROTOCOL_VERSION);
serializedAccumulator << accumulator;
// Deserialize the accumulator object
libzerocoin::Accumulator newAccumulator(params, serializedAccumulator);
// We can now continue accumulating things into the accumulator
// we just deserialized. For example, let's put in the coin
// we generated up above.
newAccumulator += pubCoinNew;
cout << "Successfully accumulated coins." << endl;
/********************************************************************/
// What is it: Coin spend
// Who does it: ZEROCOIN CLIENTS
// What it does: Create a new transaction that spends a Zerocoin.
// The user first authors a transaction specifying
// a set of destination addresses (outputs). They
// next compute an Accumulator over all coins in the
// block chain (see above) and a Witness based on the
// coin to be spent. Finally they instantiate a CoinSpend
// object that 'signs' the transaction with a special
// zero knowledge signature of knowledge over the coin
// data, Witness and Accumulator.
/********************************************************************/
// We are going to spend the coin "newCoin" that we generated at the
// top of this function.
//
// We'll use the accumulator we constructed above. This contains
// a set of coins, but does NOT include the coin "newCoin".
//
// To generate the witness, we start with this accumulator and
// add the public half of the coin we want to spend.
libzerocoin::AccumulatorWitness witness(params, accumulator, newCoin.getPublicCoin());
// Add the public half of "newCoin" to the Accumulator itself.
accumulator += newCoin.getPublicCoin();
// At this point we should generate a ZEROCOIN_SPEND transaction to
// send to the network. This network should include a set of outputs
// totalling to the value of one zerocoin (minus transaction fees).
//
// The format of this transaction is up to the implementer. Here we'll
// assume you've formatted this transaction and placed the hash into
// "transactionHash". We'll also assume "accumulatorHash" contains the
// hash of the last block whose transactions are in the accumulator.
uint256 transactionHash = DUMMY_TRANSACTION_HASH;
uint256 accumulatorID = DUMMY_ACCUMULATOR_ID;
// Place "transactionHash" and "accumulatorBlockHash" into a new
// SpendMetaData object.
libzerocoin::SpendMetaData metaData(accumulatorID, transactionHash);
// Construct the CoinSpend object. This acts like a signature on the
// transaction.
libzerocoin::CoinSpend spend(params, newCoin, accumulator, witness, metaData);
// This is a sanity check. The CoinSpend object should always verify,
// but why not check before we put it onto the wire?
if (!spend.Verify(accumulator, metaData)) {
cout << "ERROR: Our new CoinSpend transaction did not verify!" << endl;
return false;
}
// Serialize the CoinSpend object into a buffer.
CDataStream serializedCoinSpend(SER_NETWORK, PROTOCOL_VERSION);
serializedCoinSpend << spend;
cout << "Successfully generated a coin spend transaction." << endl;
/********************************************************************/
// What is it: Coin spend verification
// Who does it: ALL PARTIES
// What it does: Verifies that a CoinSpend signature is correct
// with respect to a ZEROCOIN_SPEND transaction hash.
// The client must also extract the serial number from
// the CoinSpend and verify that this serial number has
// not previously appeared in another ZEROCOIN_SPEND
// transaction.
/********************************************************************/
// Deserialize the CoinSpend intro a fresh object
libzerocoin::CoinSpend newSpend(params, serializedCoinSpend);
// Create a new metadata object to contain the hash of the received
// ZEROCOIN_SPEND transaction. If we were a real client we'd actually
// compute the hash of the received transaction here.
libzerocoin::SpendMetaData newMetadata(accumulatorID, transactionHash);
// If we were a real client we would now re-compute the Accumulator
// from the information given in the ZEROCOIN_SPEND transaction.
// For our purposes we'll just use the one we calculated above.
//
// Verify that the spend is valid with respect to the Accumulator
// and the Metadata
if (!newSpend.Verify(accumulator, newMetadata)) {
cout << "ERROR: The CoinSpend transaction did not verify!" << endl;
return false;
}
// Pull the serial number out of the CoinSpend object. If we
// were a real Zerocoin client we would now check that the serial number
// has not been spent before (in another ZEROCOIN_SPEND) transaction.
// The serial number is stored as a Bignum.
Bignum serialNumber = newSpend.getCoinSerialNumber();
cout << "Successfully verified a coin spend transaction." << endl;
cout << endl << "Coin serial number is:" << endl << serialNumber << endl;
// We're done
return true;
} catch (runtime_error &e) {
cout << e.what() << endl;
return false;
}
return false;
}
