Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Jump to bottom

Fuentes et al. algorithm for computing hard part of final exponentiation #40

Open
wants to merge 1 commit into
base: master
Choose a base branch
from
Open

Fuentes et al. algorithm for computing hard part of final exponentiation #40

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
65 changes: 35 additions & 30 deletions optate.go
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -208,6 +208,7 @@ func miller(q *twistPoint, p *curvePoint) *gfP12 {
func finalExponentiation(in *gfP12) *gfP12 {
t1 := &gfP12{}

// Easy part
// This is the p^6-Frobenius
t1.x.Neg(&in.x)
t1.y.Set(&in.y)
Expand All @@ -219,41 +220,45 @@ func finalExponentiation(in *gfP12) *gfP12 {
t2 := (&gfP12{}).FrobeniusP2(t1)
t1.Mul(t1, t2)

fp := (&gfP12{}).Frobenius(t1)
fp2 := (&gfP12{}).FrobeniusP2(t1)
fp3 := (&gfP12{}).Frobenius(fp2)

fu := (&gfP12{}).Exp(t1, u)
fu2 := (&gfP12{}).Exp(fu, u)
fu3 := (&gfP12{}).Exp(fu2, u)

y3 := (&gfP12{}).Frobenius(fu)
fu2p := (&gfP12{}).Frobenius(fu2)
fu3p := (&gfP12{}).Frobenius(fu3)
y2 := (&gfP12{}).FrobeniusP2(fu2)
// Hard Part
// Follows Fuentes et al. algorithm (https://link.springer.com/content/pdf/10.1007/978-3-642-28496-0_25.pdf)
// for computing the hard part based on the fact that f ^ d' is also a pairing and can be computed atleast as efficiently as f ^ d
// where d' is a multiple of d and d = Φk(p)/r = (p4 − p2 + 1)/r

// see algorithm 6 from https://eprint.iacr.org/2015/192.pdf
y0 := (&gfP12{}).Exp(t1, u)
y0.Conjugate(y0)
y0.Square(y0)
y1 := (&gfP12{}).Square(y0)
y1.Mul(y0, y1)
y2 := (&gfP12{}).Exp(y1, u)
y2.Conjugate(y2) // f ^ 6u²

y3 := (&gfP12{}).Conjugate(y1)
y1.Mul(y2, y3)
y3.Square(y2)
y4 := (&gfP12{}).Exp(y3, u)
y4.Conjugate(y4)
y4.Conjugate(y4)
y4.Mul(y4, y1) // f ^ (6u + 6u² + 12u²) = f ^ a₂

y0 := &gfP12{}
y0.Mul(fp, fp2).Mul(y0, fp3)
y3.Mul(y4, y0) // f ^ 4u * f ^ 6u² * f ^ 12u³ = f ^ a₁

y1 := (&gfP12{}).Conjugate(t1)
y5 := (&gfP12{}).Conjugate(fu2)
y3.Conjugate(y3)
y4 := (&gfP12{}).Mul(fu, fu2p)
y4.Conjugate(y4)
y0.Mul(y2, y4)
y0.Mul(y0, t1) // a * f ^ 6u² * f = f ^ a₀

y6 := (&gfP12{}).Mul(fu3, fu3p)
y6.Conjugate(y6)
y2.Frobenius(y3)
y0.Mul(y2, y0)
y2.FrobeniusP2(y4)
y0.Mul(y2, y0)
y2.Conjugate(t1)
y2.Mul(y2, y3) // f ^ -1 * f ^ 4u * f ^ 6u² * f ^ 12u³ = f^a₃

t0 := (&gfP12{}).Square(y6)
t0.Mul(t0, y4).Mul(t0, y5)
t1.Mul(y3, y5).Mul(t1, t0)
t0.Mul(t0, y2)
t1.Square(t1).Mul(t1, t0).Square(t1)
t0.Mul(t1, y1)
t1.Mul(t1, y0)
t0.Square(t0).Mul(t0, t1)
y2.FrobeniusP2(y2)
y2.Frobenius(y2)
y0.Mul(y2, y0) // f ^ s * (p⁴ - p² + 1 ) / r

return t0
return y0
}

func optimalAte(a *twistPoint, b *curvePoint) *gfP12 {
Expand Down