diff --git a/src/kem/kyber768/arithmetic.rs b/src/kem/kyber768/arithmetic.rs
index 74a6fd53a..4c131c46f 100644
--- a/src/kem/kyber768/arithmetic.rs
+++ b/src/kem/kyber768/arithmetic.rs
@@ -4,49 +4,31 @@ use crate::kem::kyber768::parameters::{COEFFICIENTS_IN_RING_ELEMENT, FIELD_MODUL
 
 pub(crate) type KyberFieldElement = i16;
 
-const BARRETT_SHIFT: u32 = 24; // 2 * ceil(log_2(FIELD_MODULUS))
-const BARRETT_MULTIPLIER: u64 = (1u64 << BARRETT_SHIFT) / (FIELD_MODULUS as u64);
+const BARRETT_SHIFT: i32 = 26;
+const BARRETT_R: i32 = 1i32 << BARRETT_SHIFT;
+const BARRETT_MULTIPLIER: i32 = 20159; // floor((BARRETT_R / FIELD_MODULUS) + 0.5)
 
-pub(crate) fn barrett_reduce(value: i32) -> KyberFieldElement {
-    let product: u64 = (value as u64) * BARRETT_MULTIPLIER;
-    let quotient: u32 = (product >> BARRETT_SHIFT) as u32;
+pub(crate) fn barrett_reduce(value: i16) -> KyberFieldElement {
+    let quotient = (i32::from(value) * BARRETT_MULTIPLIER) + (BARRETT_R >> 1);
+    let quotient = (quotient >> BARRETT_SHIFT) as i16;
 
-    // TODO: Justify in the comments (and subsequently in the proofs) that these
-    // operations do not lead to overflow/underflow.
-    let remainder = (value as u32) - (quotient * (FIELD_MODULUS as u32));
-    let remainder: i16 = remainder as i16;
-
-    let remainder_minus_modulus = remainder - FIELD_MODULUS;
-
-    // TODO: Check if LLVM detects this and optimizes it away into a
-    // conditional.
-    let selector = remainder_minus_modulus >> 15;
-
-    (selector & remainder) | (!selector & remainder_minus_modulus)
-}
-
-pub(crate) fn fe_add(lhs: KyberFieldElement, rhs: KyberFieldElement) -> KyberFieldElement {
-    let sum: i16 = lhs + rhs;
-    let difference: i16 = sum - FIELD_MODULUS;
-
-    let mask = difference >> 15;
-
-    (mask & sum) | (!mask & difference)
-}
-
-pub(crate) fn fe_sub(lhs: KyberFieldElement, rhs: KyberFieldElement) -> KyberFieldElement {
-    let difference = lhs - rhs;
-    let difference_plus_modulus: i16 = difference + FIELD_MODULUS;
-
-    let mask = difference >> 15;
-
-    (!mask & difference) | (mask & difference_plus_modulus)
+    value - (quotient * FIELD_MODULUS)
 }
 
 pub(crate) fn fe_mul(lhs: KyberFieldElement, rhs: KyberFieldElement) -> KyberFieldElement {
+    // TODO: This will shortly be replaced by an implementation of
+    // montgomery reduction.
     let product: i32 = i32::from(lhs) * i32::from(rhs);
 
-    barrett_reduce(product)
+    let reduced = (product % i32::from(FIELD_MODULUS)) as i16;
+
+    if reduced > FIELD_MODULUS / 2 {
+        reduced - FIELD_MODULUS
+    } else if reduced < -FIELD_MODULUS / 2 {
+        reduced + FIELD_MODULUS
+    } else {
+        reduced
+    }
 }
 
