In this article I want to explore a common problem of modern cryptographic ciphers: malleability. I will explain that problem with some handson examples, and then look in detail at how that problem is solved through the use of authenticated encryption. I will describe in particular two algorithms that provide authenticated encryption: ChaCha20Poly1305 and AESGCM, and briefly mention some of their variants.
The problem
If we want to encrypt some data, a very common approach is to use a symmetric cipher. When we use a symmetric cipher, we hold a secret key, which is generally a sequence of bits chosen at random of some fixed length (nowadays ranging from 128 to 256 bits). The symmetric cipher takes two inputs: the secret key, and the message that we want to encrypt, and produces a single output: a ciphertext. Decryption is the inverse process: it takes the secret key and the ciphertext as the input and yields back the original message as an output. With symmetric ciphers, we use the same secret key both to encrypt and decrypt messages, and this is why they are called symmetric (this is in contrast with public key cryptography, or asymmetric cryptography, where encryption and decryption are performed using two different keys: a public key and a private key).
Generally speaking, symmetric ciphers can be divided into two big families:
 stream ciphers, which can encrypt data bitbybit;
 block ciphers, which can encrypt data blockbyblock, where a block has a fixed size (usually 128 bits).
As we will discover soon, both these two families exhibit the same fundamental problem, although they slightly differ in the way this problem manifests itself. To understand this problem, let’s take a close look at how these two families of algorithms work and how we can manipulate the ciphertexts they produce.
Stream ciphers
A good way to think of a stream cipher is as a deterministic random number generator that yields a sequence of random bits. The secret key can be thought of as the seed for the random number generator. Every time we initialize the random number generator with the same secret key, we will get exactly the same sequence of random bits out of it.
The bits coming out of the random number generator can then be XORed together
with the data that we want to encrypt: ciphertext = random
sequence XOR message
, like in
the following example:
random sequence: 3bAWC5ThFSPXX1W8P94q3XV35TG6CRVTNAPW27Q69F ⊕ message: I would really like an ice cream right now = ciphertext: zB686Y0H46144HwT9RQQR6vZV1gU1779n390ZCqXV1
The XOR operator acts as a toggle that can either flip bits or keep them unchanged. Let me explain with an example:
a XOR 0 = a
a XOR 1 = NOT a
If we XOR “something” with a 0 bit, we get “something” out; if we XOR “something” with a 1 bit, we get the opposite of “something”. And if we use the same toggle twice, we return to the initial state:
a XOR b XOR b = a
This works for any a
and any b
and it’s due to the fact that b XOR b
is
always equal to 0. In more technical terms, each input is its own selfinverse
under the XOR operator.
The selfinverse property gives us a way to decrypt the message that we encrypted above: all we have to do is to replay the random sequence and XOR it together with the ciphertext:
random sequence: 3bAWC5ThFSPXX1W8P94q3XV35TG6CRVTNAPW27Q69F ⊕ ciphertext: zB686Y0H46144HwT9RQQR6vZV1gU1779n390ZCqXV1 = message: I would really like an ice cream right now
This works because ciphertext = random sequence XOR message
, therefore random sequence XOR ciphertext = random
sequence XOR random sequence XOR
message
. The two random sequence
are the same, so they
cancel each other (selfinverse), leaving only message
:
random sequence: 3bAWC5ThFSPXX1W8P94q3XV35TG6CRVTNAPW27Q69F ⊕ random sequence: 3bAWC5ThFSPXX1W8P94q3XV35TG6CRVTNAPW27Q69F ⊕ message: I would really like an ice cream right now = message: I would really like an ice cream right now
Only the owner of the secret key will be able to generate the random sequence, therefore only the owner of the secret key should, in theory, be able to recover the message using this method.
