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/*
* Elliptic curves over GF(p) Montgomery Representation
* (C) 2014,2015,2018 Jack Lloyd
* 2016 Matthias Gierlings
*
* Botan is released under the Simplified BSD License (see license.txt)
*/
#include <botan/curve_gfp.h>
#include <botan/curve_nistp.h>
#include <botan/numthry.h>
#include <botan/reducer.h>
#include <botan/internal/mp_core.h>
#include <botan/internal/mp_asmi.h>
namespace Botan {
namespace {
class CurveGFp_Montgomery final : public CurveGFp_Repr
{
public:
CurveGFp_Montgomery(const BigInt& p, const BigInt& a, const BigInt& b) :
m_p(p), m_a(a), m_b(b),
m_p_words(m_p.sig_words()),
m_p_dash(monty_inverse(m_p.word_at(0)))
{
Modular_Reducer mod_p(m_p);
m_r.set_bit(m_p_words * BOTAN_MP_WORD_BITS);
m_r = mod_p.reduce(m_r);
m_r2 = mod_p.square(m_r);
m_r3 = mod_p.multiply(m_r, m_r2);
m_a_r = mod_p.multiply(m_r, m_a);
m_b_r = mod_p.multiply(m_r, m_b);
m_a_is_zero = m_a.is_zero();
m_a_is_minus_3 = (m_a + 3 == m_p);
}
bool a_is_zero() const override { return m_a_is_zero; }
bool a_is_minus_3() const override { return m_a_is_minus_3; }
const BigInt& get_a() const override { return m_a; }
const BigInt& get_b() const override { return m_b; }
const BigInt& get_p() const override { return m_p; }
const BigInt& get_a_rep() const override { return m_a_r; }
const BigInt& get_b_rep() const override { return m_b_r; }
const BigInt& get_1_rep() const override { return m_r; }
bool is_one(const BigInt& x) const override { return x == m_r; }
size_t get_p_words() const override { return m_p_words; }
size_t get_ws_size() const override { return 2*m_p_words + 4; }
BigInt invert_element(const BigInt& x, secure_vector<word>& ws) const override;
void to_curve_rep(BigInt& x, secure_vector<word>& ws) const override;
void from_curve_rep(BigInt& x, secure_vector<word>& ws) const override;
void curve_mul_words(BigInt& z,
const word x_words[],
const size_t x_size,
const BigInt& y,
secure_vector<word>& ws) const override;
void curve_sqr_words(BigInt& z,
const word x_words[],
size_t x_size,
secure_vector<word>& ws) const override;
private:
BigInt m_p;
BigInt m_a, m_b;
BigInt m_a_r, m_b_r;
size_t m_p_words; // cache of m_p.sig_words()
// Montgomery parameters
BigInt m_r, m_r2, m_r3;
word m_p_dash;
bool m_a_is_zero;
bool m_a_is_minus_3;
};
BigInt CurveGFp_Montgomery::invert_element(const BigInt& x, secure_vector<word>& ws) const
{
// Should we use Montgomery inverse instead?
