Add LinePoly docstrings. Improve efficiency. Add assert to the last l…

…evel.

stwo_cairo_verifier/src/poly/line.cairo

@@ -1,6 +1,16 @@
 use stwo_cairo_verifier::fields::SecureField;
 use stwo_cairo_verifier::fields::m31::m31;
 
+/// A univariate polynomial represented by its coefficients in the line part of the FFT-basis {bᵢ}
+/// in bit reversed order. See Definition 4 of the Circle STARKs paper
+/// https://eprint.iacr.org/2024/278. The line part of the FFT-basis is formed by the elements bᵢ
+/// with i even.
+///
+/// Example: The line part of the FFT of order 4 in natural order is {b₀, b₂, b₄, b₆}, where
+/// b₀ = 1, b₂ = x, b₄ = 2x² − 1, and b₆ = x(2x² − 1). The same basis but in bit
+/// reversed order is {b₀, b₄, b₂, b₆}.
+/// Then, the polynomial p represented by coeffs = [a₀, a₁, a₂, a₃] and log_size = 2 is
+/// p = a₀ * b₀ + a₁ * b₄ + a₂ * b₂ + a₃ * b₆.
 #[derive(Drop, Clone)]
 pub struct LinePoly {
     pub coeffs: Array<SecureField>,
@@ -13,12 +23,14 @@ pub impl LinePolyImpl of LinePolyTrait {
         self.coeffs.len()
     }
 
+    // TODO: Consider replacing with FFT.
     fn eval_at_point(self: @LinePoly, mut x: SecureField) -> SecureField {
         let mut mappings = array![];
         let mut i = 0;
         while i < *self.log_size {
             mappings.append(x);
-            x = m31(2).into() * x * x - m31(1).into();
+            let x_square = x * x;
+            x = x_square + x_square - m31(1).into();
             i += 1;
         };
 
@@ -37,6 +49,7 @@ pub impl LinePolyImpl of LinePolyTrait {
             level = new_level;
         };
 
+        assert_eq!(level.len(), 1);
         *level[0]
     }
 }