diff --git a/src/encryption/symmetric/aes/mod.rs b/src/encryption/symmetric/aes/mod.rs
index ba0c1ec..c16347c 100644
--- a/src/encryption/symmetric/aes/mod.rs
+++ b/src/encryption/symmetric/aes/mod.rs
@@ -220,7 +220,6 @@ where [(); N / 8]:
         .try_into()
         .unwrap(),
     );
-    assert!(state != State::default(), "State is not instantiated");
 
     // Round 0 - add round key
     Self::add_round_key(&mut state, round_keys.next().unwrap());
diff --git a/src/encryption/symmetric/modes/README.md b/src/encryption/symmetric/modes/README.md
index f77b9b4..56cb671 100644
--- a/src/encryption/symmetric/modes/README.md
+++ b/src/encryption/symmetric/modes/README.md
@@ -35,6 +35,68 @@ IV4["IV||2"]-->Fk2[F_k]-->xor2["⨁"]-->c2
 m2-->xor2
 ```
 
+## GCM: Galois/Counter Mode
+
+GCM is a block cipher mode of operation that provides both confidentiality and authentication.
+To provide confidentiality, it uses Counter(CTR) mode for encryption and decryption.
+To provide authentication, it uses a universal hash function, GHASH.
+Authentication is provided not only for confidential data but also other associated data.
+
+In this section, we will give an informal overview of GCM mode for 128-bit block cipher.
+*To see the formal definition of the operations of GCM, I recommend reading the original paper. [The Galois/Counter Mode of Operation (GCM)](https://csrc.nist.rip/groups/ST/toolkit/BCM/documents/proposedmodes/gcm/gcm-revised-spec.pdf)*
+
+The two operations of GCM are authenticated encryption and authenticated decryption.
+
+Here is a figure that gives a complete overview of the authenticated encryption in GCM. 
+
+In the figure, we have taken
+- the size of plaintext is `3 * 128-bit = 384 bits or 48 bytes`, 
+- Additionally Authenticated Data(AAD) is of `2 * 128-bit = 248 bits or 32 bytes`.
+
+![GCM](./figure_full_gcm.svg)
+*Note: The yellow diamonds represent functions/algorithms, the small rectangle with a blue outline represents 128-bit blocks.*
+Also,
+- *Enc(K)*: The encryption operation of the cipher used, for example AES, under the key K.
+- *incr*: The increment function, which treats the rightmost 32-bit of the block as an unsigned integer and increments it by 1.
+
+If you look at the figure carefully, you will notice that the GCM mode is composed of two main parts:
+- Encryption: This part is the same as the CTR mode of operation, with minor changes to the counter block.
+- Authentication: In this part, we generate an authentication tag for the ciphertext and some additional data, which we refer to as Additionally Authenticated Data(AAD).
+
+The counter block is the same as in CTR mode. In general, it can be thought of as a 96-bit nonce value followed by a 32-bit counter value.
+
+The tag is generated by XOR of:
+1. Hash of ciphertext and AAD, using GHASH algorithm
+2. Encryption of Counter block 0.
+
+### GHASH
+
+The GHASH algorithm can be viewed as a series of `ADD and MULTIPLY` in $GF(2^{128})$. Mathematically put the basic operation of GHASH is,
+
+$$
+X_{i} =
+\begin{cases}
+0  & \quad i = 0 \\
+( X_{i-1} \oplus B_{i} ) * H & \quad \text{otherwise}
+\end{cases}
+$$
+
+$B_{i}$ represents blocks of AAD followed by blocks of ciphertext followed by a special length block.
+The length block consists of 64-bit lengths(in bits) of AAD and ciphertext.
+$H$ called the hash key, is the encryption of 128-bit of zeros using the chosen cipher and key.
+
+The interesting thing to note here is that the multiplication($*$) and addition($\oplus$) are operations of the Galois(finite) field of order $2^{128}$. 
+A brief summary of finite field arithmetic,
+- The elements of the field are represented as polynomials. Each bit of the 128-bit block represents coefficients of a polynomial of degree strictly less than 128.
+- Addition in a finite field is equivalent to bitwise XOR.
+- Multiplication in a finite field is the multiplication of corresponding polynomials modulo an irreducible reducing polynomial.
+
+In GCM the reducing polynomial is $f = 1 + x + x^2 + x^7 + x^{128}$
+
+If you want to read about Finite Field, the Wikipedia article on [Finite Field Arithemtic](https://en.wikipedia.org/wiki/Galois/Counter_Mode) is pretty good!
+
+The authenticated decryption operation is identical to authenticated encryption, except the tag is generated before the decryption.
+
 ## Next Steps
 Implement more modes, and subsequent attacks/vulnerabilities:
 - [ ] CFB
diff --git a/src/encryption/symmetric/modes/ctr.rs b/src/encryption/symmetric/modes/ctr.rs
index 4864c1b..132f142 100644
--- a/src/encryption/symmetric/modes/ctr.rs
+++ b/src/encryption/symmetric/modes/ctr.rs
@@ -2,20 +2,22 @@
 
 use crate::encryption::symmetric::{counter::Counter, BlockCipher};
 
-/// [`BlockCipher`] counter mode of operation
-pub struct CTR<C: BlockCipher>
-where [(); C::BLOCK_SIZE / 2]: {
-  nonce: [u8; C::BLOCK_SIZE / 2],
+/// [`BlockCipher`] counter mode of operation with two parameters:
+/// - `C`, a cipher that implements the `BlockCipher` trait.
+/// - `M`, a usize const that indicates the size of counter in bytes.
+pub struct CTR<C: BlockCipher, const M: usize>
+where [(); C::BLOCK_SIZE - M]: {
+  nonce: [u8; C::BLOCK_SIZE - M],
 }
 
-impl<C: BlockCipher> CTR<C>
-where [(); C::BLOCK_SIZE / 2]:
+impl<C: BlockCipher, const M: usize> CTR<C, M>
+where [(); C::BLOCK_SIZE - M]:
 {
   /// Create a CTR mode of operation object
   /// # Arguments
-  /// - `nonce`: *Non-repeating* IV of [`BlockCipher::BLOCK_SIZE`]/2 bytes. Nonce is concatenated
-  ///   with [`Counter`] to generate a keystream that does not repeat for a long time.
-  pub fn new(nonce: [u8; C::BLOCK_SIZE / 2]) -> Self { Self { nonce } }
+  /// - `nonce`: *Non-repeating* IV of [`BlockCipher::BLOCK_SIZE`] - M bytes. Nonce is concatenated
+  ///   with [`Counter`], of M bytes, to generate a keystream that does not repeat for a long time.
