info: quadratic residue algo

src/field/prime/mod.rs

@@ -45,6 +45,30 @@ impl<const P: usize> PrimeField<P> {
 
  /// Computes euler criterion of the field element, i.e. Returns true if the element is a quadratic
  /// residue (a square number) in the field.
+ ///
+ /// ## NOTES
+ /// By fermat's little theorem, (assume `is_congruent_to` is =)
+ /// x^(p-1) - 1 = 0 mod P
+ ///
+ /// All primes > 2 are odd, a.k.a P is odd, hence (P-1) is even.
+ /// So, we can split as follows:
+ /// (x^(p-1)/2 - 1)(x^(p-1)/2 + 1) = 0 mod P
+ /// or L * R = 0 mod P
+ ///
+ /// All quadratic residues are of the form (g^(2k)) where `g` is the
+ /// multiplicative generator and k is some natural number. All non-residues
+ /// on the other hand are of the form (g^(2k+1)).
+ ///
+ /// In case of QR, substitute x = g^2k
+ /// g^(2k)((p-1)/2) = 1 mod P
+ /// g^(p-1) = 1 mod P
+ /// which is true by fermat's little theorem
+ ///
+ /// In the other case, the same doesn't hold.
+ /// Hence, the case `L` should hold for all quadratic residues and is the
+ /// test for quadratic residuosity.
+ ///
+ /// More info here: https://www.youtube.com/watch?v=2IBPOI43jek
  pub fn euler_criterion(&self) -> bool { self.pow((P - 1) / 2).value == 1 }
 
  /// Computes the square root of a field element using the [Tonelli-Shanks algorithm](https://en.wikipedia.org/wiki/Tonelli–Shanks_algorithm).