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update InCorrectSubgroup #34
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changed gt.isValid to M. Scott's method in https://eprint.iacr.org/2021/1130 On i7-3770 CPU @ 3.40GHz BenchmarkGT_IsValid-8 250904 ns/op |
AnomalRoil
approved these changes
Feb 8, 2023
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| // A note on group membership tests for G1, G2 | ||
| // and GT on BLS pairing-friendly curves | ||
| // M. Scott | ||
| // https://eprint.iacr.org/2021/1130.pdf | ||
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| // [z]ψ^3(P) − ψ^2(P) + P = O | ||
| t0, t1 := g.New().Set(p), g.New() | ||
| // ψ^3(P) − [u]P = O | ||
| t0, t1 := g.New().Set(p), g.New().Set(p) | ||
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| g.psi(t0) | ||
| g.psi(t0) | ||
| g.Neg(t1, t0) // - ψ^2(P) | ||
| g.psi(t0) // ψ^3(P) | ||
| g.mulX(t0) // - x ψ^3(P) | ||
| g.Neg(t0, t0) | ||
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| g.Add(t0, t0, t1) | ||
| g.Add(t0, t0, p) | ||
| g.psi(t0) //ψ(P) | ||
| g.mulX(t1) //-[u]P | ||
| g.Add(t0, t0, t1) //ψ(P)-[u]P |
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This looks good to me, this method works as better proven also in https://hal.inria.fr/hal-03608264/document
Note that u == x but it's just a question of notation.
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| g.fp12.frobeniusMap1(r0) | ||
| r1.set(r0) | ||
| g.fp12.frobeniusMap1(r0) | ||
| r2.set(r0) | ||
| g.fp12.frobeniusMap2(r0) | ||
| g.Mul(r0, r0, e) |
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Suggested change
| g.fp12.frobeniusMap1(r0) | |
| r1.set(r0) | |
| g.fp12.frobeniusMap1(r0) | |
| r2.set(r0) | |
| g.fp12.frobeniusMap2(r0) | |
| g.Mul(r0, r0, e) | |
| g.fp12.frobeniusMap1(r0) // r0 = e^p | |
| r1.set(r0) // r1 = e^p | |
| g.fp12.frobeniusMap1(r0) // r0 = e^(p^2) | |
| r2.set(r0) // r2 = e^(p^2) | |
| g.fp12.frobeniusMap2(r0) // r0 = e^(p^4) | |
| g.Mul(r0, r0, e) // r0 = e·e^(p^4) |
| r2.set(r0) | ||
| g.fp12.frobeniusMap2(r0) | ||
| g.Mul(r0, r0, e) | ||
| if !r0.Equal(r2) { |
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Suggested change
| if !r0.Equal(r2) { | |
| // cyclotomic test | |
| if !r0.Equal(r2) { |
| g.Exp(r0, e, bigFromHex("0xd201000000010000")) | ||
| g.Mul(r0, r0, r1) | ||
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| return r0.IsOne() |
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So, you are doing the cyclotomic test e·e^(p^4) == e^(p^2) and then you test if e^p = e^u as in Scott paper.
Suggested change
| return r0.IsOne() | |
| // e^(p-u) = e^(p+1-t) == 1 | |
| return r0.IsOne() |
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| g.Exp(r0, e, bigFromHex("0xd201000000010000")) | ||
| g.Mul(r0, r0, r1) |
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Suggested change
| g.Exp(r0, e, bigFromHex("0xd201000000010000")) | |
| g.Mul(r0, r0, r1) | |
| g.Exp(r0, e, bigFromHex("0xd201000000010000")) // r0 = e^-u | |
| g.Mul(r0, r0, r1) // r0 = e^-u · e^p = e^(p-u) |
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Changed InCorrectSubgroup to M. Scott's method in https://eprint.iacr.org/2021/1130
On i7-3770 CPU @ 3.40GHz
BenchmarkG2SubgroupCheck-8 153520 ns/op
BenchmarkG2SubgroupCheckOld-8 167288 ns/op