diff --git a/stwo_cairo_verifier/src/circle.cairo b/stwo_cairo_verifier/src/circle.cairo
index e386054d..7acd9720 100644
--- a/stwo_cairo_verifier/src/circle.cairo
+++ b/stwo_cairo_verifier/src/circle.cairo
@@ -1,9 +1,10 @@
 use core::num::traits::one::One;
 use core::num::traits::zero::Zero;
 use core::num::traits::{WrappingAdd, WrappingSub, WrappingMul};
+use stwo_cairo_verifier::channel::{Channel, ChannelImpl};
 use stwo_cairo_verifier::fields::cm31::CM31;
 use stwo_cairo_verifier::fields::m31::{M31, M31Impl};
-use stwo_cairo_verifier::fields::qm31::{QM31Impl, QM31, QM31Trait};
+use stwo_cairo_verifier::fields::qm31::{QM31Impl, QM31One, QM31, QM31Trait};
 use super::utils::pow;
 
 /// A generator for the circle group over [`M31`].
@@ -99,10 +100,38 @@ impl CirclePointAdd<F, +Add<F>, +Sub<F>, +Mul<F>, +Drop<F>, +Copy<F>> of Add<Cir
     }
 }
 
+impl CirclePointNeg<F, +Neg<F>, +Drop<F>, +Copy<F>> of Neg<CirclePoint<F>> {
+    fn neg(a: CirclePoint<F>) -> CirclePoint<F> {
+        CirclePoint { x: a.x, y: -a.y }
+    }
+}
+
 pub impl CirclePointM31Impl of CirclePointTrait<M31> {}
 
+#[generate_trait]
+pub impl CirclePointQM31AddCirclePointM31Impl of CirclePointQM31AddCirclePointM31Trait {
+    fn add_circle_point_m31(self: CirclePoint<QM31>, rhs: CirclePoint<M31>) -> CirclePoint<QM31> {
+        CirclePoint {
+            x: self.x.mul_m31(rhs.x) - self.y.mul_m31(rhs.y),
+            y: self.x.mul_m31(rhs.y) + self.y.mul_m31(rhs.x)
+        }
+    }
+}
+
 pub impl CirclePointQM31Impl of CirclePointTrait<QM31> {}
 
+#[generate_trait]
+pub impl ChannelGetRandomCirclePointImpl of ChannelGetRandomCirclePointTrait {
+    fn get_random_point(ref self: Channel) -> CirclePoint<QM31> {
+        let t = self.draw_felt();
+        let t_squared = t * t;
+        let t_squared_plus_1_inv = (t_squared + QM31One::one()).inverse();
+        let x = (QM31One::one() - t_squared) * t_squared_plus_1_inv;
+        let y = (t + t) * t_squared_plus_1_inv;
+        CirclePoint { x, y }
+    }
+}
+
 impl CirclePointQM31PartialOrd of PartialOrd<CirclePoint<QM31>> {
     fn lt(lhs: CirclePoint<QM31>, rhs: CirclePoint<QM31>) -> bool {
         lhs.x < rhs.x || (lhs.x == rhs.x && lhs.y < rhs.y)