 #[derive(Debug, Clone, Copy, PartialEq, Eq)]
@@ -58,14 +40,6 @@ impl KyberPolynomialRingElement {
     pub const ZERO: Self = Self {
         coefficients: [0i16; COEFFICIENTS_IN_RING_ELEMENT],
     };
-
-    pub fn new(coefficients: [KyberFieldElement; COEFFICIENTS_IN_RING_ELEMENT]) -> Self {
-        Self { coefficients }
-    }
-
-    pub fn coefficients(&self) -> &[KyberFieldElement; COEFFICIENTS_IN_RING_ELEMENT] {
-        &self.coefficients
-    }
 }
 
 impl Index<usize> for KyberPolynomialRingElement {
@@ -97,7 +71,7 @@ impl ops::Add for KyberPolynomialRingElement {
     fn add(self, other: Self) -> Self {
         let mut result = KyberPolynomialRingElement::ZERO;
         for i in 0..COEFFICIENTS_IN_RING_ELEMENT {
-            result.coefficients[i] = fe_add(self.coefficients[i], other.coefficients[i]);
+            result.coefficients[i] = self.coefficients[i] + other.coefficients[i];
         }
         result
     }
@@ -109,7 +83,7 @@ impl ops::Sub for KyberPolynomialRingElement {
     fn sub(self, other: Self) -> Self {
         let mut result = KyberPolynomialRingElement::ZERO;
         for i in 0..COEFFICIENTS_IN_RING_ELEMENT {
-            result.coefficients[i] = fe_sub(self.coefficients[i], other.coefficients[i]);
+            result.coefficients[i] = self.coefficients[i] - other.coefficients[i];
         }
         result
     }
diff --git a/src/kem/kyber768/compress.rs b/src/kem/kyber768/compress.rs
index def07849a..e46858d33 100644
--- a/src/kem/kyber768/compress.rs
+++ b/src/kem/kyber768/compress.rs
@@ -1,26 +1,26 @@
 use crate::kem::kyber768::{
     arithmetic::{KyberFieldElement, KyberPolynomialRingElement},
-    parameters,
+    parameters::{self, FIELD_MODULUS},
 };
 
 pub fn compress(
-    re: KyberPolynomialRingElement,
+    mut re: KyberPolynomialRingElement,
     bits_per_compressed_coefficient: usize,
 ) -> KyberPolynomialRingElement {
-    KyberPolynomialRingElement::new(
-        re.coefficients()
-            .map(|coefficient| compress_q(coefficient, bits_per_compressed_coefficient)),
-    )
+    re.coefficients = re
+        .coefficients
+        .map(|coefficient| compress_q(coefficient, bits_per_compressed_coefficient));
+    re
 }
 
 pub fn decompress(
-    re: KyberPolynomialRingElement,
+    mut re: KyberPolynomialRingElement,
     bits_per_compressed_coefficient: usize,
 ) -> KyberPolynomialRingElement {
-    KyberPolynomialRingElement::new(
-        re.coefficients()
-            .map(|coefficient| decompress_q(coefficient, bits_per_compressed_coefficient)),
-    )
+    re.coefficients = re
+        .coefficients
+        .map(|coefficient| decompress_q(coefficient, bits_per_compressed_coefficient));
+    re
 }
 
 fn compress_q(fe: KyberFieldElement, to_bit_size: usize) -> KyberFieldElement {
@@ -28,7 +28,11 @@ fn compress_q(fe: KyberFieldElement, to_bit_size: usize) -> KyberFieldElement {
 
     let two_pow_bit_size = 1u32 << to_bit_size;
 
-    let mut compressed = (fe as u32) * (two_pow_bit_size << 1);
+    // Convert from canonical signed representative to canonical unsigned
+    // representative.
+    let fe_unsigned = fe + ((fe >> 15) & FIELD_MODULUS);
+
+    let mut compressed = (fe_unsigned as u32) * (two_pow_bit_size << 1);
     compressed += parameters::FIELD_MODULUS as u32;
     compressed /= (parameters::FIELD_MODULUS << 1) as u32;
 
diff --git a/src/kem/kyber768/ntt.rs b/src/kem/kyber768/ntt.rs
index 0e6d4a1a0..c45b78c80 100644
--- a/src/kem/kyber768/ntt.rs
+++ b/src/kem/kyber768/ntt.rs
@@ -1,36 +1,38 @@
-use crate::kem::kyber768::arithmetic::KyberPolynomialRingElement;
-use crate::kem::kyber768::parameters::RANK;
+use crate::kem::kyber768::{
+    arithmetic::{barrett_reduce, KyberPolynomialRingElement},
+    parameters::RANK,
+};
 
