void XTR_FindPrimesAndGenerator(RandomNumberGenerator &rng, Integer &p, Integer &q, GFP2Element &g, unsigned int pbits, unsigned int qbits)
{
CRYPTOPP_ASSERT(qbits > 9); // no primes exist for pbits = 10, qbits = 9
CRYPTOPP_ASSERT(pbits > qbits);
const Integer minQ = Integer::Power2(qbits - 1);
const Integer maxQ = Integer::Power2(qbits) - 1;
const Integer minP = Integer::Power2(pbits - 1);
const Integer maxP = Integer::Power2(pbits) - 1;
top:
Integer r1, r2;
do
{
(void)q.Randomize(rng, minQ, maxQ, Integer::PRIME, 7, 12);
// Solution always exists because q === 7 mod 12.
(void)SolveModularQuadraticEquation(r1, r2, 1, -1, 1, q);
// I believe k_i, r1 and r2 are being used slightly different than the
// paper's algorithm. I believe it is leading to the failed asserts.
// Just make the assert part of the condition.
if(!p.Randomize(rng, minP, maxP, Integer::PRIME, CRT(rng.GenerateBit() ?
r1 : r2, q, 2, 3, EuclideanMultiplicativeInverse(p, 3)), 3 * q)) { continue; }
} while (((p % 3U) != 2) || (((p.Squared() - p + 1) % q).NotZero()));
// CRYPTOPP_ASSERT((p % 3U) == 2);
// CRYPTOPP_ASSERT(((p.Squared() - p + 1) % q).IsZero());
GFP2_ONB<ModularArithmetic> gfp2(p);
GFP2Element three = gfp2.ConvertIn(3), t;
while (true)
{
g.c1.Randomize(rng, Integer::Zero(), p-1);
g.c2.Randomize(rng, Integer::Zero(), p-1);
t = XTR_Exponentiate(g, p+1, p);
if (t.c1 == t.c2)
continue;
g = XTR_Exponentiate(g, (p.Squared()-p+1)/q, p);
if (g != three)
break;
}
if (XTR_Exponentiate(g, q, p) != three)
goto top;
// CRYPTOPP_ASSERT(XTR_Exponentiate(g, q, p) == three);
}
bool XTR_DH::Agree(byte *agreedValue, const byte *privateKey, const byte *otherPublicKey, bool validateOtherPublicKey) const
{
GFP2Element w(otherPublicKey, PublicKeyLength());
if (validateOtherPublicKey)
{
GFP2_ONB<ModularArithmetic> gfp2(m_p);
GFP2Element three = gfp2.ConvertIn(3);
if (w.c1.IsNegative() || w.c2.IsNegative() || w.c1 >= m_p || w.c2 >= m_p || w == three)
return false;
if (XTR_Exponentiate(w, m_q, m_p) != three)
return false;
}
Integer s(privateKey, PrivateKeyLength());
GFP2Element z = XTR_Exponentiate(w, s, m_p);
z.Encode(agreedValue, AgreedValueLength());
return true;
}
bool XTR_DH::Validate(RandomNumberGenerator &rng, unsigned int level) const
{
bool pass = true;
pass = pass && m_p > Integer::One() && m_p.IsOdd();
pass = pass && m_q > Integer::One() && m_q.IsOdd();
GFP2Element three = GFP2_ONB<ModularArithmetic>(m_p).ConvertIn(3);
pass = pass && !(m_g.c1.IsNegative() || m_g.c2.IsNegative() || m_g.c1 >= m_p || m_g.c2 >= m_p || m_g == three);
if (level >= 1)
pass = pass && ((m_p.Squared()-m_p+1)%m_q).IsZero();
if (level >= 2)
{
pass = pass && VerifyPrime(rng, m_p, level-2) && VerifyPrime(rng, m_q, level-2);
pass = pass && XTR_Exponentiate(m_g, (m_p.Squared()-m_p+1)/m_q, m_p) != three;
pass = pass && XTR_Exponentiate(m_g, m_q, m_p) == three;
}
return pass;
}
void XTR_FindPrimesAndGenerator(RandomNumberGenerator &rng, Integer &p, Integer &q, GFP2Element &g, unsigned int pbits, unsigned int qbits)
{
assert(qbits > 9); // no primes exist for pbits = 10, qbits = 9
assert(pbits > qbits);
const Integer minQ = Integer::Power2(qbits - 1);
const Integer maxQ = Integer::Power2(qbits) - 1;
const Integer minP = Integer::Power2(pbits - 1);
const Integer maxP = Integer::Power2(pbits) - 1;
Integer r1, r2;
do
{
bool qFound = q.Randomize(rng, minQ, maxQ, Integer::PRIME, 7, 12);
CRYPTOPP_UNUSED(qFound); assert(qFound);
bool solutionsExist = SolveModularQuadraticEquation(r1, r2, 1, -1, 1, q);
CRYPTOPP_UNUSED(solutionsExist); assert(solutionsExist);
} while (!p.Randomize(rng, minP, maxP, Integer::PRIME, CRT(rng.GenerateBit()?r1:r2, q, 2, 3, EuclideanMultiplicativeInverse(p, 3)), 3*q));
assert(((p.Squared() - p + 1) % q).IsZero());
GFP2_ONB<ModularArithmetic> gfp2(p);
GFP2Element three = gfp2.ConvertIn(3), t;
while (true)
{
g.c1.Randomize(rng, Integer::Zero(), p-1);
g.c2.Randomize(rng, Integer::Zero(), p-1);
t = XTR_Exponentiate(g, p+1, p);
if (t.c1 == t.c2)
continue;
g = XTR_Exponentiate(g, (p.Squared()-p+1)/q, p);
if (g != three)
break;
}
assert(XTR_Exponentiate(g, q, p) == three);
}
void XTR_DH::GeneratePublicKey(RandomNumberGenerator &rng, const byte *privateKey, byte *publicKey) const
{
Integer x(privateKey, PrivateKeyLength());
GFP2Element y = XTR_Exponentiate(m_g, x, m_p);
y.Encode(publicKey, PublicKeyLength());
}