+  pub fn new(nonce: [u8; C::BLOCK_SIZE - M]) -> Self { Self { nonce } }
 
   /// Encrypt a plaintext with [`BlockCipher::Key`] and [`Counter`]
   /// ## Arguments
@@ -37,10 +39,10 @@ where [(); C::BLOCK_SIZE / 2]:
   /// let mut rng = thread_rng();
   /// let rand_key: [u8; 16] = rng.gen();
   /// let key = Key::<128>::new(rand_key);
-  /// let nonce: [u8; AES::<128>::BLOCK_SIZE / 2] = rng.gen();
-  /// let counter: Counter<{ AES::<128>::BLOCK_SIZE / 2 }> = Counter::from(0);
+  /// let nonce: [u8; 12] = rng.gen();
+  /// let counter: Counter<4> = Counter::from(0);
   ///
-  /// let ctr = CTR::<AES<128>>::new(nonce);
+  /// let ctr = CTR::<AES<128>, 4>::new(nonce);
   /// let plaintext = b"Hello World!";
   ///
   /// let ciphertext = ctr.encrypt(&key, &counter, plaintext).unwrap();
@@ -48,7 +50,7 @@ where [(); C::BLOCK_SIZE / 2]:
   pub fn encrypt(
     &self,
     key: &C::Key,
-    counter: &Counter<{ C::BLOCK_SIZE / 2 }>,
+    counter: &Counter<M>,
     plaintext: &[u8],
   ) -> Result<Vec<u8>, String> {
     let mut ciphertext = Vec::new();
@@ -56,8 +58,8 @@ where [(); C::BLOCK_SIZE / 2]:
 
     for chunk in plaintext.chunks(C::BLOCK_SIZE) {
       let mut block = [0u8; C::BLOCK_SIZE];
-      block[..C::BLOCK_SIZE / 2].copy_from_slice(&self.nonce);
-      block[C::BLOCK_SIZE / 2..].copy_from_slice(&cipher_counter.0);
+      block[..{ C::BLOCK_SIZE - M }].copy_from_slice(&self.nonce);
+      block[{ C::BLOCK_SIZE - M }..].copy_from_slice(&cipher_counter.0);
       cipher_counter.increment()?;
 
       let encrypted = C::encrypt_block(key, &C::Block::from(block.to_vec()));
@@ -70,7 +72,7 @@ where [(); C::BLOCK_SIZE / 2]:
     Ok(ciphertext)
   }
 
-  /// Decrypt a ciphertext with counter of size [`BlockCipher::BLOCK_SIZE`]/2 bytes
+  /// Decrypt a ciphertext with counter of size [`BlockCipher::BLOCK_SIZE`] - M bytes
   /// ## Usage
   /// ```
   /// #![allow(incomplete_features)]
@@ -86,10 +88,10 @@ where [(); C::BLOCK_SIZE / 2]:
   /// let mut rng = thread_rng();
   /// let rand_key: [u8; 16] = rng.gen();
   /// let key = Key::<128>::new(rand_key);
-  /// let nonce: [u8; AES::<128>::BLOCK_SIZE / 2] = rng.gen();
-  /// let counter: Counter<{ AES::<128>::BLOCK_SIZE / 2 }> = Counter::from(0);
+  /// let nonce: [u8; 12] = rng.gen();
+  /// let counter: Counter<4> = Counter::from(0);
   ///
-  /// let ctr = CTR::<AES<128>>::new(nonce);
+  /// let ctr = CTR::<AES<128>, 4>::new(nonce);
   /// let plaintext = b"Hello World!";
   ///
   /// let ciphertext = ctr.encrypt(&key, &counter, plaintext).unwrap();
@@ -98,7 +100,7 @@ where [(); C::BLOCK_SIZE / 2]:
   pub fn decrypt(
     &self,
     key: &C::Key,
-    counter: &Counter<{ C::BLOCK_SIZE / 2 }>,
+    counter: &Counter<M>,
     ciphertext: &[u8],
   ) -> Result<Vec<u8>, String> {
     self.encrypt(key, counter, ciphertext)
@@ -107,6 +109,8 @@ where [(); C::BLOCK_SIZE / 2]:
 
 #[cfg(test)]
 mod tests {
+  use std::{fmt::Write, num::ParseIntError};
+
   use rand::{thread_rng, Rng};
   use rstest::{fixture, rstest};
 
@@ -127,13 +131,13 @@ mod tests {
   }
 
   #[rstest]
-  fn ctr(rand_key: Key<128>) {
+  fn test_ctr_rand_key(rand_key: Key<128>) {
     for _ in 0..10 {
       let mut rng = thread_rng();
-      let nonce: [u8; AES::<128>::BLOCK_SIZE / 2] = rng.gen();
-      let counter: Counter<{ AES::<128>::BLOCK_SIZE / 2 }> = Counter::from(0);
+      let nonce: [u8; AES::<128>::BLOCK_SIZE - 4] = rng.gen();
+      let counter: Counter<4> = Counter::from(0);
 
-      let ctr = CTR::<AES<128>>::new(nonce);
+      let ctr = CTR::<AES<128>, 4>::new(nonce);
 
       let plaintext = rand_message(rng.gen_range(1000..10000));
       let ciphertext = ctr.encrypt(&rand_key, &counter, &plaintext).unwrap();
@@ -143,4 +147,127 @@ mod tests {
       assert_eq!(plaintext, decrypted);
     }
   }
+
+  /// Encode bytes to hex
+  pub fn encode_hex(bytes: &[u8]) -> String {
+    let mut s = String::with_capacity(bytes.len() * 2);
+    for &b in bytes {
+      write!(&mut s, "{:02x}", b).unwrap();
+    }
+    s
+  }
+
+  /// Decode hex to bytes
+  pub fn decode_hex(s: &str) -> Result<Vec<u8>, ParseIntError> {
+    (0..s.len()).step_by(2).map(|i| u8::from_str_radix(&s[i..i + 2], 16)).collect()
+  }
+
+  // `test_ctr_128`, `test_ctr_192`, and 'test_ctr_256' are based on test vectors given in:
+  // "Recommendation for Block Cipher Modes of Operation(NIST Special Publication 800-38A)"
+  // Link: (https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication800-38a.pdf)
+
+  // Appendix F.5.1 and F.5.2
+  #[rstest]
+  #[case(
+    "2b7e151628aed2a6abf7158809cf4f3c",
+    "f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff",
+    "6bc1bee22e409f96e93d7e117393172aae2d8a571e03ac9c9eb76fac45af8e5130c81c46a35ce411e5fbc1191a0a52eff69f2445df4f9b17ad2b417be66c3710",
+    "874d6191b620e3261bef6864990db6ce9806f66b7970fdff8617187bb9fffdff5ae4df3edbd5d35e5b4f09020db03eab1e031dda2fbe03d1792170a0f3009cee"
+  )]
+  fn test_ctr_128(
+    #[case] kx: &str,
+    #[case] ivx: &str,
+    #[case] ptx: &str,
+    #[case] expected_ctx: &str,
+  ) {
+    let k = decode_hex(kx).unwrap();
+    let iv = decode_hex(ivx).unwrap();
+    let pt = decode_hex(ptx).unwrap();
+
+    let key = Key::<128>::new(k.try_into().unwrap());
+    let nonce = &iv[..8];
+    let counter = Counter(iv[8..].try_into().unwrap());
+
+    let ctr = CTR::<AES<128>, 8>::new(nonce.try_into().unwrap());
+
+    let ct = ctr.encrypt(&key, &counter, &pt).unwrap();
+
+    let ctx = encode_hex(&ct);
+    assert_eq!(ctx, expected_ctx);
