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crypto: Specialize the Fq2 multiplication by ksi #1648

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35 changes: 20 additions & 15 deletions lib/evmone_precompiles/pairing/bn254/fields.hpp
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Original file line number Diff line number Diff line change
Expand Up @@ -37,6 +37,18 @@ struct Fq12Config
};
using Fq12 = ecc::ExtFieldElem<Fq12Config>;

/// Multiplies an Fq^2 field element by ksi, the non-residue defining the Fq^6 extension.
constexpr Fq2 mul_by_ksi(const Fq2& a) noexcept
{
// ksi = 9 + u, so (a0 + a1·u)·ksi = (9a0 − a1) + (a0 + 9a1)·u.
// TODO: The multiplications by 9 can be done with additions only, but that was not faster.
static_assert(Fq6Config::ksi == Fq2{9, 1});
const auto& nine = Fq6Config::ksi.coeffs[0];

const auto& [a0, a1] = a.coeffs;
return Fq2({a0 * nine - a1, a1 * nine + a0});
}

/// Multiplies two Fq^2 field elements
constexpr Fq2 multiply(const Fq2& a, const Fq2& b) noexcept
{
Expand All @@ -61,14 +73,12 @@ constexpr Fq6 multiply(const Fq6& a, const Fq6& b) noexcept
const auto& [a0, a1, a2] = a.coeffs;
const auto& [b0, b1, b2] = b.coeffs;

const Fq2& ksi = Fq6Config::ksi;

const auto t0 = a0 * b0;
const auto t1 = a1 * b1;
const auto t2 = a2 * b2;

const auto c0 = ((a1 + a2) * (b1 + b2) - t1 - t2) * ksi + t0;
const auto c1 = (a0 + a1) * (b0 + b1) - t0 - t1 + ksi * t2;
const auto c0 = mul_by_ksi((a1 + a2) * (b1 + b2) - t1 - t2) + t0;
const auto c1 = (a0 + a1) * (b0 + b1) - t0 - t1 + mul_by_ksi(t2);
const auto c2 = (a0 + a2) * (b0 + b2) - t0 - t2 + t1;

return Fq6({c0, c1, c2});
Expand All @@ -83,9 +93,8 @@ constexpr Fq12 multiply(const Fq12& a, const Fq12& b) noexcept
const auto t0 = a0 * b0;
const auto t1 = a1 * b1;

const Fq2& ksi = Fq6Config::ksi;

const auto c0 = t0 + Fq6({ksi * t1.coeffs[2], t1.coeffs[0], t1.coeffs[1]}); // gamma is sparse.
const auto c0 =
t0 + Fq6({mul_by_ksi(t1.coeffs[2]), t1.coeffs[0], t1.coeffs[1]}); // gamma is sparse.
const auto c1 = (a0 + a1) * (b0 + b1) - t0 - t1;

return Fq12({c0, c1});
Expand Down Expand Up @@ -113,8 +122,6 @@ inline Fq6 inverse(const Fq6& f) noexcept
{
const auto& [a0, a1, a2] = f.coeffs;

const Fq2& ksi = Fq6Config::ksi;

const auto t0 = a0 * a0;
const auto t1 = a1 * a1;
const auto t2 = a2 * a2;
Expand All @@ -123,11 +130,11 @@ inline Fq6 inverse(const Fq6& f) noexcept
const auto t4 = a0 * a2;
const auto t5 = a2 * a1;

const auto c0 = t0 - ksi * t5;
const auto c1 = ksi * t2 - t3;
const auto c0 = t0 - mul_by_ksi(t5);
const auto c1 = mul_by_ksi(t2) - t3;
const auto c2 = t1 - t4;

const auto t = a0 * c0 + (a2 * c1 + a1 * c2) * ksi;
const auto t = a0 * c0 + mul_by_ksi(a2 * c1 + a1 * c2);
const auto t6 = t.inv();

return Fq6({c0 * t6, c1 * t6, c2 * t6});
Expand All @@ -141,9 +148,7 @@ inline Fq12 inverse(const Fq12& f) noexcept
auto t0 = a0 * a0;
auto t1 = a1 * a1;

