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/*
* Point arithmetic on elliptic curves over GF(p)
*
* (C) 2007 Martin Doering, Christoph Ludwig, Falko Strenzke
* 2008-2011,2012,2014,2015,2018 Jack Lloyd
*
* Botan is released under the Simplified BSD License (see license.txt)
*/
#include <botan/point_gfp.h>
#include <botan/numthry.h>
#include <botan/rng.h>
#include <botan/internal/rounding.h>
#include <botan/internal/ct_utils.h>
namespace Botan {
PointGFp::PointGFp(const CurveGFp& curve) :
m_curve(curve),
m_coord_x(0),
m_coord_y(curve.get_1_rep()),
m_coord_z(0)
{
// Assumes Montgomery rep of zero is zero
}
PointGFp::PointGFp(const CurveGFp& curve, const BigInt& x, const BigInt& y) :
m_curve(curve),
m_coord_x(x),
m_coord_y(y),
m_coord_z(m_curve.get_1_rep())
{
if(x < 0 || x >= curve.get_p())
throw Invalid_Argument("Invalid PointGFp affine x");
if(y < 0 || y >= curve.get_p())
throw Invalid_Argument("Invalid PointGFp affine y");
secure_vector<word> monty_ws(m_curve.get_ws_size());
m_curve.to_rep(m_coord_x, monty_ws);
m_curve.to_rep(m_coord_y, monty_ws);
}
void PointGFp::randomize_repr(RandomNumberGenerator& rng)
{
secure_vector<word> ws(m_curve.get_ws_size());
randomize_repr(rng, ws);
}
void PointGFp::randomize_repr(RandomNumberGenerator& rng, secure_vector<word>& ws)
{
const BigInt mask = BigInt::random_integer(rng, 2, m_curve.get_p());
/*
* No reason to convert this to Montgomery representation first,
* just pretend the random mask was chosen as Redc(mask) and the
* random mask we generated above is in the Montgomery
* representation.
* //m_curve.to_rep(mask, ws);
*/
const BigInt mask2 = m_curve.sqr_to_tmp(mask, ws);
const BigInt mask3 = m_curve.mul_to_tmp(mask2, mask, ws);
m_coord_x = m_curve.mul_to_tmp(m_coord_x, mask2, ws);
m_coord_y = m_curve.mul_to_tmp(m_coord_y, mask3, ws);
m_coord_z = m_curve.mul_to_tmp(m_coord_z, mask, ws);
}
namespace {
inline void resize_ws(std::vector<BigInt>& ws_bn, size_t cap_size)
{
BOTAN_ASSERT(ws_bn.size() >= PointGFp::WORKSPACE_SIZE,
"Expected size for PointGFp workspace");
for(size_t i = 0; i != ws_bn.size(); ++i)
if(ws_bn[i].size() < cap_size)
ws_bn[i].get_word_vector().resize(cap_size);
}
inline word all_zeros(const word x[], size_t len)
{
word z = 0;
for(size_t i = 0; i != len; ++i)
z |= x[i];
return CT::Mask<word>::is_zero(z).value();
}
}
void PointGFp::add_affine(const word x_words[], size_t x_size,
const word y_words[], size_t y_size,
std::vector<BigInt>& ws_bn)
{
if(all_zeros(x_words, x_size) & all_zeros(y_words, y_size))
{
return;
}
if(is_zero())
{
m_coord_x.set_words(x_words, x_size);
m_coord_y.set_words(y_words, y_size);
m_coord_z = m_curve.get_1_rep();
return;
}
resize_ws(ws_bn, m_curve.get_ws_size());
secure_vector<word>& ws = ws_bn[0].get_word_vector();
secure_vector<word>& sub_ws = ws_bn[1].get_word_vector();
BigInt& T0 = ws_bn[2];
BigInt& T1 = ws_bn[3];
BigInt& T2 = ws_bn[4];
BigInt& T3 = ws_bn[5];
BigInt& T4 = ws_bn[6];
/*
simplified with Z2 = 1
*/
const BigInt& p = m_curve.get_p();
