diff --git a/plonky2/src/curve/glv.rs b/plonky2/src/curve/glv.rs
index ded0d0b3..a2786e31 100644
--- a/plonky2/src/curve/glv.rs
+++ b/plonky2/src/curve/glv.rs
@@ -1,4 +1,5 @@
 use num::rational::Ratio;
+use num::BigUint;
 use plonky2_field::field_types::Field;
 use plonky2_field::secp256k1_base::Secp256K1Base;
 use plonky2_field::secp256k1_scalar::Secp256K1Scalar;
@@ -30,26 +31,46 @@ const A2: Secp256K1Scalar = Secp256K1Scalar([6323353552219852760, 14980988506747
 
 const B2: Secp256K1Scalar = Secp256K1Scalar([16747920425669159701, 3496713202691238861, 0, 0]);
 
-pub fn decompose_secp256k1_scalar(k: Secp256K1Scalar) -> (Secp256K1Scalar, Secp256K1Scalar) {
+pub fn decompose_secp256k1_scalar(
+    k: Secp256K1Scalar,
+) -> (Secp256K1Scalar, Secp256K1Scalar, bool, bool) {
     let p = Secp256K1Scalar::order();
     let c1_biguint = Ratio::new(B2.to_biguint() * k.to_biguint(), p.clone())
         .round()
         .to_integer();
     let c1 = Secp256K1Scalar::from_biguint(c1_biguint);
-    let c2_biguint = Ratio::new(MINUS_B1.to_biguint() * k.to_biguint(), p)
+    let c2_biguint = Ratio::new(MINUS_B1.to_biguint() * k.to_biguint(), p.clone())
         .round()
         .to_integer();
     let c2 = Secp256K1Scalar::from_biguint(c2_biguint);
 
-    let k1 = k - c1 * A1 - c2 * A2;
-    let k2 = c1 * MINUS_B1 - c2 * B2;
-    debug_assert!(k1 + S * k2 == k);
-    (k1, k2)
+    let k1_raw = k - c1 * A1 - c2 * A2;
+    let k2_raw = c1 * MINUS_B1 - c2 * B2;
+    debug_assert!(k1_raw + S * k2_raw == k);
+
+    let two = BigUint::from_slice(&[2]);
+    let k1_neg = k1_raw.to_biguint() > p.clone() / two.clone();
+    let k1 = if k1_neg {
+        Secp256K1Scalar::from_biguint(p.clone() - k1_raw.to_biguint())
+    } else {
+        k1_raw
+    };
+    let k2_neg = k2_raw.to_biguint() > p.clone() / two.clone();
+    let k2 = if k2_neg {
+        Secp256K1Scalar::from_biguint(p.clone() - k2_raw.to_biguint())
+    } else {
+        k2_raw
+    };
+
+    (k1, k2, k1_neg, k2_neg)
 }
 
 pub fn glv_mul(p: ProjectivePoint<Secp256K1>, k: Secp256K1Scalar) -> ProjectivePoint<Secp256K1> {
-    let (k1, k2) = decompose_secp256k1_scalar(k);
-    assert!(k1 + S * k2 == k);
+    let (k1, k2, k1_neg, k2_neg) = decompose_secp256k1_scalar(k);
+    let one = Secp256K1Scalar::ONE;
+    /*let m1 = if k1_neg { -one } else { one };
+    let m2 = if k2_neg { -one } else { one };
+    assert!(k1 * m1 + S * k2 * m2 == k);*/
 
     let p_affine = p.to_affine();
     let sp = AffinePoint::<Secp256K1> {
@@ -58,7 +79,10 @@ pub fn glv_mul(p: ProjectivePoint<Secp256K1>, k: Secp256K1Scalar) -> ProjectiveP
         zero: p_affine.zero,
     };
 
-    msm_parallel(&[k1, k2], &[p, sp.to_projective()], 5)
+    let first = if k1_neg { p.neg() } else { p };
+    let second = if k2_neg { sp.to_projective().neg() } else { sp.to_projective() };
+
+    msm_parallel(&[k1, k2], &[first, second], 5)
 }
 
