diff --git a/libcrux-ml-dsa/src/simd/avx2/arithmetic.rs b/libcrux-ml-dsa/src/simd/avx2/arithmetic.rs
index c9b4eaaa0..12954f91b 100644
--- a/libcrux-ml-dsa/src/simd/avx2/arithmetic.rs
+++ b/libcrux-ml-dsa/src/simd/avx2/arithmetic.rs
@@ -38,20 +38,29 @@ fn simd_multiply_i32_and_return_high(lhs: Vec256, rhs: Vec256) -> Vec256 {
 }
 
 #[inline(always)]
-pub fn montgomery_multiply_by_constant(simd_unit: Vec256, constant: i32) -> Vec256 {
-    let constant = mm256_set1_epi32(constant);
+pub fn montgomery_multiply_by_constant(lhs: Vec256, constant: i32) -> Vec256 {
+    let rhs = mm256_set1_epi32(constant);
     let field_modulus = mm256_set1_epi32(FIELD_MODULUS);
     let inverse_of_modulus_mod_montgomery_r =
         mm256_set1_epi32(INVERSE_OF_MODULUS_MOD_MONTGOMERY_R as i32);
 
-    let product_low = mm256_mullo_epi32(simd_unit, constant);
+    let prod02 = mm256_mul_epi32(lhs, rhs);
+    let prod13 = mm256_mul_epi32(
+        mm256_shuffle_epi32::<0b11_11_01_01>(lhs),
+        mm256_shuffle_epi32::<0b11_11_01_01>(rhs),
+    );
 
-    let k = mm256_mullo_epi32(product_low, inverse_of_modulus_mod_montgomery_r);
+    let k02 = mm256_mul_epi32(prod02, inverse_of_modulus_mod_montgomery_r);
+    let k13 = mm256_mul_epi32(prod13, inverse_of_modulus_mod_montgomery_r);
 
-    let c = simd_multiply_i32_and_return_high(k, field_modulus);
-    let product_high = simd_multiply_i32_and_return_high(simd_unit, constant);
+    let c02 = mm256_mul_epi32(k02, field_modulus);
+    let c13 = mm256_mul_epi32(k13, field_modulus);
 
-    mm256_sub_epi32(product_high, c)
+    let res02 = mm256_sub_epi32(prod02, c02);
+    let res13 = mm256_sub_epi32(prod13, c13);
+    let res02_shifted = mm256_shuffle_epi32::<0b11_11_01_01>(res02);
+    let res = mm256_blend_epi32::<0b10101010>(res02_shifted, res13);
+    res
 }
 
 #[inline(always)]
@@ -60,14 +69,22 @@ pub fn montgomery_multiply(lhs: Vec256, rhs: Vec256) -> Vec256 {
     let inverse_of_modulus_mod_montgomery_r =
         mm256_set1_epi32(INVERSE_OF_MODULUS_MOD_MONTGOMERY_R as i32);
 
-    let product_low = mm256_mullo_epi32(lhs, rhs);
-
-    let k = mm256_mullo_epi32(product_low, inverse_of_modulus_mod_montgomery_r);
+    let prod02 = mm256_mul_epi32(lhs, rhs);
+    let prod13 = mm256_mul_epi32(
+        mm256_shuffle_epi32::<0b11_11_01_01>(lhs),
+        mm256_shuffle_epi32::<0b11_11_01_01>(rhs),
+    );
+    let k02 = mm256_mul_epi32(prod02, inverse_of_modulus_mod_montgomery_r);
+    let k13 = mm256_mul_epi32(prod13, inverse_of_modulus_mod_montgomery_r);
 
-    let c = simd_multiply_i32_and_return_high(k, field_modulus);
-    let product_high = simd_multiply_i32_and_return_high(lhs, rhs);
+    let c02 = mm256_mul_epi32(k02, field_modulus);
+    let c13 = mm256_mul_epi32(k13, field_modulus);
 
-    mm256_sub_epi32(product_high, c)
+    let res02 = mm256_sub_epi32(prod02, c02);
+    let res13 = mm256_sub_epi32(prod13, c13);
+    let res02_shifted = mm256_shuffle_epi32::<0b11_11_01_01>(res02);
+    let res = mm256_blend_epi32::<0b10101010>(res02_shifted, res13);
+    res
 }
 
 #[inline(always)]
diff --git a/libcrux-ml-dsa/src/simd/avx2/ntt.rs b/libcrux-ml-dsa/src/simd/avx2/ntt.rs
index b18887b1d..c6f302155 100644
--- a/libcrux-ml-dsa/src/simd/avx2/ntt.rs
+++ b/libcrux-ml-dsa/src/simd/avx2/ntt.rs
@@ -93,7 +93,10 @@ fn butterfly_8(a: Vec256, b: Vec256, zeta0: i32, zeta1: i32) -> (Vec256, Vec256)
     let add_terms = arithmetic::add(summands, zeta_products);
     let sub_terms = arithmetic::subtract(summands, zeta_products);
 
-    let a_out = mm256_set_m128i(mm256_castsi256_si128(sub_terms), mm256_castsi256_si128(add_terms));
+    let a_out = mm256_set_m128i(
+        mm256_castsi256_si128(sub_terms),
+        mm256_castsi256_si128(add_terms),
+    );
     let b_out = mm256_permute2x128_si256::<0b0001_0011>(sub_terms, add_terms);
 
     (a_out, b_out)
diff --git a/libcrux-ml-dsa/src/simd/portable/arithmetic.rs b/libcrux-ml-dsa/src/simd/portable/arithmetic.rs
index a57ceb241..1785d108e 100644
--- a/libcrux-ml-dsa/src/simd/portable/arithmetic.rs
+++ b/libcrux-ml-dsa/src/simd/portable/arithmetic.rs
@@ -2,7 +2,10 @@ use crate::{
     constants::BITS_IN_LOWER_PART_OF_T,
     simd::{
         portable::PortableSIMDUnit,
-        traits::{Operations, FIELD_MODULUS, INVERSE_OF_MODULUS_MOD_MONTGOMERY_R, FieldElementTimesMontgomeryR},
+        traits::{
+            FieldElementTimesMontgomeryR, Operations, FIELD_MODULUS,
+            INVERSE_OF_MODULUS_MOD_MONTGOMERY_R,
+        },
     },
 };
 
diff --git a/libcrux-ml-dsa/src/simd/portable/ntt.rs b/libcrux-ml-dsa/src/simd/portable/ntt.rs
index ae2e76284..1674c9c66 100644
--- a/libcrux-ml-dsa/src/simd/portable/ntt.rs
+++ b/libcrux-ml-dsa/src/simd/portable/ntt.rs
@@ -1,7 +1,10 @@
 use super::arithmetic;
 use crate::simd::{
     portable::PortableSIMDUnit,
-    traits::{montgomery_multiply_by_fer, COEFFICIENTS_IN_SIMD_UNIT, SIMD_UNITS_IN_RING_ELEMENT, ZETAS_TIMES_MONTGOMERY_R},
+    traits::{
+        montgomery_multiply_by_fer, COEFFICIENTS_IN_SIMD_UNIT, SIMD_UNITS_IN_RING_ELEMENT,
+        ZETAS_TIMES_MONTGOMERY_R,
+    },
 };
 
 #[inline(always)]