m_curve.sqr(T3, m_coord_z, ws); // z1^2
m_curve.mul(T4, x_words, x_size, T3, ws); // x2*z1^2
m_curve.mul(T2, m_coord_z, T3, ws); // z1^3
m_curve.mul(T0, y_words, y_size, T2, ws); // y2*z1^3
T4.mod_sub(m_coord_x, p, sub_ws); // x2*z1^2 - x1*z2^2
T0.mod_sub(m_coord_y, p, sub_ws);
if(T4.is_zero())
{
if(T0.is_zero())
{
mult2(ws_bn);
return;
}
// setting to zero:
m_coord_x.clear();
m_coord_y = m_curve.get_1_rep();
m_coord_z.clear();
return;
}
m_curve.sqr(T2, T4, ws);
m_curve.mul(T3, m_coord_x, T2, ws);
m_curve.mul(T1, T2, T4, ws);
m_curve.sqr(m_coord_x, T0, ws);
m_coord_x.mod_sub(T1, p, sub_ws);
m_coord_x.mod_sub(T3, p, sub_ws);
m_coord_x.mod_sub(T3, p, sub_ws);
T3.mod_sub(m_coord_x, p, sub_ws);
m_curve.mul(T2, T0, T3, ws);
m_curve.mul(T0, m_coord_y, T1, ws);
T2.mod_sub(T0, p, sub_ws);
m_coord_y.swap(T2);
m_curve.mul(T0, m_coord_z, T4, ws);
m_coord_z.swap(T0);
}
void PointGFp::add(const word x_words[], size_t x_size,
const word y_words[], size_t y_size,
const word z_words[], size_t z_size,
std::vector<BigInt>& ws_bn)
{
if(all_zeros(x_words, x_size) & all_zeros(z_words, z_size))
return;
if(is_zero())
{
m_coord_x.set_words(x_words, x_size);
m_coord_y.set_words(y_words, y_size);
m_coord_z.set_words(z_words, z_size);
return;
}
resize_ws(ws_bn, m_curve.get_ws_size());
secure_vector<word>& ws = ws_bn[0].get_word_vector();
secure_vector<word>& sub_ws = ws_bn[1].get_word_vector();
BigInt& T0 = ws_bn[2];
BigInt& T1 = ws_bn[3];
BigInt& T2 = ws_bn[4];
BigInt& T3 = ws_bn[5];
BigInt& T4 = ws_bn[6];
BigInt& T5 = ws_bn[7];
/*
*/
const BigInt& p = m_curve.get_p();
m_curve.sqr(T0, z_words, z_size, ws); // z2^2
m_curve.mul(T1, m_coord_x, T0, ws); // x1*z2^2
m_curve.mul(T3, z_words, z_size, T0, ws); // z2^3
m_curve.mul(T2, m_coord_y, T3, ws); // y1*z2^3
m_curve.sqr(T3, m_coord_z, ws); // z1^2
m_curve.mul(T4, x_words, x_size, T3, ws); // x2*z1^2
m_curve.mul(T5, m_coord_z, T3, ws); // z1^3
m_curve.mul(T0, y_words, y_size, T5, ws); // y2*z1^3
T4.mod_sub(T1, p, sub_ws); // x2*z1^2 - x1*z2^2
T0.mod_sub(T2, p, sub_ws);
if(T4.is_zero())
{
if(T0.is_zero())
{
mult2(ws_bn);
return;
}
// setting to zero:
m_coord_x.clear();
m_coord_y = m_curve.get_1_rep();
m_coord_z.clear();
return;
}
m_curve.sqr(T5, T4, ws);
m_curve.mul(T3, T1, T5, ws);
m_curve.mul(T1, T5, T4, ws);
m_curve.sqr(m_coord_x, T0, ws);
m_coord_x.mod_sub(T1, p, sub_ws);
m_coord_x.mod_sub(T3, p, sub_ws);
m_coord_x.mod_sub(T3, p, sub_ws);
T3.mod_sub(m_coord_x, p, sub_ws);
m_curve.mul(m_coord_y, T0, T3, ws);
m_curve.mul(T3, T2, T1, ws);
m_coord_y.mod_sub(T3, p, sub_ws);
m_curve.mul(T3, z_words, z_size, m_coord_z, ws);
m_curve.mul(m_coord_z, T3, T4, ws);
}
void PointGFp::mult2i(size_t iterations, std::vector<BigInt>& ws_bn)
{
if(iterations == 0)
return;
if(m_coord_y.is_zero())
{
*this = PointGFp(m_curve); // setting myself to zero
return;
}
/*
TODO we can save 2 squarings per iteration by computing
a*Z^4 using values cached from previous iteration
*/
for(size_t i = 0; i != iterations; ++i)