const BigInt inv = inverse_mod(x, m_p);
BigInt res;
curve_mul(res, inv, m_r3, ws);
return res;
}
void CurveGFp_Montgomery::to_curve_rep(BigInt& x, secure_vector<word>& ws) const
{
const BigInt tx = x;
curve_mul(x, tx, m_r2, ws);
}
void CurveGFp_Montgomery::from_curve_rep(BigInt& z, secure_vector<word>& ws) const
{
if(ws.size() < get_ws_size())
ws.resize(get_ws_size());
const size_t output_size = 2*m_p_words + 2;
if(z.size() < output_size)
z.grow_to(output_size);
bigint_monty_redc(z.mutable_data(),
m_p.data(), m_p_words, m_p_dash,
ws.data(), ws.size());
}
void CurveGFp_Montgomery::curve_mul_words(BigInt& z,
const word x_w[],
size_t x_size,
const BigInt& y,
secure_vector<word>& ws) const
{
BOTAN_DEBUG_ASSERT(y.sig_words() <= m_p_words);
if(ws.size() < get_ws_size())
ws.resize(get_ws_size());
const size_t output_size = 2*m_p_words + 2;
if(z.size() < output_size)
z.grow_to(output_size);
bigint_mul(z.mutable_data(), z.size(),
x_w, x_size, std::min(m_p_words, x_size),
y.data(), y.size(), std::min(m_p_words, y.size()),
ws.data(), ws.size());
bigint_monty_redc(z.mutable_data(),
m_p.data(), m_p_words, m_p_dash,
ws.data(), ws.size());
}
void CurveGFp_Montgomery::curve_sqr_words(BigInt& z,
const word x[],
size_t x_size,
secure_vector<word>& ws) const
{
if(ws.size() < get_ws_size())
ws.resize(get_ws_size());
const size_t output_size = 2*m_p_words + 2;
if(z.size() < output_size)
z.grow_to(output_size);
bigint_sqr(z.mutable_data(), z.size(),
x, x_size, std::min(m_p_words, x_size),
ws.data(), ws.size());
bigint_monty_redc(z.mutable_data(),
m_p.data(), m_p_words, m_p_dash,
ws.data(), ws.size());
}
class CurveGFp_NIST : public CurveGFp_Repr
{
public:
CurveGFp_NIST(size_t p_bits, const BigInt& a, const BigInt& b) :
m_1(1), m_a(a), m_b(b), m_p_words((p_bits + BOTAN_MP_WORD_BITS - 1) / BOTAN_MP_WORD_BITS)
{
// All Solinas prime curves are assumed a == -3
}
bool a_is_zero() const override { return false; }
bool a_is_minus_3() const override { return true; }
const BigInt& get_a() const override { return m_a; }
const BigInt& get_b() const override { return m_b; }
const BigInt& get_1_rep() const override { return m_1; }
size_t get_p_words() const override { return m_p_words; }
size_t get_ws_size() const override { return 2*m_p_words + 4; }
const BigInt& get_a_rep() const override { return m_a; }
const BigInt& get_b_rep() const override { return m_b; }
bool is_one(const BigInt& x) const override { return x == 1; }
void to_curve_rep(BigInt& x, secure_vector<word>& ws) const override
{ redc_mod_p(x, ws); }
void from_curve_rep(BigInt& x, secure_vector<word>& ws) const override
{ redc_mod_p(x, ws); }
virtual void redc_mod_p(BigInt& z, secure_vector<word>& ws) const = 0;
BigInt invert_element(const BigInt& x, secure_vector<word>& ws) const override;
void curve_mul_words(BigInt& z,
const word x_words[],
const size_t x_size,
const BigInt& y,
secure_vector<word>& ws) const override;
void curve_mul_tmp(BigInt& x, const BigInt& y, BigInt& tmp, secure_vector<word>& ws) const
{
curve_mul(tmp, x, y, ws);
x.swap(tmp);
}
void curve_sqr_tmp(BigInt& x, BigInt& tmp, secure_vector<word>& ws) const
{
curve_sqr(tmp, x, ws);
x.swap(tmp);
}
void curve_sqr_words(BigInt& z,
const word x_words[],
size_t x_size,
secure_vector<word>& ws) const override;
private:
// Curve parameters
BigInt m_1;
BigInt m_a, m_b;
size_t m_p_words; // cache of m_p.sig_words()
};
BigInt CurveGFp_NIST::invert_element(const BigInt& x, secure_vector<word>& ws) const
{
BOTAN_UNUSED(ws);
return inverse_mod(x, get_p());
}
void CurveGFp_NIST::curve_mul_words(BigInt& z,
const word x_w[],
size_t x_size,
const BigInt& y,
secure_vector<word>& ws) const
{
BOTAN_DEBUG_ASSERT(y.sig_words() <= m_p_words);