 #[cfg(test)]
@@ -74,9 +98,12 @@ mod tests {
     #[test]
     fn test_glv_decompose() -> Result<()> {
         let k = Secp256K1Scalar::rand();
-        let (k1, k2) = decompose_secp256k1_scalar(k);
+        let (k1, k2, k1_neg, k2_neg) = decompose_secp256k1_scalar(k);
+        let one = Secp256K1Scalar::ONE;
+        let m1 = if k1_neg { -one } else { one };
+        let m2 = if k2_neg { -one } else { one };
 
-        assert!(k1 + S * k2 == k);
+        assert!(k1 * m1 + S * k2 * m2 == k);
 
         Ok(())
     }
diff --git a/plonky2/src/gadgets/curve.rs b/plonky2/src/gadgets/curve.rs
index 2b27c120..aa8c4112 100644
--- a/plonky2/src/gadgets/curve.rs
+++ b/plonky2/src/gadgets/curve.rs
@@ -71,6 +71,24 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         }
     }
 
+    pub fn curve_conditional_neg<C: Curve>(
+        &mut self,
+        p: &AffinePointTarget<C>,
+        b: BoolTarget,
+    ) -> AffinePointTarget<C> {
+        let not_b = self.not(b);
+        let neg = self.curve_neg(p);
+        let x_if_true = self.mul_nonnative_by_bool(&neg.x, b);
+        let y_if_true = self.mul_nonnative_by_bool(&neg.y, b);
+        let x_if_false = self.mul_nonnative_by_bool(&p.x, not_b);
+        let y_if_false = self.mul_nonnative_by_bool(&p.y, not_b);
+
+        let x = self.add_nonnative(&x_if_true, &x_if_false);
+        let y = self.add_nonnative(&y_if_true, &y_if_false);
+
+        AffinePointTarget { x, y }
+    }
+
     pub fn curve_double<C: Curve>(&mut self, p: &AffinePointTarget<C>) -> AffinePointTarget<C> {
         let AffinePointTarget { x, y } = p;
         let double_y = self.add_nonnative(y, y);
diff --git a/plonky2/src/gadgets/curve_windowed_mul.rs b/plonky2/src/gadgets/curve_windowed_mul.rs
index 2f0516dd..f4cebe0e 100644
--- a/plonky2/src/gadgets/curve_windowed_mul.rs
+++ b/plonky2/src/gadgets/curve_windowed_mul.rs
@@ -18,18 +18,18 @@ use crate::plonk::config::{GenericHashOut, Hasher};
 const WINDOW_SIZE: usize = 4;
 
 impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
-    // TODO: fix if p is the generator
     pub fn precompute_window<C: Curve>(
         &mut self,
         p: &AffinePointTarget<C>,
     ) -> Vec<AffinePointTarget<C>> {
+        let g = (CurveScalar(C::ScalarField::rand()) * C::GENERATOR_PROJECTIVE).to_affine();
         let neg = {
-            let mut g = C::GENERATOR_AFFINE;
-            g.y = -g.y;
-            self.constant_affine_point(g)
+            let mut neg = g;
+            neg.y = -neg.y;
+            self.constant_affine_point(neg)
         };
 
-        let mut multiples = vec![self.constant_affine_point(C::GENERATOR_AFFINE)];
+        let mut multiples = vec![self.constant_affine_point(g)];
         for i in 1..1 << WINDOW_SIZE {
             multiples.push(self.curve_add(p, &multiples[i - 1]));
         }
diff --git a/plonky2/src/gadgets/glv.rs b/plonky2/src/gadgets/glv.rs
index 5ac3fb93..5614de55 100644
--- a/plonky2/src/gadgets/glv.rs
+++ b/plonky2/src/gadgets/glv.rs
@@ -10,7 +10,7 @@ use crate::gadgets::curve::AffinePointTarget;
 use crate::gadgets::nonnative::NonNativeTarget;
 use crate::hash::hash_types::RichField;
 use crate::iop::generator::{GeneratedValues, SimpleGenerator};
-use crate::iop::target::Target;
+use crate::iop::target::{BoolTarget, Target};
 use crate::iop::witness::{PartitionWitness, Witness};
 use crate::plonk::circuit_builder::CircuitBuilder;
 