mult2(ws_bn);
}
// *this *= 2
void PointGFp::mult2(std::vector<BigInt>& ws_bn)
{
if(is_zero())
return;
if(m_coord_y.is_zero())
{
*this = PointGFp(m_curve); // setting myself to zero
return;
}
resize_ws(ws_bn, m_curve.get_ws_size());
secure_vector<word>& ws = ws_bn[0].get_word_vector();
secure_vector<word>& sub_ws = ws_bn[1].get_word_vector();
BigInt& T0 = ws_bn[2];
BigInt& T1 = ws_bn[3];
BigInt& T2 = ws_bn[4];
BigInt& T3 = ws_bn[5];
BigInt& T4 = ws_bn[6];
/*
*/
const BigInt& p = m_curve.get_p();
m_curve.sqr(T0, m_coord_y, ws);
m_curve.mul(T1, m_coord_x, T0, ws);
T1.mod_mul(4, p, sub_ws);
if(m_curve.a_is_zero())
{
// if a == 0 then 3*x^2 + a*z^4 is just 3*x^2
m_curve.sqr(T4, m_coord_x, ws); // x^2
T4.mod_mul(3, p, sub_ws); // 3*x^2
}
else if(m_curve.a_is_minus_3())
{
/*
if a == -3 then
3*x^2 + a*z^4 == 3*x^2 - 3*z^4 == 3*(x^2-z^4) == 3*(x-z^2)*(x+z^2)
*/
m_curve.sqr(T3, m_coord_z, ws); // z^2
// (x-z^2)
T2 = m_coord_x;
T2.mod_sub(T3, p, sub_ws);
// (x+z^2)
T3.mod_add(m_coord_x, p, sub_ws);
m_curve.mul(T4, T2, T3, ws); // (x-z^2)*(x+z^2)
T4.mod_mul(3, p, sub_ws); // 3*(x-z^2)*(x+z^2)
}
else
{
m_curve.sqr(T3, m_coord_z, ws); // z^2
m_curve.sqr(T4, T3, ws); // z^4
m_curve.mul(T3, m_curve.get_a_rep(), T4, ws); // a*z^4
m_curve.sqr(T4, m_coord_x, ws); // x^2
T4.mod_mul(3, p, sub_ws);
T4.mod_add(T3, p, sub_ws); // 3*x^2 + a*z^4
}
m_curve.sqr(T2, T4, ws);
T2.mod_sub(T1, p, sub_ws);
T2.mod_sub(T1, p, sub_ws);
m_curve.sqr(T3, T0, ws);
T3.mod_mul(8, p, sub_ws);
T1.mod_sub(T2, p, sub_ws);
m_curve.mul(T0, T4, T1, ws);
T0.mod_sub(T3, p, sub_ws);
m_coord_x.swap(T2);
m_curve.mul(T2, m_coord_y, m_coord_z, ws);
T2.mod_mul(2, p, sub_ws);
m_coord_y.swap(T0);
m_coord_z.swap(T2);
}
// arithmetic operators
PointGFp& PointGFp::operator+=(const PointGFp& rhs)
{
std::vector<BigInt> ws(PointGFp::WORKSPACE_SIZE);
add(rhs, ws);
return *this;
}
PointGFp& PointGFp::operator-=(const PointGFp& rhs)
{
PointGFp minus_rhs = PointGFp(rhs).negate();
if(is_zero())
*this = minus_rhs;
else
*this += minus_rhs;
return *this;
}
PointGFp& PointGFp::operator*=(const BigInt& scalar)
{
*this = scalar * *this;
return *this;
}
PointGFp operator*(const BigInt& scalar, const PointGFp& point)
{
BOTAN_DEBUG_ASSERT(point.on_the_curve());
const size_t scalar_bits = scalar.bits();
std::vector<BigInt> ws(PointGFp::WORKSPACE_SIZE);
PointGFp R[2] = { point.zero(), 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_DEBUG_ASSERT(R[0].on_the_curve());
return R[0];
}
//static
void PointGFp::force_all_affine(std::vector<PointGFp>& points,
secure_vector<word>& ws)
{
if(points.size() <= 1)
{
for(size_t i = 0; i != points.size(); ++i)
points[i].force_affine();
return;
}
for(size_t i = 0; i != points.size(); ++i)
{
if(points[i].is_zero())
throw Invalid_State("Cannot convert zero ECC point to affine");
}
/*
For >= 2 points use Montgomery's trick
See Algorithm 2.26 in "Guide to Elliptic Curve Cryptography"
(Hankerson, Menezes, Vanstone)
TODO is it really necessary to save all k points in c?