if(ws.size() < get_ws_size())
ws.resize(get_ws_size());
const size_t output_size = 2*m_p_words + 2;
if(z.size() < output_size)
z.grow_to(output_size);
bigint_mul(z.mutable_data(), z.size(),
x_w, x_size, std::min(m_p_words, x_size),
y.data(), y.size(), std::min(m_p_words, y.size()),
ws.data(), ws.size());
this->redc_mod_p(z, ws);
}
void CurveGFp_NIST::curve_sqr_words(BigInt& z, const word x[], size_t x_size,
secure_vector<word>& ws) const
{
if(ws.size() < get_ws_size())
ws.resize(get_ws_size());
const size_t output_size = 2*m_p_words + 2;
if(z.size() < output_size)
z.grow_to(output_size);
bigint_sqr(z.mutable_data(), output_size,
x, x_size, std::min(m_p_words, x_size),
ws.data(), ws.size());
this->redc_mod_p(z, ws);
}
/**
* The NIST P-192 curve
*/
class CurveGFp_P192 final : public CurveGFp_NIST
{
public:
CurveGFp_P192(const BigInt& a, const BigInt& b) : CurveGFp_NIST(192, a, b) {}
const BigInt& get_p() const override { return prime_p192(); }
private:
void redc_mod_p(BigInt& x, secure_vector<word>& ws) const override { redc_p192(x, ws); }
};
/**
* The NIST P-224 curve
*/
class CurveGFp_P224 final : public CurveGFp_NIST
{
public:
CurveGFp_P224(const BigInt& a, const BigInt& b) : CurveGFp_NIST(224, a, b) {}
const BigInt& get_p() const override { return prime_p224(); }
private:
void redc_mod_p(BigInt& x, secure_vector<word>& ws) const override { redc_p224(x, ws); }
};
/**
* The NIST P-256 curve
*/
class CurveGFp_P256 final : public CurveGFp_NIST
{
public:
CurveGFp_P256(const BigInt& a, const BigInt& b) : CurveGFp_NIST(256, a, b) {}
const BigInt& get_p() const override { return prime_p256(); }
private:
void redc_mod_p(BigInt& x, secure_vector<word>& ws) const override { redc_p256(x, ws); }
BigInt invert_element(const BigInt& x, secure_vector<word>& ws) const override;
};
BigInt CurveGFp_P256::invert_element(const BigInt& x, secure_vector<word>& ws) const
{
BigInt r, p2, p4, p8, p16, p32, tmp;
curve_sqr(r, x, ws);
curve_mul(p2, r, x, ws);
curve_sqr(r, p2, ws);
curve_sqr_tmp(r, tmp, ws);
curve_mul(p4, r, p2, ws);
curve_sqr(r, p4, ws);
for(size_t i = 0; i != 3; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul(p8, r, p4, ws);
curve_sqr(r, p8, ws);
for(size_t i = 0; i != 7; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul(p16, r, p8, ws);
curve_sqr(r, p16, ws);
for(size_t i = 0; i != 15; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul(p32, r, p16, ws);
curve_sqr(r, p32, ws);
for(size_t i = 0; i != 31; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x, tmp, ws);
for(size_t i = 0; i != 32*4; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, p32, tmp, ws);
for(size_t i = 0; i != 32; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, p32, tmp, ws);
for(size_t i = 0; i != 16; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, p16, tmp, ws);
for(size_t i = 0; i != 8; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, p8, tmp, ws);
for(size_t i = 0; i != 4; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, p4, tmp, ws);
for(size_t i = 0; i != 2; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, p2, tmp, ws);
for(size_t i = 0; i != 2; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x, tmp, ws);
return r;
}
/**
* The NIST P-384 curve
*/
class CurveGFp_P384 final : public CurveGFp_NIST
{
public:
CurveGFp_P384(const BigInt& a, const BigInt& b) : CurveGFp_NIST(384, a, b) {}
const BigInt& get_p() const override { return prime_p384(); }
private:
void redc_mod_p(BigInt& x, secure_vector<word>& ws) const override { redc_p384(x, ws); }
BigInt invert_element(const BigInt& x, secure_vector<word>& ws) const override;
};
BigInt CurveGFp_P384::invert_element(const BigInt& x, secure_vector<word>& ws) const
{
BigInt r, x2, x3, x15, x30, tmp, rl;