 use self::kyber_polynomial_ring_element_mod::ntt_multiply;
 
 pub(crate) mod kyber_polynomial_ring_element_mod {
     use crate::kem::kyber768::{
-        arithmetic::{fe_add, fe_mul, fe_sub, KyberFieldElement, KyberPolynomialRingElement},
-        parameters::{self, COEFFICIENTS_IN_RING_ELEMENT},
+        arithmetic::{barrett_reduce, fe_mul, KyberFieldElement, KyberPolynomialRingElement},
+        parameters::COEFFICIENTS_IN_RING_ELEMENT,
     };
 
     const ZETAS: [i16; 128] = [
-        1, 1729, 2580, 3289, 2642, 630, 1897, 848, 1062, 1919, 193, 797, 2786, 3260, 569, 1746,
-        296, 2447, 1339, 1476, 3046, 56, 2240, 1333, 1426, 2094, 535, 2882, 2393, 2879, 1974, 821,
-        289, 331, 3253, 1756, 1197, 2304, 2277, 2055, 650, 1977, 2513, 632, 2865, 33, 1320, 1915,
-        2319, 1435, 807, 452, 1438, 2868, 1534, 2402, 2647, 2617, 1481, 648, 2474, 3110, 1227, 910,
-        17, 2761, 583, 2649, 1637, 723, 2288, 1100, 1409, 2662, 3281, 233, 756, 2156, 3015, 3050,
-        1703, 1651, 2789, 1789, 1847, 952, 1461, 2687, 939, 2308, 2437, 2388, 733, 2337, 268, 641,
-        1584, 2298, 2037, 3220, 375, 2549, 2090, 1645, 1063, 319, 2773, 757, 2099, 561, 2466, 2594,
-        2804, 1092, 403, 1026, 1143, 2150, 2775, 886, 1722, 1212, 1874, 1029, 2110, 2935, 885,
-        2154,
+        1, -1600, -749, -40, -687, 630, -1432, 848, 1062, -1410, 193, 797, -543, -69, 569, -1583,
+        296, -882, 1339, 1476, -283, 56, -1089, 1333, 1426, -1235, 535, -447, -936, -450, -1355,
+        821, 289, 331, -76, -1573, 1197, -1025, -1052, -1274, 650, -1352, -816, 632, -464, 33,
+        1320, -1414, -1010, 1435, 807, 452, 1438, -461, 1534, -927, -682, -712, 1481, 648, -855,
+        -219, 1227, 910, 17, -568, 583, -680, 1637, 723, -1041, 1100, 1409, -667, -48, 233, 756,
+        -1173, -314, -279, -1626, 1651, -540, -1540, -1482, 952, 1461, -642, 939, -1021, -892,
+        -941, 733, -992, 268, 641, 1584, -1031, -1292, -109, 375, -780, -1239, 1645, 1063, 319,
+        -556, 757, -1230, 561, -863, -735, -525, 1092, 403, 1026, 1143, -1179, -554, 886, -1607,
+        1212, -1455, 1029, -1219, -394, 885, -1175,
     ];
 
