Update QuadExtField::sqrt for better performance

Currently, the QuadExtField uses an expensive runtime inverse of two
calculation defined as `P::BaseField::one().double().inverse()`.

With the proposed change, we compute the BigInt `(p+1)/2` that is ~15%
cheaper than the previous method.

Alternatively, we could require a compile-time constant provided by the
user that represents `1/2`. However, having a constant requirement to
satisfy a single use-case isn't ideal.

Closes #210

CHANGELOG.md

@@ -22,6 +22,7 @@
 ### Improvements
 
 - [\#339](https://github.com/arkworks-rs/algebra/pull/339) (ark-ff) Remove duplicated code from `test_field` module and replace its usage with `ark-test-curves` crate.
+- [\#352](https://github.com/arkworks-rs/algebra/pull/352) (ark-ff) Update `QuadExtField::sqrt` for better performance.
 
 ### Bugfixes
 

ff/src/fields/models/quadratic_extension.rs

@@ -19,8 +19,9 @@ use ark_std::rand::{
 };
 
 use crate::{
+    biginteger::BigInteger,
     bytes::{FromBytes, ToBytes},
-    fields::{Field, LegendreSymbol, PrimeField, SquareRootField},
+    fields::{Field, FpParameters, LegendreSymbol, PrimeField, SquareRootField},
     ToConstraintField, UniformRand,
 };
 
@@ -331,6 +332,7 @@ impl<P: QuadExtParameters> Field for QuadExtField<P> {
 impl<'a, P: QuadExtParameters> SquareRootField for QuadExtField<P>
 where
     P::BaseField: SquareRootField,
+    P::BaseField: From<<<P as QuadExtParameters>::BasePrimeField as PrimeField>::BigInt>,
 {
     fn legendre(&self) -> LegendreSymbol {
         // The LegendreSymbol in a field of order q for an element x can be
@@ -355,11 +357,18 @@ where
         // Try computing the square root
         // Check at the end of the algorithm if it was a square root
         let alpha = self.norm();
-        // TODO: Precompute this
-        let two_inv = P::BaseField::one()
-            .double()
-            .inverse()
-            .expect("Two should always have an inverse");
+
+        // Compute `(p+1)/2` as `1/2`.
+        // This is cheaper than `P::BaseField::one().double().inverse()`
+        let mut two_inv = <<P as QuadExtParameters>::BasePrimeField as PrimeField>::Params::MODULUS;
+
+        // TODO use from_limbs API
+        // https://github.com/arkworks-rs/algebra/issues/351
+        two_inv.add_nocarry(&1u64.into());
+        two_inv.div2();
+
+        let two_inv = P::BaseField::from(two_inv);
+
         alpha.sqrt().and_then(|alpha| {
             let mut delta = (alpha + &self.c0) * &two_inv;
             if delta.legendre().is_qnr() {