+
+    let _pt = ctr.decrypt(&key, &counter, &ct).unwrap();
+
+    let _ptx = encode_hex(&_pt);
+    assert_eq!(_ptx, ptx);
+  }
+
+  // Appendix F.5.3 and F.5.4
+  #[rstest]
+  #[case(
+    "8e73b0f7da0e6452c810f32b809079e562f8ead2522c6b7b",
+    "f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff",
+    "6bc1bee22e409f96e93d7e117393172aae2d8a571e03ac9c9eb76fac45af8e5130c81c46a35ce411e5fbc1191a0a52eff69f2445df4f9b17ad2b417be66c3710",
+    "1abc932417521ca24f2b0459fe7e6e0b090339ec0aa6faefd5ccc2c6f4ce8e941e36b26bd1ebc670d1bd1d665620abf74f78a7f6d29809585a97daec58c6b050"
+  )]
+  fn test_ctr_192(
+    #[case] kx: &str,
+    #[case] ivx: &str,
+    #[case] ptx: &str,
+    #[case] expected_ctx: &str,
+  ) {
+    let k = decode_hex(kx).unwrap();
+    let iv = decode_hex(ivx).unwrap();
+    let pt = decode_hex(ptx).unwrap();
+
+    let key = Key::<192>::new(k.try_into().unwrap());
+    let nonce = &iv[..8];
+    let counter = Counter(iv[8..].try_into().unwrap());
+
+    let ctr = CTR::<AES<192>, 8>::new(nonce.try_into().unwrap());
+
+    let ct = ctr.encrypt(&key, &counter, &pt).unwrap();
+
+    let ctx = encode_hex(&ct);
+    assert_eq!(ctx, expected_ctx);
+
+    let _pt = ctr.decrypt(&key, &counter, &ct).unwrap();
+
+    let _ptx = encode_hex(&_pt);
+    assert_eq!(_ptx, ptx);
+  }
+
+  // Appendix F.5.3 and F.5.4
+  #[rstest]
+  #[case(
+    "603deb1015ca71be2b73aef0857d77811f352c073b6108d72d9810a30914dff4",
+    "f0f1f2f3f4f5f6f7f8f9fafbfcfdfeff",
+    "6bc1bee22e409f96e93d7e117393172aae2d8a571e03ac9c9eb76fac45af8e5130c81c46a35ce411e5fbc1191a0a52eff69f2445df4f9b17ad2b417be66c3710",
+    "601ec313775789a5b7a7f504bbf3d228f443e3ca4d62b59aca84e990cacaf5c52b0930daa23de94ce87017ba2d84988ddfc9c58db67aada613c2dd08457941a6"
+  )]
+  fn test_ctr_256(
+    #[case] kx: &str,
+    #[case] ivx: &str,
+    #[case] ptx: &str,
+    #[case] expected_ctx: &str,
+  ) {
+    let k = decode_hex(kx).unwrap();
+    let iv = decode_hex(ivx).unwrap();
+    let pt = decode_hex(ptx).unwrap();
+
+    let key = Key::<256>::new(k.try_into().unwrap());
+    let nonce = &iv[..8];
+    let counter = Counter(iv[8..].try_into().unwrap());
+
+    let ctr = CTR::<AES<256>, 8>::new(nonce.try_into().unwrap());
+
+    let ct = ctr.encrypt(&key, &counter, &pt).unwrap();
+
+    let ctx = encode_hex(&ct);
+    assert_eq!(ctx, expected_ctx);
+
+    let _pt = ctr.decrypt(&key, &counter, &ct).unwrap();
+
+    let _ptx = encode_hex(&_pt);
+    assert_eq!(_ptx, ptx);
+  }
 }
diff --git a/src/encryption/symmetric/modes/figure_full_gcm.svg b/src/encryption/symmetric/modes/figure_full_gcm.svg
new file mode 100644
index 0000000..17a378c
--- /dev/null
+++ b/src/encryption/symmetric/modes/figure_full_gcm.svg
@@ -0,0 +1,13 @@
+<svg version="1.1" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1618.333333333334 1053.3333333333333" width="1618.333333333334" height="1053.3333333333333">
+  <!-- svg-source:excalidraw -->
+  
+  <defs>
+    <style class="style-fonts">
+      @font-face {
+        font-family: Virgil;
+        src: 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35 C4.18 35.97, 1.6 32.52, 0 26.25 M8.75 35 C4.2 36.83, -1.11 33.36, 0 26.25 M0 26.25 C0.53 21.03, 0.36 13.81, 0 8.75 M0 26.25 C0.79 21.22, 0.26 17.7, 0 8.75 M0 8.75 C-1.48 3.71, 1.81 0.38, 8.75 0 M0 8.75 C0.34 4.58, 4.99 0.62, 8.75 0" stroke="#1971c2" stroke-width="2" fill="none"></path></g><g transform="translate(997.0702365693593 561.3690476190479) rotate(0 62.775001525878906 12.5)"><text x="62.775001525878906" y="17.619999999999997" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#1971c2" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">Ciphertext 2</text></g><g stroke-linecap="round" transform="translate(1336.4722222222222 556.7261904761908) rotate(0 99.44444444444446 17.5)"><path d="M8.75 0 C50.33 0.04, 94.83 0.52, 190.14 0 C193.11 3.23, 195.32 5, 198.89 8.75 C199.74 15.75, 198.59 21.39, 198.89 26.25 C198.16 33.63, 192.74 38.48, 190.14 35 C127.05 31.84, 62.72 28.96, 8.75 35 C5.03 35.71, -3.37 32.95, 0 26.25 C2.97 22.21, 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35 C3.65 34.28, 1.57 31.29, 0 26.25 M0 26.25 C1.53 21.54, -1.13 15.28, 0 8.75 M0 26.25 C0.7 19.97, -0.09 12.21, 0 8.75 M0 8.75 C-1.89 3.76, 3.36 1.19, 8.75 0 M0 8.75 C-0.43 2.03, 0.86 0.88, 8.75 0" stroke="#1971c2" stroke-width="2" fill="none"></path></g><g transform="translate(914.6380952380954 975.1190476190477) rotate(0 23.600000000000023 12.5)"><text x="23.6" y="17.619999999999997" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#1971c2" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">TAG</text></g><g transform="translate(823.3333333333335 18.333333333333314) rotate(0 169.03333333333336 17.5)"><text x="0" y="24.668" font-family="Virgil, Segoe UI Emoji" font-size="28px" fill="#e03131" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">Part 1: Encryption (CTR)</text></g><g transform="translate(53.33333333333337 974.1666666666665) rotate(0 220.63333333333333 17.5)"><text x="0" y="24.668" font-family="Virgil, Segoe UI Emoji" font-size="28px" fill="#2f9e44" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">Part 2: Authentication (GHASH)</text></g></svg>
\ No newline at end of file
diff --git a/src/encryption/symmetric/modes/gcm.rs b/src/encryption/symmetric/modes/gcm.rs
new file mode 100644
index 0000000..2a6dc9f
--- /dev/null
+++ b/src/encryption/symmetric/modes/gcm.rs
@@ -0,0 +1,372 @@
+//! Implementation of GCM cipher mode of operation based on NIST GCM specification.