     const MOD_ROOTS: [i16; 128] = [
-        17, 3312, 2761, 568, 583, 2746, 2649, 680, 1637, 1692, 723, 2606, 2288, 1041, 1100, 2229,
-        1409, 1920, 2662, 667, 3281, 48, 233, 3096, 756, 2573, 2156, 1173, 3015, 314, 3050, 279,
-        1703, 1626, 1651, 1678, 2789, 540, 1789, 1540, 1847, 1482, 952, 2377, 1461, 1868, 2687,
-        642, 939, 2390, 2308, 1021, 2437, 892, 2388, 941, 733, 2596, 2337, 992, 268, 3061, 641,
-        2688, 1584, 1745, 2298, 1031, 2037, 1292, 3220, 109, 375, 2954, 2549, 780, 2090, 1239,
-        1645, 1684, 1063, 2266, 319, 3010, 2773, 556, 757, 2572, 2099, 1230, 561, 2768, 2466, 863,
-        2594, 735, 2804, 525, 1092, 2237, 403, 2926, 1026, 2303, 1143, 2186, 2150, 1179, 2775, 554,
-        886, 2443, 1722, 1607, 1212, 2117, 1874, 1455, 1029, 2300, 2110, 1219, 2935, 394, 885,
-        2444, 2154, 1175,
+        17, -17, -568, 568, 583, -583, -680, 680, 1637, -1637, 723, -723, -1041, 1041, 1100, -1100,
+        1409, -1409, -667, 667, -48, 48, 233, -233, 756, -756, -1173, 1173, -314, 314, -279, 279,
+        -1626, 1626, 1651, -1651, -540, 540, -1540, 1540, -1482, 1482, 952, -952, 1461, -1461,
+        -642, 642, 939, -939, -1021, 1021, -892, 892, -941, 941, 733, -733, -992, 992, 268, -268,
+        641, -641, 1584, -1584, -1031, 1031, -1292, 1292, -109, 109, 375, -375, -780, 780, -1239,
+        1239, 1645, -1645, 1063, -1063, 319, -319, -556, 556, 757, -757, -1230, 1230, 561, -561,
+        -863, 863, -735, 735, -525, 525, 1092, -1092, 403, -403, 1026, -1026, 1143, -1143, -1179,
+        1179, -554, 554, 886, -886, -1607, 1607, 1212, -1212, -1455, 1455, 1029, -1029, -1219,
+        1219, -394, 394, 885, -885, -1175, 1175,
     ];
 
     const NTT_LAYERS: [usize; 7] = [2, 4, 8, 16, 32, 64, 128];
@@ -43,21 +45,18 @@ pub(crate) mod kyber_polynomial_ring_element_mod {
 
                 for j in offset..offset + layer {
                     let t = fe_mul(re[j + layer], ZETAS[zeta_i]);
-                    re[j + layer] = fe_sub(re[j], t);
-                    re[j] = fe_add(re[j], t);
+                    re[j + layer] = re[j] - t;
+                    re[j] += t;
                 }
             }
         }
+        re.coefficients = re.coefficients.map(barrett_reduce);
+
         re
     }
 
-    pub fn invert_ntt(re: KyberPolynomialRingElement) -> KyberPolynomialRingElement {
-        let inverse_of_2: i16 = (parameters::FIELD_MODULUS + 1) >> 1;
-
-        let mut out = KyberPolynomialRingElement::ZERO;
-        for i in 0..COEFFICIENTS_IN_RING_ELEMENT {
-            out[i] = re[i];
-        }
+    pub fn invert_ntt(mut re: KyberPolynomialRingElement) -> KyberPolynomialRingElement {
+        let inverse_of_2: i16 = -1664;
 
         let mut zeta_i = COEFFICIENTS_IN_RING_ELEMENT / 2;
 
@@ -66,14 +65,15 @@ pub(crate) mod kyber_polynomial_ring_element_mod {
                 zeta_i -= 1;
 
                 for j in offset..offset + layer {
-                    let a_minus_b = fe_sub(out[j + layer], out[j]);
-                    out[j] = fe_mul(fe_add(out[j], out[j + layer]), inverse_of_2);
-                    out[j + layer] = fe_mul(fe_mul(a_minus_b, ZETAS[zeta_i]), inverse_of_2);
+                    let a_minus_b = re[j + layer] - re[j];
+                    re[j] = fe_mul(re[j] + re[j + layer], inverse_of_2);
+                    re[j + layer] = fe_mul(fe_mul(a_minus_b, ZETAS[zeta_i]), inverse_of_2);
                 }
             }
         }
+        re.coefficients = re.coefficients.map(barrett_reduce);
 