Playing with stream ciphers
The selfinverse property not only allows us to recover the message from the random sequence and the ciphertext, but it also allows us to recover the random sequence if can correctly guess the message:
message: I would really like an ice cream right now ⊕ ciphertext: zB686Y0H46144HwT9RQQR6vZV1gU1779n390ZCqXV1 = random sequence: 3bAWC5ThFSPXX1W8P94q3XV35TG6CRVTNAPW27Q69F
This “feature” opens the door to at least two serious problems. If we are able to correctly guess the message or a portion of it, then we can:
 decrypt other ciphertexts produced by the same secret key (or at least portions of them, depending on what portions of the random sequence we were able to recover);
 modify ciphertexts.
And we can do all of this without any knowledge of the secret key.
The first problem implies that key reuse is forbidden with stream ciphers. Every time we want to encrypt something with a stream cipher, we need a new key. This problem is easily solved by the use of a nonce (also known as initialization vector, IV, or starting variable, SV): a random value that is generated before every encryption, and that is combined in some way with the secret key to produce a new value to initialize the random number generator. If the nonce is unique per encryption, then we can be sufficiently confident that the random sequence generated will also be unique. The nonce value does not necessarily need to be kept secret, and needs to be known at decryption time. Nonces are usually generated at random at encryption time and stored alongside the ciphertext.
The second problem is a bit more subtle: if we have a ciphertext and we can correctly guess the original message that produced it, we can modify it using the XOR operator to “cancel” the original message and “insert” a new message, like in this example:
ciphertext: zB686Y0H46144HwT9RQQR6vZV1gU1779n390ZCqXV1 ⊕ message: I would really like an ice cream right now ⊕ altered message: I would really like to go to bed right now = tampered ciphertext: zB686Y0H46144HwT9RQQG7vTZt3Yc030n390ZCqXV1
This message, when correctly decrypted with the secret key, will return the tampered ciphertext without detection!
Note that I do not need to know the full message to carry out this technique, in fact, the following example (where unknown parts have been replaced by hyphens) produces the same result as the above one:
ciphertext: zB686Y0H46144HwT9RQQR6vZV1gU1779n390ZCqXV1 ⊕ message: an ice cream ⊕ altered message: to go to bed = tampered ciphertext: zB686Y0H46144HwT9RQQG7vTZt3Yc030n390ZCqXV1
This problem is known as malleability, and it’s a serious issue in the real world because most of the messages that we exchange are in practice relatively easy to guess.
Suppose for example that I have control over a WiFi network, and I can inspect and alter the internet traffic that passes through it. Suppose that I know that a person connected to my WiFi network is visiting an ecommerce website and that they’re interested in a particular item. The traffic that your browser exchanges with the ecommerce website may be encrypted, and therefore I won’t be able to decrypt its contents, but I might be able to guess certain parts of it, like the HTTP headers sent by the website, or some parts of the HTML that are common to all pages on that website, or even the name and the price of the item you want to buy. If I can guess that information (which is public information, not a secret, and it’s generally easy to guess), then I might be able to alter some parts of the web page, showing you false information, and altering the price that you see in an attempt to trick you into buying that item.
An example of malleability using ChaCha20 with OpenSSL
Here’s a practical example of how we can take the output of a stream cipher, and alter it as we wish without knowledge of the secret key. I’m going to use the OpenSSL command line interface to encrypt a message with a stream cipher: ChaCha20. This is a modern, fast, stream cipher with a good reputation:
openssl enc chacha20 \
K 0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef \
iv 0123456789abcdef0123456789abcdef \
in <(echo 'I would really like an ice cream right now') \
out ciphertext.bin
The K
option specifies the key in hexadecimal format (256 bits, or 32 bytes,
or 64 hex characters), the iv
is the nonce, also known as initialization
vector (128 bits, or 16 bytes, or 32 hex characters).