*/
const CurveGFp& curve = points[0].m_curve;
const BigInt& rep_1 = curve.get_1_rep();
if(ws.size() < curve.get_ws_size())
ws.resize(curve.get_ws_size());
std::vector<BigInt> c(points.size());
c[0] = points[0].m_coord_z;
for(size_t i = 1; i != points.size(); ++i)
{
curve.mul(c[i], c[i-1], points[i].m_coord_z, ws);
}
BigInt s_inv = curve.invert_element(c[c.size()-1], ws);
BigInt z_inv, z2_inv, z3_inv;
for(size_t i = points.size() - 1; i != 0; i--)
{
PointGFp& point = points[i];
curve.mul(z_inv, s_inv, c[i-1], ws);
s_inv = curve.mul_to_tmp(s_inv, point.m_coord_z, ws);
curve.sqr(z2_inv, z_inv, ws);
curve.mul(z3_inv, z2_inv, z_inv, ws);
point.m_coord_x = curve.mul_to_tmp(point.m_coord_x, z2_inv, ws);
point.m_coord_y = curve.mul_to_tmp(point.m_coord_y, z3_inv, ws);
point.m_coord_z = rep_1;
}
curve.sqr(z2_inv, s_inv, ws);
curve.mul(z3_inv, z2_inv, s_inv, ws);
points[0].m_coord_x = curve.mul_to_tmp(points[0].m_coord_x, z2_inv, ws);
points[0].m_coord_y = curve.mul_to_tmp(points[0].m_coord_y, z3_inv, ws);
points[0].m_coord_z = rep_1;
}
void PointGFp::force_affine()
{
if(is_zero())
throw Invalid_State("Cannot convert zero ECC point to affine");
secure_vector<word> ws;
const BigInt z_inv = m_curve.invert_element(m_coord_z, ws);
const BigInt z2_inv = m_curve.sqr_to_tmp(z_inv, ws);
const BigInt z3_inv = m_curve.mul_to_tmp(z_inv, z2_inv, ws);
m_coord_x = m_curve.mul_to_tmp(m_coord_x, z2_inv, ws);
m_coord_y = m_curve.mul_to_tmp(m_coord_y, z3_inv, ws);
m_coord_z = m_curve.get_1_rep();
}
bool PointGFp::is_affine() const
{
return m_curve.is_one(m_coord_z);
}
BigInt PointGFp::get_affine_x() const
{
if(is_zero())
throw Illegal_Transformation("Cannot convert zero point to affine");
secure_vector<word> monty_ws;
if(is_affine())
return m_curve.from_rep_to_tmp(m_coord_x, monty_ws);
BigInt z2 = m_curve.sqr_to_tmp(m_coord_z, monty_ws);
z2 = m_curve.invert_element(z2, monty_ws);
BigInt r;
m_curve.mul(r, m_coord_x, z2, monty_ws);
m_curve.from_rep(r, monty_ws);
return r;
}
BigInt PointGFp::get_affine_y() const
{
if(is_zero())
throw Illegal_Transformation("Cannot convert zero point to affine");
secure_vector<word> monty_ws;
if(is_affine())
return m_curve.from_rep_to_tmp(m_coord_y, monty_ws);
const BigInt z2 = m_curve.sqr_to_tmp(m_coord_z, monty_ws);
const BigInt z3 = m_curve.mul_to_tmp(m_coord_z, z2, monty_ws);
const BigInt z3_inv = m_curve.invert_element(z3, monty_ws);
BigInt r;
m_curve.mul(r, m_coord_y, z3_inv, monty_ws);
m_curve.from_rep(r, monty_ws);
return r;
}
bool PointGFp::on_the_curve() const
{
/*
Is the point still on the curve?? (If everything is correct, the
point is always on its curve; then the function will return true.
If somehow the state is corrupted, which suggests a fault attack
(or internal computational error), then return false.
*/
if(is_zero())
return true;
secure_vector<word> monty_ws;
const BigInt y2 = m_curve.from_rep_to_tmp(m_curve.sqr_to_tmp(m_coord_y, monty_ws), monty_ws);
const BigInt x3 = m_curve.mul_to_tmp(m_coord_x, m_curve.sqr_to_tmp(m_coord_x, monty_ws), monty_ws);
const BigInt ax = m_curve.mul_to_tmp(m_coord_x, m_curve.get_a_rep(), monty_ws);
const BigInt z2 = m_curve.sqr_to_tmp(m_coord_z, monty_ws);
if(m_coord_z == z2) // Is z equal to 1 (in Montgomery form)?