-        out
+        re
     }
 
     fn ntt_multiply_binomials(
@@ -82,8 +82,8 @@ pub(crate) mod kyber_polynomial_ring_element_mod {
         zeta: i16,
     ) -> (KyberFieldElement, KyberFieldElement) {
         (
-            fe_add(fe_mul(a0, b0), fe_mul(fe_mul(a1, b1), zeta)),
-            fe_add(fe_mul(a0, b1), fe_mul(a1, b0)),
+            fe_mul(a0, b0) + fe_mul(fe_mul(a1, b1), zeta),
+            fe_mul(a0, b1) + fe_mul(a1, b0),
         )
     }
 
@@ -110,6 +110,8 @@ pub(crate) mod kyber_polynomial_ring_element_mod {
             out[i + 2] = product.0;
             out[i + 3] = product.1;
         }
+        out.coefficients = out.coefficients.map(barrett_reduce);
+
         out
     }
 }
@@ -125,7 +127,9 @@ pub(crate) fn multiply_matrix_by_column(
             let product = ntt_multiply(matrix_element, &vector[j]);
             result[i] = result[i] + product;
         }
+        result[i].coefficients = result[i].coefficients.map(barrett_reduce);
     }
+
     result
 }
 
@@ -139,5 +143,7 @@ pub(crate) fn multiply_row_by_column(
         result = result + ntt_multiply(row_element, column_element);
     }
 
+    result.coefficients = result.coefficients.map(barrett_reduce);
+
     result
 }
diff --git a/src/kem/kyber768/sampling.rs b/src/kem/kyber768/sampling.rs
index ac21aaebc..996d05fa7 100644
--- a/src/kem/kyber768/sampling.rs
+++ b/src/kem/kyber768/sampling.rs
@@ -1,5 +1,5 @@
 use crate::kem::kyber768::{
-    arithmetic::{fe_sub, KyberPolynomialRingElement},
+    arithmetic::KyberPolynomialRingElement,
     parameters::{COEFFICIENTS_IN_RING_ELEMENT, FIELD_MODULUS, REJECTION_SAMPLING_SEED_SIZE},
     BadRejectionSamplingRandomnessError,
 };
@@ -11,21 +11,19 @@ pub fn sample_from_uniform_distribution(
     let mut out: KyberPolynomialRingElement = KyberPolynomialRingElement::ZERO;
 
     for bytes in randomness.chunks(3) {
-        let b = i16::from(bytes[0]);
-        let b1 = i16::from(bytes[1]);
-        let b2 = i16::from(bytes[2]);
+        let b1 = i16::from(bytes[0]);
+        let b2 = i16::from(bytes[1]);
+        let b3 = i16::from(bytes[2]);
 
-        let d1 = b + (256 * (b1 % 16));
-
-        // Integer division is flooring in Rust.
-        let d2 = (b1 / 16) + (16 * b2);
+        let d1 = ((b2 & 0xF) << 8) | b1;
+        let d2 = (b3 << 4) | (b2 >> 4);
 
         if d1 < FIELD_MODULUS && sampled_coefficients < COEFFICIENTS_IN_RING_ELEMENT {
-            out[sampled_coefficients] = d1 as i16;
+            out[sampled_coefficients] = d1;
             sampled_coefficients += 1
         }
         if d2 < FIELD_MODULUS && sampled_coefficients < COEFFICIENTS_IN_RING_ELEMENT {
-            out[sampled_coefficients] = d2 as i16;
+            out[sampled_coefficients] = d2;
             sampled_coefficients += 1;
         }
 
@@ -89,7 +87,7 @@ pub fn sample_from_binomial_distribution_2(randomness: [u8; 128]) -> KyberPolyno
             let outcome_2 = ((coin_toss_outcomes >> (outcome_set + 2)) & 0x3) as i16;
 
             let offset = (outcome_set >> 2) as usize;
-            sampled[8 * chunk_number + offset] = fe_sub(outcome_1, outcome_2);
+            sampled[8 * chunk_number + offset] = outcome_1 - outcome_2;
         }
     }
 
diff --git a/src/kem/kyber768/serialize.rs b/src/kem/kyber768/serialize.rs
index 962261d6f..576374864 100644
--- a/src/kem/kyber768/serialize.rs
+++ b/src/kem/kyber768/serialize.rs
@@ -1,6 +1,6 @@
 use crate::kem::kyber768::{
     arithmetic::KyberPolynomialRingElement,
-    parameters::{BYTES_PER_RING_ELEMENT, COEFFICIENTS_IN_RING_ELEMENT},
+    parameters::{BYTES_PER_RING_ELEMENT, COEFFICIENTS_IN_RING_ELEMENT, FIELD_MODULUS},
 };
 