This trivial Python script can tamper with the ciphertext:
with open('ciphertext.bin', 'rb') as file:
ciphertext = file.read()
guessed_message = b'an ice cream\n'
replacement_message = b'to go to bed\n'
tampered_ciphertext = bytes(x ^ y ^ z for (x, y, z) in
zip(ciphertext, guessed_message, replacement_message))
with open('tamperedciphertext.bin', 'wb') as file:
file.write(tampered_ciphertext)
This script is using partial knowledge of the message. It knows (thanks to an educated guess) that the original message contained the words “an ice cream” at a specific offset, and uses that knowledge to replace those words with new ones (“to go to bed”) which add up to the same length. Note that this technique cannot be used to remove or add parts from the message, only to modify them without changing their length.
Now if we run this script and we decrypt the tamperedciphertext.bin
file
with the same key and nonce as before, we get “to go to bed” instead of “an ice
cream”, without any error indicating that tampering occurred:
openssl enc d chacha20 \
K 0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef \
iv 0123456789abcdef0123456789abcdef \
in tamperedciphertext.bin
Block ciphers
We have seen that stream ciphers alone have a serious problem (malleability) that allows anyone to modify arbitrary portions of ciphertexts without detection. Let’s take a look at the alternative: block ciphers. Will they have the same problem?
While a stream cipher can encrypt variable amounts of data, a block cipher can only take as input a block of data of a fixed size, and produce as output another block of data. A good block cipher produces an output that is indistinguishable from random.
The block size is generally small, usually 128 bits (16 bytes), so if we want to encrypt larger amounts of data, we have to split the data into multiple blocks, and encrypt each block individually. If the data is too short to fit in a block, the data will also need to be padded.
message: The cat is on th e table......... ______________ ______________ block #1 block #2 (padded) ciphertext: c2TNPW3r09hZ6f1P Vc32VX41XSy579Y9
This approach however has a problem: if we encrypt multiple blocks with the same secret key, then portions of messages that are the repeated will produce the same output. This gives the ability to analyze a ciphertext and find patterns in it without knowledge of the secret key. This problem is famously evident when encrypting pictures:
How this image was generated
Before jumping into how I encrypted the image, let me spend a few words on how I did NOT encrypt the image: I did not use a modern image format. Modern image formats are very sophisticated, they’re not a simple sequence of RGB values. Instead, they have some control structures mixed in the image, they implement compression to reduce the image size, etc. This complexity means that if I simply take an image in any format and encrypt it, the result won’t be visualizable by an image viewer: the image viewer would just throw an error because it would find invalid data structures.
Note that this does not imply that encrypting modern image formats is more secure: people can still analyze patterns in them, but it simply means that a modern image format, once encrypted, would not produce the sensational visualization that I showed above.
In order to produce this visualization I had to find an uncompressed image format without too much metadata in it. Thankfully the Wikipedia article on image file formats provided a list, which included the Netpbm family of formats (something I never heard of before). Among the formats in this family, I chose PPM, because it’s the one that supports colors.
The PPM file format is very simple: it has 3 lines of metadata, followed by the RGB values for each pixel. No compression. Definitely the right format for this kind of experiment!
So here’s what I did: first of all I downloaded an image (the Ubuntu “Circle of Friends” logo, obtained from Wikipedia) and converted it to PPM with ImageMagick:
convert UbuntuCoF.png img.ppm
I separated the header from the RGB values:
head n3 img.ppm > ppmheader
tail n+4 img.ppm > ppmimage
The reason why I separated the header from RGB values is that I won’t encrypt the header. If I did, then the image won’t be visualizable by an image viewer, just like if I used a modern image format. In a realworld scenario, a person would be able to easily guess the header if it was encrypted.
I encrypted the RGB values with AES256, a modern, strong block cipher with a good reputation:
openssl enc aes256ecb \
K 0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef \
in ppmimage \
out ppmimageencrypted
Then I joined the header and the encrypted RGB values in a PPM file:
cat ppmheader ppmimageencrypted > imgencrypted.ppm
This results in a randomized image that can be viewed without problems on an image viewer. It’s interesting to see that, if you change the encryption key, you will get a different randomized image!
