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Faster final exponentiation for BLS12 #211

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merged 12 commits into from
Mar 17, 2021
Merged

Faster final exponentiation for BLS12 #211

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187da46
New final exp for BLS12
Pratyush Feb 6, 2021
6327c66
Format
Pratyush Feb 6, 2021
3c76a62
Comments
Pratyush Feb 6, 2021
620bbc3
Merge branch 'master' into faster-bls12-final-exp
Pratyush Feb 10, 2021
c524e64
Update CHANGELOG.md
Pratyush Feb 10, 2021
30d5250
Operations by reference instead of by value
Pratyush Mar 8, 2021
332bf37
Merge branch 'master' into faster-bls12-final-exp
Pratyush Mar 8, 2021
5bbbe92
In place ops
Pratyush Mar 10, 2021
1ab9d1e
Fix
Pratyush Mar 11, 2021
425dad0
Fix
Pratyush Mar 11, 2021
e64b5f9
Merge branch 'master' into faster-bls12-final-exp
Pratyush Mar 11, 2021
e370624
Merge branch 'master' into faster-bls12-final-exp
Pratyush Mar 17, 2021
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73 changes: 41 additions & 32 deletions ec/src/models/bls12/mod.rs
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Original file line number Diff line number Diff line change
Expand Up @@ -122,10 +122,8 @@ impl<P: Bls12Parameters> PairingEngine for Bls12<P> {

fn final_exponentiation(f: &Self::Fqk) -> Option<Self::Fqk> {
// Computing the final exponentation following
// https://eprint.iacr.org/2016/130.pdf.
// We don't use their "faster" formula because it is difficult to make
// it work for curves with odd `P::X`.
// Hence we implement the algorithm from Table 1 below.
// https://eprint.iacr.org/2020/875
// Adapted from the implementation in https://github.com/ConsenSys/gurvy/pull/29

// f1 = r.conjugate() = f^(p^6)
let mut f1 = *f;
Expand All @@ -145,35 +143,46 @@ impl<P: Bls12Parameters> PairingEngine for Bls12<P> {
// r = f^((p^6 - 1)(p^2 + 1))
r *= &f2;

// Hard part of the final exponentation is below:
// From https://eprint.iacr.org/2016/130.pdf, Table 1
// Hard part of the final exponentation:
// t[0].CyclotomicSquare(&result)
let mut y0 = r.cyclotomic_square();
y0.conjugate();

let mut y5 = Self::exp_by_x(r);

let mut y1 = y5.cyclotomic_square();
let mut y3 = y0 * &y5;
y0 = Self::exp_by_x(y3);
let y2 = Self::exp_by_x(y0);
let mut y4 = Self::exp_by_x(y2);
y4 *= &y1;
y1 = Self::exp_by_x(y4);
y3.conjugate();
y1 *= &y3;
y1 *= &r;
y3 = r;
y3.conjugate();
y0 *= &r;
y0.frobenius_map(3);
y4 *= &y3;
y4.frobenius_map(1);
y5 *= &y2;
y5.frobenius_map(2);
y5 *= &y0;
y5 *= &y4;
y5 *= &y1;
y5
// t[1].Expt(&result)
let mut y1 = Self::exp_by_x(r);
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// t[2].InverseUnitary(&result)
let mut y2 = r;
y2.conjugate();
// t[1].Mul(&t[1], &t[2])
y1 *= y2;
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// t[2].Expt(&t[1])
y2 = Self::exp_by_x(y1);
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I would be wary if copies of Fp12 elements, in my experience the compiler produce a bench of movaps, movups instructions, you might want an out-of-place pow_by_x

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@Pratyush Pratyush Mar 8, 2021 •
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To clarify, you mean that exp_by_x should take y1 by reference? Or do you mean that it should mutate y2 in place?

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@mratsim mratsim Mar 10, 2021 •
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I mean mutate y2 in-place. Does Rust take large stack parameter by reference by default? If not it's needed as well but given that Option and Result work well in Rust I assume it does the right thing.

// t[1].InverseUnitary(&t[1])
y1.conjugate();
// t[1].Mul(&t[1], &t[2])
y1 *= y2;
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// t[2].Expt(&t[1])
y2 = Self::exp_by_x(y1);
// t[1].Frobenius(&t[1])
y1.frobenius_map(1);
// t[1].Mul(&t[1], &t[2])
y1 *= &y2;
// result.Mul(&result, &t[0])
r *= y0;
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// t[0].Expt(&t[1])
y0 = Self::exp_by_x(y1);
// t[2].Expt(&t[0])
y2 = Self::exp_by_x(y0);
// t[0].FrobeniusSquare(&t[1])
y0 = y1;
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Ditto for the copy here

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Hm how do I avoid the copy of y1 here?

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@mratsim mratsim Mar 10, 2021 •
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This is how I did it

  # (x+p)
  v1.pow_x(v0)                 # v1 = f^((x-1)².x)
  v0.frobenius_map(v0)         # v0 = f^((x-1)².p)
  v0 *= v1                     # v0 = f^((x-1)².(x+p))

  # + 3
  f *= v2                      # f = f³

  # (x²+p²−1)
  v2.pow_x(v0, invert = false)
  v1.pow_x(v2, invert = false) # v1 = f^((x-1)².(x+p).x²)
  v2.frobenius_map(v0, 2)      # v2 = f^((x-1)².(x+p).p²)
  v0.cyclotomic_inv()          # v0 = f^((x-1)².(x+p).-1)
  v0 *= v1                     # v0 = f^((x-1)².(x+p).(x²-1))
  v0 *= v2                     # v0 = f^((x-1)².(x+p).(x²+p²-1))

  # (x−1)².(x+p).(x²+p²−1) + 3
  f *= v0

y0.frobenius_map(2);
// t[1].InverseUnitary(&t[1])
y1.conjugate();
// t[1].Mul(&t[1], &t[2])
y1 *= y2;
// t[1].Mul(&t[1], &t[0])
y1 *= y0;
// result.Mul(&result, &t[1])
r *= y1;
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r
})
}
}