+//! [The Galois/Counter Mode of Operation (GCM)](http://www.csrc.nist.gov/groups/ST/toolkit/BCM/documents/proposedmodes/gcm/gcm-revised-spec.pdf)
+
+use super::ctr::CTR;
+use crate::{
+  encryption::symmetric::{counter::Counter, BlockCipher},
+  hashes::ghash::GHASH,
+};
+
+/// GCM (Galois/Counter Mode) struct holding a GHASH instance and a block cipher key.
+///
+/// This structure implements encryption and decryption operations using the GCM
+/// authenticated encryption mode, combining the CTR (counter mode) for encryption
+/// and GHASH for authentication.
+///
+/// # Generics
+/// - `C`: A block cipher that implements the `BlockCipher` trait.
+pub struct GCM<C: BlockCipher> {
+  ghash: GHASH,
+  key:   C::Key,
+}
+
+impl<C: BlockCipher> GCM<C>
+where [(); C::BLOCK_SIZE - 4]:
+{
+  /// Constructs a new `GCM` instance with the given key.
+  ///
+  /// # Parameters
+  /// - `key`: A key compatible with the block cipher `C`.
+  ///
+  /// # Returns
+  /// A `GCM` instance ready for encryption or decryption.
+  pub fn new(key: C::Key) -> Self {
+    assert_eq!(C::BLOCK_SIZE, 16, "GCM only supports 128-bit block size.");
+    // The GHASH algorithms requires the encryption of 128-bits of zeros.
+    let zero_string = C::Block::from(vec![0u8; 16]);
+    let hash_key = C::encrypt_block(&key, &zero_string);
+    let ghash = GHASH::new(hash_key.as_ref());
+    GCM { ghash, key }
+  }
+
+  /// Encrypts the given plaintext using the specified nonce and additional authenticated data
+  /// (AAD).
+  ///
+  /// # Parameters
+  /// - `nonce`: A unique nonce for the encryption process (should be '12' bytes long).
+  /// - `plaintext`: The data to be encrypted.
+  /// - `aad`: Additional authenticated data (AAD) that will be included in the authentication tag
+  ///   but not encrypted.
+  ///
+  /// # Returns
+  /// A tuple `(ciphertext, tag)`:
+  /// - `ciphertext`: The encrypted data.
+  /// - `tag`: The authentication tag that verifies the integrity and authenticity of the ciphertext
+  ///   and AAD.
+  ///
+  /// A `String` error if encryption fails.
+  pub fn encrypt(
+    &self,
+    nonce: &[u8],
+    plaintext: &[u8],
+    aad: &[u8],
+  ) -> Result<(Vec<u8>, Vec<u8>), String> {
+    // The `initial_block` is the first block that is encrypted.
+    // This is used in the tag generation step.
+    // It consists of two parts, a 96-bit nonce value and 32-bit counter value, which is
+    // incremented for each block to be encrypted.
+    let mut initial_block;
+    // `counter_start` is the start value for the 32-bit counter.
+    let counter_start: [u8; 4];
+    // `new_nonce` is the 96-bit nonce value.
+    // That is C::BLOCK_SIZE(=16 bytes) - 4 bytes = 12 bytes = 96-bits!
+    // Also NOTE: the compiler is not happy if I write 12 bytes instead of `C::BLOCK_SIZE - 4`. :(
+    let new_nonce: [u8; C::BLOCK_SIZE - 4];
+
+    // The GCM specification recommends a 96-bit(or 12-byte) nonce, but it may not be 96-bit.
+    if nonce.len() != 12 {
+      // If the nonce is not 96-bit, we GHASH the nonce, which outputs a 128-bit value.
+      // The first 96-bit is used as new nonce(hence named `new_nonce`).
+      // The rest of the 32-bits are used as the start of counter value!
+      initial_block = self.ghash.digest(&[], nonce);
+      new_nonce = initial_block[..12].try_into().unwrap();
+      counter_start = initial_block[12..].try_into().unwrap();
+    } else {
+      new_nonce = nonce.try_into().unwrap();
+      // The counter starts with `1`.
+      counter_start = [0, 0, 0, 1];
+      initial_block = [0u8; 16];
+      initial_block[..12].copy_from_slice(&new_nonce);
+      initial_block[12..].copy_from_slice(&counter_start);
+    }
+
+    // Step 1: Increment the counter
+    let mut counter = Counter(counter_start);
+    counter.increment()?;
+
+    // Step 2: Encrypt the plaintext using the `CTR` object.
+    let ctr = CTR::<C, 4>::new(new_nonce.try_into().unwrap());
+    let ciphertext = ctr.encrypt(&self.key, &counter, plaintext)?;
+
+    // Step3: Generate Tag
+    // The tag is the XOR of the `initial_block` and ghash of ciphertext.