The problem we have just seen is known as lack of diffusion. This is kinda analogous to the first problem we identified with stream ciphers: at the root of both problems there is key reuse. We solved this problem for stream ciphers by combining a key with a random nonce. We could use the same strategy here, would be an expensive approach, as initializing a block cipher with a new key is a relatively expensive operation. It’s much cheaper to initialize the block cipher once, and reuse it for every block encryption. We need a way to “link” blocks to each other, so that if two linked blocks contain the same plaintext, their encryption will give different results.
There are various strategies do that. These strategies are known as mode of operation of block ciphers. Let’s take a look at two of them: Cipher Block Chaining (CBC) and Counter Mode (CTR).
Cipher Block Chaining (CBC)
This mode of operation, as the name suggests, ‘chains’ each block to the next one. The way it works is by using the XOR operator in the following way:

First of all, a random nonce is generated. The purpose of the nonce is the same as before (with stream ciphers): ensuring that using the same secret key to perform multiple encryptions yields different results each time, so that secret information or patterns are not revealed.
The nonce does not need to be kept secret and is normally stored alongside the ciphertext so that it can be easily used during decryption. It is however important that the nonce is unique.

The first block of message
m[0]
is XORed with the nonce, and then encrypted the block cipher, producing the first block of ciphertextc[0] = block_encrypt(m[0] XOR nonce)

The second block of message
m[1]
is XORed withc[0]
, and then encrypted with the block cipher:c[1] = block_encrypt(m2 XOR c[0])

…

The last block of message
m[n]
is XORed together withc[n1]
, and then encrypted with the block cipher:c[n] = block_encrypt(m[n] XOR c[n1])
The XOR operator is back! With stream ciphers, the XOR operator was allowing us to tamper with ciphertexts. Can we do the same thing here? Yes of course! The approach is slightly different though: instead of acting directly on the block that we want to change, we will act on the block that precedes it.
For example, if we want to change the sentence “I came home in the afternoon and the cat was on the table” so that it reads ‘dog’ instead of ‘cat’, we would need to change the block right before the one that contains the word ‘cat’. If we want to change the very first block, for example to change the word ‘came’ to ‘left’, we would need to change the nonce instead.
nonce+ciphertext: yzURZRbP6X1w3ZRL XRDnPbEkx3JUP2Fv C2ZWt19EdAXDi76H pkbk8qTgaSdzerbF 8CWYqscBqE6cSLmx ⊕ message (shifted 1 block to the left): I came home in t he afternoon and the cat was on the table....... ⊕ altered message (shifted 1 block to the left): I left home in t he afternoon and the dog was on the table....... = tampered nonce+ciphertext: yzZVQCbP6X1w3ZRL XRDnPbEkx3JUP2Fv C2ZWt67VdAXDi76H pkbk8qTgaSdzerbF 8CWYqscBqE6cSLmx
If we do the above, and then decrypt the tampered ciphertext, we will get something like this:
I left home in t���������������the dog was on the table
How to get this result using AESCBC with OpenSSL
Here’s a stepbystep guide on how to tamper with a ciphertext encrypted with AES256 in CBC mode.
First, generate a valid ciphertext:
openssl enc aes256cbc \
K 0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef \
iv 0123456789abcdef0123456789abcdef \
in <(echo 'I came home in the afternoon and the cat was on the table') \
out ciphertext.bin
Like in the stream cipher example, K
is the key in hexadecimal format (256
bits), while iv
is the nonce (128 bits).
We can perform the tampering with this Python script:
with open('ciphertext.bin', 'rb') as file:
ciphertext = file.read()
guessed_message = b'cat'
replacement_message = b'dog'
tampered_ciphertext = bytes(x ^ y ^ z for (x, y, z) in
zip(ciphertext, guessed_message, replacement_message))
with open('tamperedciphertext.bin', 'wb') as file:
file.write(tampered_ciphertext)
Note that OpenSSL does not store the nonce along with the ciphertext, but instead expects it to be passed as a command line argument. We need to modify it separately, so here’s another Python script just for the nonce:
nonce = bytes.fromhex('0123456789abcdef0123456789abcdef')
guessed_message = b'came'
replacement_message = b'left'
tampered_nonce = bytes(x ^ y ^ z for (x, y, z) in
zip(nonce, guessed_message, replacement_message))
print(tampered_nonce.hex())
If we run that script, we get: 01234a6382bacdef0123456789abcdef
.