@@ -25,18 +25,24 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
     ) -> (
         NonNativeTarget<Secp256K1Scalar>,
         NonNativeTarget<Secp256K1Scalar>,
+        BoolTarget,
+        BoolTarget,
     ) {
-        let k1 = self.add_virtual_nonnative_target::<Secp256K1Scalar>();
-        let k2 = self.add_virtual_nonnative_target::<Secp256K1Scalar>();
+        let k1 = self.add_virtual_nonnative_target_sized::<Secp256K1Scalar>(4);
+        let k2 = self.add_virtual_nonnative_target_sized::<Secp256K1Scalar>(4);
+        let k1_neg = self.add_virtual_bool_target();
+        let k2_neg = self.add_virtual_bool_target();
 
         self.add_simple_generator(GLVDecompositionGenerator::<F, D> {
             k: k.clone(),
             k1: k1.clone(),
             k2: k2.clone(),
+            k1_neg: k1_neg.clone(),
+            k2_neg: k2_neg.clone(),
             _phantom: PhantomData,
         });
 
-        (k1, k2)
+        (k1, k2, k1_neg, k2_neg)
     }
 
     pub fn glv_mul(
@@ -44,7 +50,7 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         p: &AffinePointTarget<Secp256K1>,
         k: &NonNativeTarget<Secp256K1Scalar>,
     ) -> AffinePointTarget<Secp256K1> {
-        let (k1, k2) = self.decompose_secp256k1_scalar(k);
+        let (k1, k2, k1_neg, k2_neg) = self.decompose_secp256k1_scalar(k);
 
         let beta = self.secp256k1_glv_beta();
         let beta_px = self.mul_nonnative(&beta, &p.x);
@@ -54,9 +60,11 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         };
 
         let part1 = self.curve_scalar_mul_windowed(p, &k1);
+        let part1_neg = self.curve_conditional_neg(&part1, k1_neg);
         let part2 = self.curve_scalar_mul_windowed(&sp, &k2);
+        let part2_neg = self.curve_conditional_neg(&part2, k2_neg);
 
-        self.curve_add(&part1, &part2)
+        self.curve_add(&part1_neg, &part2_neg)
     }
 }
 
@@ -65,6 +73,8 @@ struct GLVDecompositionGenerator<F: RichField + Extendable<D>, const D: usize> {
     k: NonNativeTarget<Secp256K1Scalar>,
     k1: NonNativeTarget<Secp256K1Scalar>,
     k2: NonNativeTarget<Secp256K1Scalar>,
+    k1_neg: BoolTarget,
+    k2_neg: BoolTarget,
     _phantom: PhantomData<F>,
 }
 
@@ -77,10 +87,12 @@ impl<F: RichField + Extendable<D>, const D: usize> SimpleGenerator<F>
 
     fn run_once(&self, witness: &PartitionWitness<F>, out_buffer: &mut GeneratedValues<F>) {
         let k = witness.get_nonnative_target(self.k.clone());
-        let (k1, k2) = decompose_secp256k1_scalar(k);
+        let (k1, k2, k1_neg, k2_neg) = decompose_secp256k1_scalar(k);
 
         out_buffer.set_nonnative_target(self.k1.clone(), k1);
         out_buffer.set_nonnative_target(self.k2.clone(), k2);
+        out_buffer.set_bool_target(self.k1_neg.clone(), k1_neg);
+        out_buffer.set_bool_target(self.k2_neg.clone(), k2_neg);
     }
 }
 
@@ -100,7 +112,7 @@ mod tests {
     use crate::plonk::verifier::verify;
 
     #[test]
-    fn test_glv() -> Result<()> {
+    fn test_glv_gadget() -> Result<()> {
         const D: usize = 2;
         type C = PoseidonGoldilocksConfig;
         type F = <C as GenericConfig<D>>::F;
diff --git a/plonky2/src/gadgets/nonnative.rs b/plonky2/src/gadgets/nonnative.rs
index 44969ac9..046931d2 100644
--- a/plonky2/src/gadgets/nonnative.rs
+++ b/plonky2/src/gadgets/nonnative.rs
@@ -63,6 +63,18 @@ impl<F: RichField + Extendable<D>, const D: usize> CircuitBuilder<F, D> {
         }
     }
 
+    pub fn add_virtual_nonnative_target_sized<FF: Field>(
+        &mut self,
+        num_limbs: usize,
+    ) -> NonNativeTarget<FF> {
+        let value = self.add_virtual_biguint_target(num_limbs);
+
+        NonNativeTarget {
+            value,
+            _phantom: PhantomData,
+        }
+    }
+
     pub fn add_nonnative<FF: PrimeField>(
         &mut self,
         a: &NonNativeTarget<FF>,