 /// This file contains instantiations of the functions
@@ -135,8 +135,11 @@ pub fn serialize_little_endian_12(re: KyberPolynomialRingElement) -> [u8; BYTES_
     let mut serialized = [0u8; BYTES_PER_RING_ELEMENT];
 
     for (i, chunks) in re.coefficients.chunks_exact(2).enumerate() {
-        let coefficient1 = chunks[0];
-        let coefficient2 = chunks[1];
+        let mut coefficient1 = chunks[0];
+        coefficient1 += (coefficient1 >> 15) & FIELD_MODULUS;
+
+        let mut coefficient2 = chunks[1];
+        coefficient2 += (coefficient2 >> 15) & FIELD_MODULUS;
 
         serialized[3 * i] = (coefficient1 & 0xFF) as u8;
         serialized[3 * i + 1] = ((coefficient1 >> 8) | ((coefficient2 & 0xF) << 4)) as u8;
diff --git a/tests/kyber768.rs b/tests/kyber768.rs
index 6a7cdb8f2..2a37115a4 100644
--- a/tests/kyber768.rs
+++ b/tests/kyber768.rs
@@ -27,7 +27,10 @@ fn modified_ciphertext() {
     let mut drbg = Drbg::new(digest::Algorithm::Sha256).unwrap();
 
     let random_u32 = drbg.next_u32();
-    let random_byte: u8 = (random_u32 & 0xFF).try_into().unwrap();
+    let mut random_byte: u8 = (random_u32 & 0xFF).try_into().unwrap();
+    if random_byte == 0 {
+        random_byte += 1;
+    }
 
     let ciphertext_position: usize = (random_u32 % 1088).try_into().unwrap();
 
@@ -53,7 +56,10 @@ fn modified_secret_key() {
 
     let random_u32 = drbg.next_u32();
 
-    let random_byte: u8 = (random_u32 & 0xFF).try_into().unwrap();
+    let mut random_byte: u8 = (random_u32 & 0xFF).try_into().unwrap();
+    if random_byte == 0 {
+        random_byte += 1;
+    }
 
     let secret_key_position: usize = ((random_u32 >> 8) % (2400 - 32)).try_into().unwrap();
 
@@ -78,11 +84,21 @@ fn modified_ciphertext_and_implicit_rejection_value() {
     let mut drbg = Drbg::new(digest::Algorithm::Sha256).unwrap();
 
     let random_u32 = drbg.next_u32();
-    let random_byte_for_ciphertext: u8 = (random_u32 & 0xFF).try_into().unwrap();
+
+    let mut random_byte_for_ciphertext: u8 = (random_u32 & 0xFF).try_into().unwrap();
+    if random_byte_for_ciphertext == 0 {
+        random_byte_for_ciphertext += 1;
+    }
+
     let ciphertext_position: usize = ((random_u32 >> 8) % 1088).try_into().unwrap();
 
     let random_u32 = drbg.next_u32();
-    let random_byte_for_secret_key: u8 = (random_u32 & 0xFF).try_into().unwrap();
+
+    let mut random_byte_for_secret_key: u8 = (random_u32 & 0xFF).try_into().unwrap();
+    if random_byte_for_secret_key == 0 {
+        random_byte_for_secret_key += 1;
+    }
+
     let secret_key_position: usize = ((random_u32 >> 8) % 32).try_into().unwrap();
 
     if let Ok((mut secret_key, public_key)) = kem::key_gen(Algorithm::Kyber768, &mut drbg) {