Now to decrypt the tampered ciphertext with the tampered nonce:
openssl enc d aes256cbc \
K 0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef \
iv 01234a6382bacdef0123456789abcdef \
in tamperedciphertext.bin
It’s interesting to see that we were successful in changing the word ‘cat’ to ‘dog’ but in doing so we had to sacrifice a block, which, when decrypted, resulted in random bytes.
In a real world scenario, seeing some random bytes could raise some suspicion, and maybe generate some errors in applications, however that’s not always the case (how many times have we seen garbled text on our monitors, and we never worried that somebody was tampering with our communications). Also, when dealing with formats like HTML, one could conceal tampering attempts using comment blocks, or using JavaScript. One example of what I’m describing is the EFAIL vulnerability: discovered in 2017, it affected some popular email clients including Gmail, it targeted the use of AES in CBC mode (as well as another mode very similar to it: Cipher Feedback, CFB), and allowed the injection of malicious content in HTML emails.
We can conclude that block ciphers in CBC mode, just like stream ciphers, are also malleable.
Counter Mode (CTR)
Are other modes of operation all malleable like CBC, or will they be different? Let’s take a look at another, very common, mode of operation: Counter Mode (CTR), so that we can get a better sense of how the problem of malleability can affect the world of block ciphers.
The mechanism behind Counter Mode is very simple:

A random nonce is generated. The purpose of the nonce is the usual one: make sure that repeated encryptions using the same key produce different results.

A counter (usually an integer) is initialized to 1 (or any other starting value of your choice).

The nonce is concatenated with the counter, and encrypted using the block cipher:
r[0] = block_encrypt(nonce  counter)
.Because the block cipher can only accept as input a block of a fixed size, it follows that the length of the nonce plus the length of the counter must be equal to the block size. For example, for a 128bit block cipher, a common choice is to have a 96bit nonce and a 32bit counter.

The counter is incremented:
counter = counter + 1
(the increment does not necessarily need to be by 1, but that’s a common choice). The nonce and the new counter are concatenated again, and encrypted using the block cipher:r[1] = block_encrypt(nonce  counter)
. 
The counter is incremented again (
counter = counter + 1
), and a new block is encrypted, just like before:r[2] = block_encrypt(nonce  counter)
. 
…
This mechanism produces a sequence of blocks r[0]
, r[1]
, r[2]
, … which
are indistinguishable from random. This sequence of random blocks can be XORed
with the message to produce the ciphertext.
It’s important that the values for the counter never repeat. If, for example, we’re using a 32bit counter, the counter will “reset” (go back to the starting value) after 2^{32} iterations, and will start repeating the same sequence of random blocks as it did at the beginning. This introduces the problem of lack of diffusion that we have seen before, just at a larger scale. If we’re using a 32bit counter with a 128bit block cipher, we cannot encrypt more than 128·2^{32} bits = 64 GiB of data at once. This is a very important detail: exceeding these limits may allow the decryption of portions of ciphertext without knowledge of the secret key.
What Counter Mode is doing is effectively turning a block cipher into a stream cipher. As such, a block cipher in Counter Mode has the exact same malleability problems of stream ciphers that we have seen before.
An example of malleability using AESCTR with OpenSSL
This example is going to be very similar (almost identical) to the example with ChaCha20 that I showed in the stream cipher section, just that this time I’m going to use AES256 in CTR mode.