{
if(y2 != m_curve.from_rep_to_tmp(x3 + ax + m_curve.get_b_rep(), monty_ws))
return false;
}
const BigInt z3 = m_curve.mul_to_tmp(m_coord_z, z2, monty_ws);
const BigInt ax_z4 = m_curve.mul_to_tmp(ax, m_curve.sqr_to_tmp(z2, monty_ws), monty_ws);
const BigInt b_z6 = m_curve.mul_to_tmp(m_curve.get_b_rep(), m_curve.sqr_to_tmp(z3, monty_ws), monty_ws);
if(y2 != m_curve.from_rep_to_tmp(x3 + ax_z4 + b_z6, monty_ws))
return false;
return true;
}
// swaps the states of *this and other, does not throw!
void PointGFp::swap(PointGFp& other)
{
m_curve.swap(other.m_curve);
m_coord_x.swap(other.m_coord_x);
m_coord_y.swap(other.m_coord_y);
m_coord_z.swap(other.m_coord_z);
}
bool PointGFp::operator==(const PointGFp& other) const
{
if(m_curve != other.m_curve)
return false;
// If this is zero, only equal if other is also zero
if(is_zero())
return other.is_zero();
return (get_affine_x() == other.get_affine_x() &&
get_affine_y() == other.get_affine_y());
}
// encoding and decoding
std::vector<uint8_t> PointGFp::encode(PointGFp::Compression_Type format) const
{
if(is_zero())
return std::vector<uint8_t>(1); // single 0 byte
const size_t p_bytes = m_curve.get_p().bytes();
const BigInt x = get_affine_x();
const BigInt y = get_affine_y();
std::vector<uint8_t> result;
if(format == PointGFp::UNCOMPRESSED)
{
result.resize(1 + 2*p_bytes);
result[0] = 0x04;
BigInt::encode_1363(&result[1], p_bytes, x);
BigInt::encode_1363(&result[1+p_bytes], p_bytes, y);
}
else if(format == PointGFp::COMPRESSED)
{
result.resize(1 + p_bytes);
result[0] = 0x02 | static_cast<uint8_t>(y.get_bit(0));
BigInt::encode_1363(&result[1], p_bytes, x);
}
else if(format == PointGFp::HYBRID)
{
result.resize(1 + 2*p_bytes);
result[0] = 0x06 | static_cast<uint8_t>(y.get_bit(0));
BigInt::encode_1363(&result[1], p_bytes, x);
BigInt::encode_1363(&result[1+p_bytes], p_bytes, y);
}
else
throw Invalid_Argument("EC2OSP illegal point encoding");
return result;
}
namespace {
BigInt decompress_point(bool yMod2,
const BigInt& x,
const BigInt& curve_p,
const BigInt& curve_a,
const BigInt& curve_b)
{
BigInt xpow3 = x * x * x;
BigInt g = curve_a * x;
g += xpow3;
g += curve_b;
g = g % curve_p;
BigInt z = ressol(g, curve_p);
if(z < 0)
throw Illegal_Point("error during EC point decompression");
if(z.get_bit(0) != yMod2)
z = curve_p - z;
return z;
}
}
PointGFp OS2ECP(const uint8_t data[], size_t data_len,
const CurveGFp& curve)
{
// Should we really be doing this?
if(data_len <= 1)
return PointGFp(curve); // return zero
std::pair<BigInt, BigInt> xy = OS2ECP(data, data_len, curve.get_p(), curve.get_a(), curve.get_b());
PointGFp point(curve, xy.first, xy.second);
if(!point.on_the_curve())
throw Illegal_Point("OS2ECP: Decoded point was not on the curve");
return point;
}
std::pair<BigInt, BigInt> OS2ECP(const uint8_t data[], size_t data_len,
const BigInt& curve_p,
const BigInt& curve_a,
const BigInt& curve_b)
{
if(data_len <= 1)
throw Decoding_Error("OS2ECP invalid point");
const uint8_t pc = data[0];
BigInt x, y;
if(pc == 2 || pc == 3)
{
//compressed form
x = BigInt::decode(&data[1], data_len - 1);
const bool y_mod_2 = ((pc & 0x01) == 1);
y = decompress_point(y_mod_2, x, curve_p, curve_a, curve_b);
}
else if(pc == 4)
{
const size_t l = (data_len - 1) / 2;
// uncompressed form
x = BigInt::decode(&data[1], l);
y = BigInt::decode(&data[l+1], l);
}
else if(pc == 6 || pc == 7)
{
const size_t l = (data_len - 1) / 2;
// hybrid form
x = BigInt::decode(&data[1], l);
y = BigInt::decode(&data[l+1], l);
const bool y_mod_2 = ((pc & 0x01) == 1);
if(decompress_point(y_mod_2, x, curve_p, curve_a, curve_b) != y)
throw Illegal_Point("OS2ECP: Decoding error in hybrid format");
}
else
throw Invalid_Argument("OS2ECP: Unknown format type " + std::to_string(pc));
return std::make_pair(x, y);
}
}