+    let y0_block = C::Block::from(initial_block.to_vec());
+    let y0_enc = C::encrypt_block(&self.key, &y0_block);
+    let hash = self.ghash.digest(aad, ciphertext.as_ref());
+    let mut tag = Vec::new();
+
+    // XOR the ghash of ciphertext and `initial_block`
+    for (x, y) in hash.iter().zip(y0_enc.as_ref()) {
+      tag.push(x ^ y);
+    }
+
+    Ok((ciphertext.to_vec(), tag))
+  }
+
+  /// Decrypt the given ciphertext using the specified nonce and additional authenticated data
+  /// (AAD).
+  ///
+  /// # Parameters
+  /// - `nonce`: A unique nonce used during encryption (should be 12 bytes long).
+  /// - `ciphertext`: The encrypted data to be decrypted.
+  /// - `aad`: Additional authenticated data (AAD) used during encryption.
+  ///
+  /// # Returns
+  /// A tuple `(plaintext, tag)`:
+  /// - `plaintext`: The decrypted data.
+  /// - `tag`: The authentication tag to verify the integrity and authenticity of the decrypted
+  ///   data.
+  ///
+  /// A `String` error if decryption fails.
+  pub fn decrypt(
+    &self,
+    nonce: &[u8],
+    ciphertext: &[u8],
+    aad: &[u8],
+  ) -> Result<(Vec<u8>, Vec<u8>), String> {
+    // The decryption algorithm is the same as the encryption algorithm but the tag is generated
+    // first and then is move on to the encryption.
+
+    let counter_start: [u8; 4];
+    let new_nonce: [u8; C::BLOCK_SIZE - 4];
+    let mut counter_block;
+
+    if nonce.len() != 12 {
+      counter_block = self.ghash.digest(&[], nonce);
+      new_nonce = counter_block[..12].try_into().unwrap();
+      counter_start = counter_block[12..].try_into().unwrap();
+    } else {
+      new_nonce = nonce.try_into().unwrap();
+      counter_start = [0, 0, 0, 1];
+      counter_block = [0u8; 16];
+      counter_block[..12].copy_from_slice(&new_nonce);
+      counter_block[12..].copy_from_slice(&counter_start);
+    }
+
+    // Step 1: Generate Tag (same as the encryption)
+    let y0 = C::Block::from(counter_block.to_vec());
+    let y0_enc = C::encrypt_block(&self.key, &y0);
+    let hash = self.ghash.digest(aad, ciphertext.as_ref());
+    let mut tag = Vec::new();
+
+    for (x, y) in hash.iter().zip(y0_enc.as_ref()) {
+      tag.push(x ^ y);
+    }
+
+    // Step 2: Increment Counter
+    let mut counter = Counter(counter_start);
+    counter.increment()?;
+
+    // Step 3: Decrypt ciphertext.
+    let ctr = CTR::<C, 4>::new(new_nonce.try_into().unwrap());
+    let plaintext = ctr.decrypt(&self.key, &counter, ciphertext)?;
+
+    Ok((plaintext.to_vec(), tag))
+  }
+}
+
+#[cfg(test)]
+mod tests {
+
+  use std::{fmt::Write, num::ParseIntError};
+
+  use rstest::rstest;
+
+  use super::*;
+  use crate::encryption::symmetric::aes::{Key, AES};
+
+  /// Encode bytes to hex
+  pub fn encode_hex(bytes: &[u8]) -> String {
+    let mut s = String::with_capacity(bytes.len() * 2);
+    for &b in bytes {
+      write!(&mut s, "{:02x}", b).unwrap();
+    }
+    s
+  }
+
+  pub fn decode_hex(s: &str) -> Result<Vec<u8>, ParseIntError> {
+    (0..s.len()).step_by(2).map(|i| u8::from_str_radix(&s[i..i + 2], 16)).collect()
+  }
+
+  #[rstest]
+  // NIST Test Case 1
+  #[case(
+    "00000000000000000000000000000000",
+    "000000000000000000000000",
+    "",
+    "",
+    "",
+    "58e2fccefa7e3061367f1d57a4e7455a"
+  )]
+  // NIST Test Case 2
+  #[case(
+    "00000000000000000000000000000000",
+    "000000000000000000000000",
+    "00000000000000000000000000000000",
+    "",
+    "0388dace60b6a392f328c2b971b2fe78",
+    "ab6e47d42cec13bdf53a67b21257bddf"
+  )]
+  // NIST Test Case 3
+  #[case(
+    "feffe9928665731c6d6a8f9467308308",
+    "cafebabefacedbaddecaf888",
+    "d9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b391aafd255",
+    "",
+    "42831ec2217774244b7221b784d0d49ce3aa212f2c02a4e035c17e2329aca12e21d514b25466931c7d8f6a5aac84aa051ba30b396a0aac973d58e091473f5985",
+    "4d5c2af327cd64a62cf35abd2ba6fab4"
+    )]
+  // NIST Test Case 4
+  #[case(
+    "feffe9928665731c6d6a8f9467308308",
+    "cafebabefacedbaddecaf888",
+    "d9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39",
+    "feedfacedeadbeeffeedfacedeadbeefabaddad2",
+    "42831ec2217774244b7221b784d0d49ce3aa212f2c02a4e035c17e2329aca12e21d514b25466931c7d8f6a5aac84aa051ba30b396a0aac973d58e091",
+    "5bc94fbc3221a5db94fae95ae7121a47"
+    )]
+  // NIST Test Case 5
+  #[case(
+    "feffe9928665731c6d6a8f9467308308",
+    "cafebabefacedbad",
+    "d9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39",
+    "feedfacedeadbeeffeedfacedeadbeefabaddad2",
+    "61353b4c2806934a777ff51fa22a4755699b2a714fcdc6f83766e5f97b6c742373806900e49f24b22b097544d4896b424989b5e1ebac0f07c23f4598",
+    "3612d2e79e3b0785561be14aaca2fccb"
+    )]
+  // NIST Test Case 6
+  #[case(
+    "feffe9928665731c6d6a8f9467308308",
+    "9313225df88406e555909c5aff5269aa6a7a9538534f7da1e4c303d2a318a728c3c0c95156809539fcf0e2429a6b525416aedbf5a0de6a57a637b39b",
