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 operator*(const BigInt& scalar, const PointGFp& point)
{
//BOTAN_ASSERT(point.on_the_curve(), "Input is on the curve");
const CurveGFp& curve = point.get_curve();
const size_t scalar_bits = scalar.bits();
std::vector<BigInt> ws(9);
if(scalar_bits <= 2)
{
const byte abs_val = scalar.byte_at(0);
if(abs_val == 0)
return PointGFp::zero_of(curve);
PointGFp result = point;
if(abs_val == 2)
result.mult2(ws);
if(scalar.is_negative())
result.negate();
return result;
}
PointGFp R[2] = { PointGFp(curve), point };
for(size_t i = scalar_bits; i > 0; i--)
{
const size_t b = scalar.get_bit(i - 1);
R[b ^ 1].add(R[b], ws);
R[b].mult2(ws);
}
if(scalar.is_negative())
R[0].negate();
//BOTAN_ASSERT(R[0].on_the_curve(), "Output is on the curve");
return R[0];
}
PointGFp operator*(const BigInt& scalar, const PointGFp& point)
{
const CurveGFp& curve = point.get_curve();
if(scalar.is_zero())
return PointGFp(curve); // zero point
std::vector<BigInt> ws(9);
if(scalar.abs() <= 2) // special cases for small values
{
byte value = scalar.abs().byte_at(0);
PointGFp result = point;
if(value == 2)
result.mult2(ws);
if(scalar.is_negative())
result.negate();
return result;
}
const size_t scalar_bits = scalar.bits();
#if 0
PointGFp x1 = PointGFp(curve);
PointGFp x2 = point;
size_t bits_left = scalar_bits;
// Montgomery Ladder
while(bits_left)
{
const bool bit_set = scalar.get_bit(bits_left - 1);
if(bit_set)
{
x1.add(x2, ws);
x2.mult2(ws);
}
else
{
x2.add(x1, ws);
x1.mult2(ws);
}
--bits_left;
}
if(scalar.is_negative())
x1.negate();
return x1;
#else
const size_t window_size = 4;
std::vector<PointGFp> Ps(1 << window_size);
Ps[0] = PointGFp(curve);
Ps[1] = point;
for(size_t i = 2; i != Ps.size(); ++i)
{
Ps[i] = Ps[i-1];
Ps[i].add(point, ws);
}
PointGFp H(curve); // create as zero
size_t bits_left = scalar_bits;
while(bits_left >= window_size)
{
for(size_t i = 0; i != window_size; ++i)
H.mult2(ws);
const u32bit nibble = scalar.get_substring(bits_left - window_size,
window_size);
H.add(Ps[nibble], ws);
bits_left -= window_size;
}
while(bits_left)
{
H.mult2(ws);
if(scalar.get_bit(bits_left-1))
H.add(point, ws);
--bits_left;
}
if(scalar.is_negative())
H.negate();
return H;
#endif
}
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;
}