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Fix sqrt for QuadExtField #358

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merged 6 commits into from
Dec 6, 2021
Merged

Fix sqrt for QuadExtField #358

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52e86e3
Split cases for c0.is_qr and c0.is_nqr
mmagician Nov 30, 2021
bf528b1
Fix bug for sqrt on QuadExtField when c1 = 0 & c0.legendre.is_qnr()
mmagician Dec 5, 2021
cc730fc
Merge branch 'master' into HEAD
mmagician Dec 5, 2021
e402d8a
Remove the changes to tests.rs file
mmagician Dec 5, 2021
84eca74
Add a line to changelog
mmagician Dec 5, 2021
1316edc
Merge branch 'master' into sqrt-testing
mmagician Dec 6, 2021
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1 change: 1 addition & 0 deletions CHANGELOG.md
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Expand Up @@ -27,6 +27,7 @@
### Bugfixes

- [\#350](https://github.com/arkworks-rs/algebra/pull/350) (ark-serialize) Fix issues with santiation whenever a non-standard `Result` type is in scope.
- [\#358](https://github.com/arkworks-rs/algebra/pull/358) (ark-ff) Fix the bug for `QuadExtField::sqrt` when `c1 = 0 && c0.legendre.is_qnr()`

## v0.3.0

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15 changes: 14 additions & 1 deletion ff/src/fields/models/quadratic_extension.rs
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Expand Up @@ -350,7 +350,20 @@ where
// Square root based on the complex method. See
// https://eprint.iacr.org/2012/685.pdf (page 15, algorithm 8)
if self.c1.is_zero() {
return self.c0.sqrt().map(|c0| Self::new(c0, P::BaseField::zero()));
// for c = c0 + c1 * x, we have c1 = 0
// sqrt(c) == sqrt(c0) is an element of Fp2, i.e. sqrt(c0) = a = a0 + a1 * x for some a0, a1 in Fp
// squaring both sides: c0 = a0^2 + a1^2 * x^2 + (2 * a0 * a1 * x) = a0^2 + (a1^2 * P::NONRESIDUE)
// since there are no `x` terms on LHS, a0 * a1 = 0
// so either a0 = sqrt(c0) or a1 = sqrt(c0/P::NONRESIDUE)
if self.c0.legendre().is_qr() {
// either c0 is a valid sqrt in the base field
return self.c0.sqrt().map(|c0| Self::new(c0, P::BaseField::zero()));
} else {
// or we need to compute sqrt(c0/P::NONRESIDUE)
return (self.c0.div(P::NONRESIDUE))
.sqrt()
.map(|res| Self::new(P::BaseField::zero(), res));
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I wonder if there's a way to precompute P::NONRESIDUE.inverse() here

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One simple solution is to add NONRESIDUE_INV into the trait of the field parameters.

}
}
// Try computing the square root
// Check at the end of the algorithm if it was a square root
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