+    "d9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39",
+    "feedfacedeadbeeffeedfacedeadbeefabaddad2",
+    "8ce24998625615b603a033aca13fb894be9112a5c3a211a8ba262a3cca7e2ca701e4a9a4fba43c90ccdcb281d48c7c6fd62875d2aca417034c34aee5",
+    "619cc5aefffe0bfa462af43c1699d050"
+    )]
+  fn test_gcm_128(
+    #[case] kx: &str,
+    #[case] ivx: &str,
+    #[case] ptx: &str,
+    #[case] aadx: &str,
+    #[case] expected_ctx: &str,
+    #[case] expected_tagx: &str,
+  ) {
+    let k = decode_hex(kx).unwrap();
+    let iv = decode_hex(ivx).unwrap();
+    let pt = decode_hex(ptx).unwrap();
+    let aad = decode_hex(aadx).unwrap();
+
+    let key = Key::<128>::new(k.try_into().unwrap());
+    let gcm = GCM::<AES<128>>::new(key);
+
+    let (ct, tag) = gcm.encrypt(&iv, &pt, &aad).unwrap();
+
+    let ctx = encode_hex(&ct);
+    assert_eq!(ctx, expected_ctx);
+
+    let tagx = encode_hex(&tag);
+    assert_eq!(tagx, expected_tagx);
+
+    let (_pt, _tag) = gcm.decrypt(&iv, &ct, &aad).unwrap();
+
+    let _ptx = encode_hex(&_pt);
+    assert_eq!(ptx, _ptx);
+
+    let _tagx = encode_hex(&_tag);
+    assert_eq!(_tagx, expected_tagx);
+  }
+
+  #[rstest]
+  // NIST Test Case 7
+  #[case(
+    "000000000000000000000000000000000000000000000000",
+    "000000000000000000000000",
+    "",
+    "",
+    "",
+    "cd33b28ac773f74ba00ed1f312572435"
+  )]
+  // NIST Test Case 8
+  #[case(
+    "000000000000000000000000000000000000000000000000",
+    "000000000000000000000000",
+    "00000000000000000000000000000000",
+    "",
+    "98e7247c07f0fe411c267e4384b0f600",
+    "2ff58d80033927ab8ef4d4587514f0fb"
+  )]
+  // NIST Test Case 9
+  #[case(
+    "feffe9928665731c6d6a8f9467308308feffe9928665731c",
+    "cafebabefacedbaddecaf888",
+    "d9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b391aafd255",
+    "",
+    "3980ca0b3c00e841eb06fac4872a2757859e1ceaa6efd984628593b40ca1e19c7d773d00c144c525ac619d18c84a3f4718e2448b2fe324d9ccda2710acade256",
+    "9924a7c8587336bfb118024db8674a14")]
+  // NIST Test Case 10
+  #[case(
+    "feffe9928665731c6d6a8f9467308308feffe9928665731c",
+    "cafebabefacedbaddecaf888",
+    "d9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39",
+    "feedfacedeadbeeffeedfacedeadbeefabaddad2",
+    "3980ca0b3c00e841eb06fac4872a2757859e1ceaa6efd984628593b40ca1e19c7d773d00c144c525ac619d18c84a3f4718e2448b2fe324d9ccda2710",
+    "2519498e80f1478f37ba55bd6d27618c")]
+  // NIST Test Case 11
+  #[case(
+    "feffe9928665731c6d6a8f9467308308feffe9928665731c",
+    "cafebabefacedbad",
+    "d9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39",
+    "feedfacedeadbeeffeedfacedeadbeefabaddad2",
+    "0f10f599ae14a154ed24b36e25324db8c566632ef2bbb34f8347280fc4507057fddc29df9a471f75c66541d4d4dad1c9e93a19a58e8b473fa0f062f7",
+    "65dcc57fcf623a24094fcca40d3533f8")]
+  // NIST Test Case 12
+  #[case(
+    "feffe9928665731c6d6a8f9467308308feffe9928665731c",
+    "9313225df88406e555909c5aff5269aa6a7a9538534f7da1e4c303d2a318a728c3c0c95156809539fcf0e2429a6b525416aedbf5a0de6a57a637b39b",
+    "d9313225f88406e5a55909c5aff5269a86a7a9531534f7da2e4c303d8a318a721c3c0c95956809532fcf0e2449a6b525b16aedf5aa0de657ba637b39",
+    "feedfacedeadbeeffeedfacedeadbeefabaddad2",
+    "d27e88681ce3243c4830165a8fdcf9ff1de9a1d8e6b447ef6ef7b79828666e4581e79012af34ddd9e2f037589b292db3e67c036745fa22e7e9b7373b",
+    "dcf566ff291c25bbb8568fc3d376a6d9")]
+  fn test_gcm_192(
+    #[case] kx: &str,
+    #[case] ivx: &str,
+    #[case] ptx: &str,
+    #[case] aadx: &str,
+    #[case] expected_ctx: &str,
+    #[case] expected_tagx: &str,
+  ) {
+    let k = decode_hex(kx).unwrap();
+    let iv = decode_hex(ivx).unwrap();
+    let pt = decode_hex(ptx).unwrap();
+    let aad = decode_hex(aadx).unwrap();
+
+    let key = Key::new(k.try_into().unwrap());
+    let gcm = GCM::<AES<192>>::new(key);
+
+    let (ct, tag) = gcm.encrypt(&iv, &pt, &aad).unwrap();
+
+    let ctx = encode_hex(ct.as_ref());
+    assert_eq!(ctx, expected_ctx);
+
+    let tagx = encode_hex(&tag);
+    assert_eq!(tagx, expected_tagx);
+
+    let (_pt, _tag) = gcm.decrypt(&iv, &ct, &aad).unwrap();
+
+    let _ptx = encode_hex(&_pt);
+    assert_eq!(ptx, _ptx);
+
+    let _tagx = encode_hex(&_tag);
+    assert_eq!(_tagx, expected_tagx);
+  }
+}
diff --git a/src/encryption/symmetric/modes/mod.rs b/src/encryption/symmetric/modes/mod.rs
index b991fe1..17ee1d1 100644
--- a/src/encryption/symmetric/modes/mod.rs
+++ b/src/encryption/symmetric/modes/mod.rs
@@ -6,3 +6,4 @@
 #![doc = include_str!("./README.md")]
 pub mod cbc;
 pub mod ctr;
+pub mod gcm;
diff --git a/src/hashes/constants.rs b/src/hashes/constants.rs
new file mode 100644
index 0000000..3326ed5
--- /dev/null
+++ b/src/hashes/constants.rs
@@ -0,0 +1,136 @@
+//! Constants used in hashes module, such as coefficients of polynomial of degree 129!