const Fq2& ksi = Fq6Config::ksi;

t0 = t0 - Fq6({ksi * t1.coeffs[2], t1.coeffs[0], t1.coeffs[1]}); // gamma is sparse.
t0 = t0 - Fq6({mul_by_ksi(t1.coeffs[2]), t1.coeffs[0], t1.coeffs[1]}); // gamma is sparse.
t1 = t0.inv();

const auto c0 = a0 * t1;
Expand Down
5 changes: 2 additions & 3 deletions lib/evmone_precompiles/pairing/bn254/pairing.cpp
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Expand Up @@ -16,13 +16,12 @@ constexpr void multiply_by_lin_func_value(
Fq12& fr, const std::array<Fq2, 3>& t, const Fq& x, const Fq& y) noexcept
{
const Fq12 f = fr;
const auto& ksi = Fq6Config::ksi;

const auto t0y = t[0] * y;
const auto t1x = t[1] * x;
const auto t2ksi = t[2] * ksi;
const auto t2ksi = mul_by_ksi(t[2]);

fr.coeffs[0].coeffs[0] = f.coeffs[0].coeffs[0] * t0y + f.coeffs[1].coeffs[2] * t1x * ksi +
fr.coeffs[0].coeffs[0] = f.coeffs[0].coeffs[0] * t0y + mul_by_ksi(f.coeffs[1].coeffs[2] * t1x) +
f.coeffs[1].coeffs[1] * t2ksi;
fr.coeffs[0].coeffs[1] =
f.coeffs[0].coeffs[1] * t0y + f.coeffs[1].coeffs[0] * t1x + f.coeffs[1].coeffs[2] * t2ksi;
Expand Down
10 changes: 4 additions & 6 deletions lib/evmone_precompiles/pairing/bn254/utils.hpp
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Expand Up @@ -359,16 +359,14 @@ constexpr void lin_func(
/// Computes f^2 for f in Fq12. For more ref https://eprint.iacr.org/2010/354.pdf Algorithm 22
[[nodiscard]] constexpr Fq12 square(const Fq12& f) noexcept
{
const Fq2& ksi = Fq6Config::ksi;

const auto& a0 = f.coeffs[0];
const auto& a1 = f.coeffs[1];
auto c0 = a0 - a1;
auto c3 = a0 - Fq6({ksi * a1.coeffs[2], a1.coeffs[0], a1.coeffs[1]});
auto c3 = a0 - Fq6({mul_by_ksi(a1.coeffs[2]), a1.coeffs[0], a1.coeffs[1]});
auto c2 = a0 * a1;
c0 = c0 * c3 + c2;
const auto c1 = c2 + c2;
c2 = Fq6({ksi * c2.coeffs[2], c2.coeffs[0], c2.coeffs[1]});
c2 = Fq6({mul_by_ksi(c2.coeffs[2]), c2.coeffs[0], c2.coeffs[1]});
c0 = c0 + c2;

return Fq12({c0, c1});
Expand All @@ -384,7 +382,7 @@ constexpr std::pair<Fq2, Fq2> fq4_square(const std::pair<Fq2, Fq2>& a) noexcept
const auto t0 = a0 * a0;
const auto t1 = a1 * a1;

const auto c0 = t1 * Fq6Config::ksi + t0;
const auto c0 = mul_by_ksi(t1) + t0;
auto c1 = a0 + a1;
c1 = c1 * c1 - t0 - t1;

Expand All @@ -410,7 +408,7 @@ constexpr Fq12 cyclotomic_square(const Fq12& c) noexcept
const auto [t01, t12] = fq4_square({h0, g2}); // Typo in paper t01 <-> t12
const auto [t02, aux] = fq4_square({g1, h2});

const auto t10 = aux * Fq6Config::ksi;
const auto t10 = mul_by_ksi(aux);

const auto c00 = (t00 + t00 + t00) - (g0 + g0);
const auto c01 = (t01 + t01 + t01) - (g1 + g1);
Expand Down