diff --git a/libcrux-ml-dsa/src/sample.rs b/libcrux-ml-dsa/src/sample.rs
index 05fd2c9f9..9e093b9ea 100644
--- a/libcrux-ml-dsa/src/sample.rs
+++ b/libcrux-ml-dsa/src/sample.rs
@@ -4,28 +4,34 @@ use crate::{
     hash_functions::XOF,
 };
 
-fn sample_from_uniform_distribution_next(
+fn rejection_sample_less_than_field_modulus(
     randomness: &[u8],
+    sampled: &mut usize,
     out: &mut PolynomialRingElement,
-) -> usize {
-    let mut sampled = 0;
+) -> bool {
+    let mut done = false;
 
     for bytes in randomness.chunks(3) {
-        let b0 = bytes[0] as i32;
-        let b1 = bytes[1] as i32;
-        let b2 = bytes[2] as i32;
+        if !done {
+            let b0 = bytes[0] as i32;
+            let b1 = bytes[1] as i32;
+            let b2 = bytes[2] as i32;
+
+            let potential_coefficient = ((b2 << 16) | (b1 << 8) | b0) & 0x00_7F_FF_FF;
 
-        let potential_coefficient = ((b2 << 16) | (b1 << 8) | b0) & 0x00_7F_FF_FF;
+            if potential_coefficient < FIELD_MODULUS && *sampled < COEFFICIENTS_IN_RING_ELEMENT {
+                out.coefficients[*sampled] = potential_coefficient;
+                *sampled += 1;
+            }
 
-        if potential_coefficient < FIELD_MODULUS && sampled < COEFFICIENTS_IN_RING_ELEMENT {
-            out.coefficients[sampled] = potential_coefficient;
-            sampled += 1;
+            if *sampled == COEFFICIENTS_IN_RING_ELEMENT {
+                done = true;
+            }
         }
     }
 
-    sampled
+    done
 }
-
 #[allow(non_snake_case)]
 pub(crate) fn sample_ring_element_uniform(seed: [u8; 34]) -> PolynomialRingElement {
     let mut state = XOF::new(seed);
@@ -33,11 +39,60 @@ pub(crate) fn sample_ring_element_uniform(seed: [u8; 34]) -> PolynomialRingEleme
 
     let mut out = PolynomialRingElement::ZERO;
 
-    let mut sampled = sample_from_uniform_distribution_next(&randomness, &mut out);
+    let mut sampled = 0;
+    let mut done = rejection_sample_less_than_field_modulus(&randomness, &mut sampled, &mut out);
+
+    while !done {
+        let randomness = XOF::squeeze_next_block(&mut state);
+        done = rejection_sample_less_than_field_modulus(&randomness, &mut sampled, &mut out);
+    }
+
+    out
+}
+
+fn rejection_sample_less_than_eta_equals_4(
+    randomness: &[u8],
+    sampled: &mut usize,
+    out: &mut PolynomialRingElement,
+) -> bool {
+    let mut done = false;
+
+    for byte in randomness {
+        if !done {
+            let try_0 = byte & 0xF;
+            let try_1 = byte >> 4;
+
+            if try_0 < (2 * 4) + 1 && *sampled < COEFFICIENTS_IN_RING_ELEMENT {
+                out.coefficients[*sampled] = 4 - (try_0 as i32);
+                *sampled += 1;
+            }
+
+            if try_1 < (2 * 4) + 1 && *sampled < COEFFICIENTS_IN_RING_ELEMENT {
+                out.coefficients[*sampled] = 4 - (try_1 as i32);
+                *sampled += 1;
+            }
+
+            if *sampled == COEFFICIENTS_IN_RING_ELEMENT {
+                done = true;
+            }
+        }
+    }
+
+    done
+}
+#[allow(non_snake_case)]
+pub(crate) fn sample_error_ring_element_uniform(seed: [u8; 34]) -> PolynomialRingElement {
+    let mut state = XOF::new(seed);
+    let randomness = XOF::squeeze_next_block(&mut state);
+
+    let mut out = PolynomialRingElement::ZERO;
+
+    let mut sampled = 0;
+    let mut done = rejection_sample_less_than_eta_equals_4(&randomness, &mut sampled, &mut out);
 
-    while sampled < COEFFICIENTS_IN_RING_ELEMENT {
+    while !done {
         let randomness = XOF::squeeze_next_block(&mut state);
-        sampled += sample_from_uniform_distribution_next(&randomness, &mut out);
+        done = rejection_sample_less_than_eta_equals_4(&randomness, &mut sampled, &mut out);
     }
 
     out