+use crate::{algebra::field::prime::AESField, Field};
+
+/// Represents the coefficients of field polynomial used in GCM
+/// f = 1 + α + α^2 + α^7 + α^128
+pub const GCMF_IRREDUCIBLE_POLYNOMIAL_COEFFICIENTS: [AESField; 129] = [
+  AESField::ONE,
+  AESField::ONE,
+  AESField::ONE,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ONE,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ZERO,
+  AESField::ONE,
+];
diff --git a/src/hashes/ghash.rs b/src/hashes/ghash.rs
new file mode 100644
index 0000000..1bc4f73
--- /dev/null
+++ b/src/hashes/ghash.rs
@@ -0,0 +1,263 @@
+//! Implementation of [`GHASH`] algorithm which is used in AES-GCM.
+//! Based on GCM specification given by NIST:
+//! [The Galois/Counter Mode of Operation (GCM)](http://www.csrc.nist.gov/groups/ST/toolkit/BCM/documents/proposedmodes/gcm/gcm-revised-spec.pdf)
+
+use core::array;
+
+use super::constants::GCMF_IRREDUCIBLE_POLYNOMIAL_COEFFICIENTS;
+use crate::{
+  algebra::field::{extension::GaloisField, prime::AESField},
+  polynomial::{Monomial, Polynomial},
+  Field,
+};
+
+type GCMField = GaloisField<128, 2>;
+/// Note: ['AESField'] is an alias for [`PrimeField<2>`]
+
+/// Helper function which turns [`num`], u64, value to vector of [`AESField`] elements
+/// The result is aligned to [`length`] by padding with AESField::ZERO.
+/// For example:
+/// If num = 5 and length = 8, the result is:
+/// {ZERO, ZERO, ZERO, ZERO, ZERO, ONE, ZERO, ONE}
+/// where the rightmost value is the least significant bit(=2^0)
+/// and `ZERO` and `ONE` are `AESField` elements.
+fn to_bool_vec(mut num: u64, length: usize) -> Vec<AESField> {
+  let mut result = vec![AESField::ZERO; length];
+  let mut idx = length;
+  while num > 0 {
+    result[idx - 1] = match num & 1 {
+      0 => AESField::ZERO,
+      _ => AESField::ONE,
+    };
+    num >>= 1;
+    idx -= 1;
+  }
+  result
+}
+
+/// Convert bytes(a slice of u8 value) to GCMField element.
+impl From<&[u8]> for GCMField {
+  fn from(value: &[u8]) -> Self {
+    let mut result = Vec::<AESField>::new();
+    for &b in value.iter() {
+      let b_f: Vec<AESField> = to_bool_vec(b as u64, 8);
+      result.extend(b_f);
+    }
+    result.extend(std::iter::repeat(AESField::ZERO).take(128 - value.len() * 8));
+    Self { coeffs: result.try_into().unwrap() }
+  }
+}
+
+/// Convert a GCMField element to a vector of u8 values.
+impl From<GCMField> for Vec<u8> {
+  fn from(value: GCMField) -> Self {
+    let mut bytes = Vec::new();
+    for block in value.coeffs.chunks(8) {
+      let mut byte: u8 = 0;
+      for i in 0..8 {
+        if block[i] == AESField::ONE {
+          byte += (1 << (7 - i)) as u8;
+        }
+      }
+      bytes.push(byte);
+    }
+    bytes
+  }
+}
+
+/// Represents the GHASH object which holds `hash_key`, a GCMField element, which in
+/// context of AES-GCM is equal to E(K, 0^128), where E is AES encryption and K, the key.
+pub struct GHASH {
+  hash_key: GCMField,
+}
+
+impl GHASH {
+  /// Construct GHASH object using 'h', the hash key(in bytes).
+  /// In context of AES-GCM, h = AES(K, 0^128), that is enc 128-bit string of zeros
+  /// encrypted using AES.
+  pub fn new(h: &[u8]) -> Self {
+    if h.len() != 16 {
+      panic!("The hash key should be 128-bits, or 16 u8 values! Got {} u8 vals", h.len());
+    }
+    Self { hash_key: GCMField::from(h) }
+  }
+
+  /// Get the hash digest using using two u8 slices `a` and `c`.
+  /// In context of AES-GCM they correspond to,
+  /// 'aad' - additional authenticated data(AAD)
+  /// 'ct' - ciphertext encrypted using AES.
+  ///
+  /// The result is a 128-bit digest, returned as an `[u8; 16]` array.
+  pub fn digest(&self, aad: &[u8], ct: &[u8]) -> [u8; 16] {
+    // The variable where the final digest is accumulated.
+    let mut j = GCMField::default();
+
+    // Process each 16-byte block from the aad
+    for block in aad.chunks(16) {
+      let a = GCMField::from(block);
+      // ADD a block of `aad` into the result and then MULTIPLY with hash_key.
+      // ADD interestingly is same as XOR in our field.
+      // MULTIPLY is a polynomial multiplication modulo a fixed field polynomial. See
+      // `poly_multiply`
+      j = Self::poly_multiply(a + j, self.hash_key);
+    }
+
+    // Do the same stuff as the previous loop but with ciphertext.
+    for block in ct.chunks(16) {
+      let a = GCMField::from(block);
+      j = Self::poly_multiply(a + j, self.hash_key);
+    }
+
+    // The final block consists of length of `aad` in bits and length of `ct` in bits, each in
+    // 64-bit format.
+    let mut aad_len = to_bool_vec((aad.len() * 8) as u64, 64);
+    let mut ct_len = to_bool_vec((ct.len() * 8) as u64, 64);
+    aad_len.append(&mut ct_len);
+    let len = GCMField { coeffs: aad_len.try_into().unwrap() };
+    j = Self::poly_multiply(j + len, self.hash_key);
+
+    let j_bytes: Vec<u8> = j.into();
+    j_bytes.try_into().unwrap()
+  }
+
+  /// Returns the result of multiplication of two GCMField elements,
+  /// modulo the field polynomial, f = 1 + α + α^2 + α^7 + α^128
+  fn poly_multiply(x: GCMField, y: GCMField) -> GCMField {
+    let x_coeffs: [AESField; 128] = x.coeffs.try_into().unwrap();
+    let y_coeffs: [AESField; 128] = y.coeffs.try_into().unwrap();
+    let poly_x = Polynomial::<Monomial, AESField, 128>::from(x_coeffs);
+    let poly_y = Polynomial::<Monomial, AESField, 128>::from(y_coeffs);
+    let poly_f =
+      Polynomial::<Monomial, AESField, 129>::from(GCMF_IRREDUCIBLE_POLYNOMIAL_COEFFICIENTS);
+
+    let poly_z = (poly_x * poly_y) % poly_f;
+    let res: [AESField; 128] =
+      array::from_fn(|i| poly_z.coefficients.get(i).cloned().unwrap_or(AESField::ZERO));
+
+    GCMField::new(res)
+  }
+
+  /// Multiply two elements in GCMField modulo the field polynomial f.