r = x;
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x, tmp, ws);
x2 = r;
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x, tmp, ws);
x3 = r;
for(size_t i = 0; i != 3; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x3, tmp, ws);
rl = r;
for(size_t i = 0; i != 6; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, rl, tmp, ws);
for(size_t i = 0; i != 3; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x3, tmp, ws);
x15 = r;
for(size_t i = 0; i != 15; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x15, tmp, ws);
x30 = r;
for(size_t i = 0; i != 30; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x30, tmp, ws);
rl = r;
for(size_t i = 0; i != 60; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, rl, tmp, ws);
rl = r;
for(size_t i = 0; i != 120; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, rl, tmp, ws);
for(size_t i = 0; i != 15; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x15, tmp, ws);
for(size_t i = 0; i != 31; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x30, tmp, ws);
for(size_t i = 0; i != 2; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x2, tmp, ws);
for(size_t i = 0; i != 94; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x30, tmp, ws);
for(size_t i = 0; i != 2; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x, tmp, ws);
return r;
}
/**
* The NIST P-521 curve
*/
class CurveGFp_P521 final : public CurveGFp_NIST
{
public:
CurveGFp_P521(const BigInt& a, const BigInt& b) : CurveGFp_NIST(521, a, b) {}
const BigInt& get_p() const override { return prime_p521(); }
private:
void redc_mod_p(BigInt& x, secure_vector<word>& ws) const override { redc_p521(x, ws); }
BigInt invert_element(const BigInt& x, secure_vector<word>& ws) const override;
};
BigInt CurveGFp_P521::invert_element(const BigInt& x, secure_vector<word>& ws) const
{
BigInt r;
BigInt rl;
BigInt a7;
BigInt tmp;
curve_sqr(r, x, ws);
curve_mul_tmp(r, x, tmp, ws);
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x, tmp, ws);
rl = r;
for(size_t i = 0; i != 3; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, rl, tmp, ws);
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x, tmp, ws);
a7 = r; // need this value later
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x, tmp, ws);
rl = r;
for(size_t i = 0; i != 8; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, rl, tmp, ws);
rl = r;
for(size_t i = 0; i != 16; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, rl, tmp, ws);
rl = r;
for(size_t i = 0; i != 32; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, rl, tmp, ws);
rl = r;
for(size_t i = 0; i != 64; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, rl, tmp, ws);
rl = r;
for(size_t i = 0; i != 128; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, rl, tmp, ws);
rl = r;
for(size_t i = 0; i != 256; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, rl, tmp, ws);
for(size_t i = 0; i != 7; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, a7, tmp, ws);
for(size_t i = 0; i != 2; ++i)
curve_sqr_tmp(r, tmp, ws);
curve_mul_tmp(r, x, tmp, ws);
return r;
}
}
std::shared_ptr<CurveGFp_Repr>
CurveGFp::choose_repr(const BigInt& p, const BigInt& a, const BigInt& b)
{
if(p == prime_p192())
return std::shared_ptr<CurveGFp_Repr>(new CurveGFp_P192(a, b));
if(p == prime_p224())
return std::shared_ptr<CurveGFp_Repr>(new CurveGFp_P224(a, b));
if(p == prime_p256())
return std::shared_ptr<CurveGFp_Repr>(new CurveGFp_P256(a, b));
if(p == prime_p384())
return std::shared_ptr<CurveGFp_Repr>(new CurveGFp_P384(a, b));
if(p == prime_p521())
return std::shared_ptr<CurveGFp_Repr>(new CurveGFp_P521(a, b));
return std::shared_ptr<CurveGFp_Repr>(new CurveGFp_Montgomery(p, a, b));
}
}