PointGFp multi_exponentiate(const PointGFp& x, const BigInt& z1,
const PointGFp& y, const BigInt& z2)
{
const size_t z_bits = round_up(std::max(z1.bits(), z2.bits()), 2);
std::vector<BigInt> ws(PointGFp::WORKSPACE_SIZE);
PointGFp x2 = x;
x2.mult2(ws);
const PointGFp x3(x2.plus(x, ws));
PointGFp y2 = y;
y2.mult2(ws);
const PointGFp y3(y2.plus(y, ws));
const PointGFp M[16] = {
x.zero(), // 0000
x, // 0001
x2, // 0010
x3, // 0011
y, // 0100
y.plus(x, ws), // 0101
y.plus(x2, ws), // 0110
y.plus(x3, ws), // 0111
y2, // 1000
y2.plus(x, ws), // 1001
y2.plus(x2, ws), // 1010
y2.plus(x3, ws), // 1011
y3, // 1100
y3.plus(x, ws), // 1101
y3.plus(x2, ws), // 1110
y3.plus(x3, ws), // 1111
};
PointGFp H = x.zero();
for(size_t i = 0; i != z_bits; i += 2)
{
if(i > 0)
{
H.mult2(ws);
H.mult2(ws);
}
const uint8_t z1_b = z1.get_substring(z_bits - i - 2, 2);
const uint8_t z2_b = z2.get_substring(z_bits - i - 2, 2);
const uint8_t z12 = (4*z2_b) + z1_b;
H.add(M[z12], ws);
}
if(z1.is_negative() != z2.is_negative())
H.negate();
return H;
}
PointGFp Blinded_Point_Multiply::blinded_multiply(const BigInt& scalar_in,
RandomNumberGenerator& rng)
{
if(scalar_in.is_negative())
throw std::invalid_argument("Blinded_Point_Multiply scalar must be positive");
#if BOTAN_POINTGFP_SCALAR_BLINDING_BITS > 0
// Choose a small mask m and use k' = k + m*order (Coron's 1st countermeasure)
const BigInt mask(rng, BOTAN_POINTGFP_SCALAR_BLINDING_BITS, false);
const BigInt scalar = scalar_in + m_order * mask;
#else
const BigInt& scalar = scalar_in;
#endif
const size_t scalar_bits = scalar.bits();
// Randomize each point representation (Coron's 3rd countermeasure)
for(size_t i = 0; i != m_U.size(); ++i)
m_U[i].randomize_repr(rng);
#if BOTAN_POINTGFP_BLINDED_MULTIPLY_USE_MONTGOMERY_LADDER
PointGFp R = m_U.at(3*m_h + 2); // base point
int32_t alpha = 0;
R.randomize_repr(rng);
/*
Algorithm 7 from "Randomizing the Montgomery Powering Ladder"
Duc-Phong Le, Chik How Tan and Michael Tunstall
http://eprint.iacr.org/2015/657
It takes a random walk through (a subset of) the set of addition
chains that end in k.
*/
for(size_t i = scalar_bits; i > 0; i--)
{
const int32_t ki = scalar.get_bit(i);
// choose gamma from -h,...,h
const int32_t gamma = static_cast<int32_t>((rng.next_byte() % (2*m_h))) - m_h;
const int32_t l = gamma - 2*alpha + ki - (ki ^ 1);
R.mult2(m_ws);
R.add(m_U.at(3*m_h + 1 + l), m_ws);
alpha = gamma;
}
const int32_t k0 = scalar.get_bit(0);
R.add(m_U[3*m_h + 1 - alpha - (k0 ^ 1)], m_ws);
#else
// N-bit windowing exponentiation:
size_t windows = round_up(scalar_bits, m_h) / m_h;
PointGFp R = m_U[0];
if(windows > 0)
{
windows--;
const u32bit nibble = scalar.get_substring(windows*m_h, m_h);
R.add(m_U[nibble], m_ws);
/*
Randomize after adding the first nibble as before the addition R
is zero, and we cannot effectively randomize the point
representation of the zero point.
*/
R.randomize_repr(rng);
while(windows)
{
for(size_t i = 0; i != m_h; ++i)
R.mult2(m_ws);
const u32bit nibble = scalar.get_substring((windows-1)*m_h, m_h);
R.add(m_U[nibble], m_ws);
windows--;
}
}
#endif
//BOTAN_ASSERT(R.on_the_curve(), "Output is on the curve");
return R;
}