+  /// The field polynomial is fixed and is given by, f = 1 + α + α^2 + α^7 + α^128
+  /// The function uses the algorithm given in the GCM spec.
+  /// NOTE: We do not use this function in our GHASH algorithm, it is for reference only.
+  #[allow(dead_code)]
+  fn poly_multiply_spec(x: GCMField, y: GCMField) -> GCMField {
+    let mut r_coeffs = to_bool_vec(0xe1, 128);
+    r_coeffs.rotate_left(120);
+    let r = GCMField { coeffs: r_coeffs.try_into().unwrap() };
+
+    let mut z = GCMField::from(0 as usize);
+    let mut v = y;
+
+    for bit in x.coeffs {
+      if bit == AESField::ONE {
+        z = z + v;
+      }
+
+      let mut v1 = v.coeffs.to_vec();
+      let v1_bit = v1.pop().unwrap();
+      v1.push(AESField::ZERO);
+      v1.rotate_right(1);
+
+      v = GCMField { coeffs: v1.try_into().unwrap() };
+
+      if v1_bit == AESField::ONE {
+        v = v + r;
+      }
+    }
+
+    z
+  }
+}
+
+#[cfg(test)]
+mod tests {
+  use core::str;
+  use std::{fmt::Write, num::ParseIntError};
+
+  use rstest::rstest;
+
+  use super::*;
+  pub fn decode_hex(s: &str) -> Result<Vec<u8>, ParseIntError> {
+    (0..s.len()).step_by(2).map(|i| u8::from_str_radix(&s[i..i + 2], 16)).collect()
+  }
+
+  pub fn encode_hex(bytes: &[u8]) -> String {
+    let mut s = String::with_capacity(bytes.len() * 2);
+    for &b in bytes {
+      write!(&mut s, "{:02x}", b).unwrap();
+    }
+    s
+  }
+
+  // Tests against NIST test vectors in [GCM spec](https://csrc.nist.rip/groups/ST/toolkit/BCM/documents/proposedmodes/gcm/gcm-revised-spec.pdf)
+  #[rstest]
+  // NIST Test Case #1
+  #[case("66e94bd4ef8a2c3b884cfa59ca342b2e", "", "", "00000000000000000000000000000000")]
+  // NIST Test Case #2
+  #[case(
+    "66e94bd4ef8a2c3b884cfa59ca342b2e",
+    "",
+    "0388dace60b6a392f328c2b971b2fe78",
+    "f38cbb1ad69223dcc3457ae5b6b0f885"
+  )]
+  // NIST Test Case #3
+  #[case(
+    "b83b533708bf535d0aa6e52980d53b78",
+    "",
+    "42831ec2217774244b7221b784d0d49ce3aa212f2c02a4e035c17e2329aca12e21d514b25466931c7d8f6a5aac84aa051ba30b396a0aac973d58e091473f5985",
+    "7f1b32b81b820d02614f8895ac1d4eac"
+    )]
+  // NIST Test Case #4
+  #[case(
+    "b83b533708bf535d0aa6e52980d53b78",
+    "feedfacedeadbeeffeedfacedeadbeefabaddad2",
+    "42831ec2217774244b7221b784d0d49ce3aa212f2c02a4e035c17e2329aca12e21d514b25466931c7d8f6a5aac84aa051ba30b396a0aac973d58e091",
+    "698e57f70e6ecc7fd9463b7260a9ae5f",
+    )]
+  // NIST Test Case #6
+  #[case(
+    "b83b533708bf535d0aa6e52980d53b78",
+    "feedfacedeadbeeffeedfacedeadbeefabaddad2",
+    "8ce24998625615b603a033aca13fb894be9112a5c3a211a8ba262a3cca7e2ca701e4a9a4fba43c90ccdcb281d48c7c6fd62875d2aca417034c34aee5",
+    "1c5afe9760d3932f3c9a878aac3dc3de",
+      )]
+  fn test_ghash(#[case] hx: &str, #[case] ax: &str, #[case] cx: &str, #[case] expected: &str) {
+    let h = decode_hex(hx).unwrap();
+    let a = decode_hex(ax).unwrap();
+    let c = decode_hex(cx).unwrap();
+
+    let gh = GHASH::new(&h);
+    let digest = gh.digest(&a, &c);
+
+    let digest_hex = encode_hex(&digest);
+
+    println!("GHASH:{}\nExpected:{}", digest_hex, expected);
+    assert!(digest_hex == expected);
+  }
+
+  #[rstest]
+  #[case(1, 1)]
+  #[case(2, 3)]
+  #[case(113, 117)]
+  #[case(0xca, 0xfe)]
+  fn test_poly_multiply(#[case] x: u64, #[case] y: u64) {
+    let x_coeffs: Vec<AESField> = to_bool_vec(x, 128).into_iter().rev().collect();
+    let xf = GCMField { coeffs: x_coeffs.try_into().unwrap() };
+    let y_coeffs = to_bool_vec(y, 128);
+    let yf = GCMField { coeffs: y_coeffs.try_into().unwrap() };
+
+    let zf = GHASH::poly_multiply(xf, yf);
+
+    let z_coeffs: Vec<u8> = zf.try_into().unwrap();
+    let z_hex = encode_hex(&z_coeffs);
+
+    let expected_zf = GHASH::poly_multiply_spec(xf, yf);
+    let expected_z_coeffs: Vec<u8> = expected_zf.try_into().unwrap();
+    let expected_z_hex = encode_hex(&expected_z_coeffs);
+
+    println!("Got: {z_hex}\nExp: {expected_z_hex}");
+    assert!(expected_z_hex == z_hex);
+  }
+}
diff --git a/src/hashes/mod.rs b/src/hashes/mod.rs
index 2c50760..b94c7ab 100644
--- a/src/hashes/mod.rs
+++ b/src/hashes/mod.rs
@@ -5,6 +5,8 @@
 #![doc = include_str!("./README.md")]
 pub mod sha256;
 use crate::Field;
+pub mod constants;
+pub mod ghash;
 pub mod poseidon;
 
 /// Sponge trait defining absorb and squeeze behavior of sponge based hash function.