Let’s produce a valid ciphertext:
openssl enc aes256ctr \
K 0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef \
iv 0123456789abcdef0123456700000001 \
in <(echo 'Can you give me a ride to the party?') \
in <(echo 'Do not give me a ride to the party!') \
out ciphertext.bin
Tamper it with Python:
with open('ciphertext.bin', 'rb') as file:
ciphertext = file.read()
guessed_message = b'an ice cream\n'
replacement_message = b'to go to bed\n'
tampered_ciphertext = bytes(x ^ y ^ z for (x, y, z) in
zip(ciphertext, guessed_message, replacement_message))
with open('tamperedciphertext.bin', 'wb') as file:
file.write(tampered_ciphertext)
And now we can decrypt it:
openssl enc d aes256ctr \
K 0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef \
iv 0123456789abcdef0123456700000001 \
in tamperedciphertext.bin
The solution: Authenticated Encryption (AE)
We have seen that stream ciphers and block ciphers (in their mode of operation) both exhibit the same problem (in different flavors): malleability. I’ve shown some examples of how this problem can be exploited with modern ciphers like ChaCha20 and AES. These ciphers, alone, cannot guarantee the integrity or authenticity of encrypted data.
In this context, integrity is the assurance that data is not corrupted or modified in any way. Authenticity can be thought of as a stronger version of integrity, and it’s the assurance that a given ciphertext was produced only with knowledge of the secret key.
Does this mean that modern ciphers, like ChaCha20 and AES, should be considered insecure and avoided? Absolutely not! The correct answer is that those ciphers cannot be used alone. You should think of them as basic building blocks, and you need some additional building blocks in order to construct a complete and secure cryptosystem. One of these additional building block, that we are going to explore in this article, is an algorithm that provides integrity and authentication: welcome Authenticated Encryption (AE).
When using authenticated encryption, an adversary may be able to modify a ciphertext using the techniques described above, but such modification would be detected by the authentication algorithm, and decryption will fail with an error. The decrypted message at that point should be discarded, preventing the use of tampered data.
There are many different methods to implement authenticated encryption. The most common approach is to use an authentication algorithm to authenticate the ciphertext produced by a cipher. Here I will describe two very popular authentication algorithms:
 Poly1305, which is often used in conjunction with the stream cipher ChaCha20 to form ChaCha20Poly1305;
 and Galois/Counter Mode (GCM), which is often used with the block cipher AES to form AESGCM.
These authentication algorithms work by computing a hash of the ciphertext, which is then stored alongside the ciphertext. This hash is not a regular hash, but it’s a keyed hash. A regular hash is a function that takes as input some data and returns a fixedsize bit string:
$$\operatorname{hash}: data \rightarrow bits$$
A keyed hash instead takes two inputs: a secret key and some data, and produces a fixedsize bit string:
$$\operatorname{keyedhash}: (key, data) \rightarrow bits$$
The output of the keyed hash is more often called Message Authentication Code (MAC), or authentication tag, or even just tag.
During decryption, the same authentication algorithm is run again on the ciphertext, and a new tag is produced. If the new tag matches the original tag (that was stored alongside the ciphertext), then decryption succeeds. Else, if the tags don’t match, it means that the ciphertext was modified (or the tag was modified), and decryption fails. This gives us a way to detect tampering and gives us the opportunity to reject ciphertexts that were not produced by the secret key.
The secret key passed to the keyed hash function is not necessarily the same secret key used for the encryption. In fact, both ChaCha20Poly1305 and AESGCM operate on a subkey derived from the key used for encryption.
Poly1305
Poly1305 is a keyed hash function proposed by Daniel J. Bernstein in 2004. It works by using polynomials evaluated modulo the prime 2^{130}  5, hence the name.
The key to Poly1305 is a 256bit string, and it’s split into two halves:
 the first half (128 bits) is called $r$;
 the second half (128 bits) is called $s$.
We’ll see later how this key is generated when Poly1305 is used to implement authenticated encryption. For now, let’s assume that the key is a random (unpredictable) bit string provided as an input.
The first half $r$ is also clamped by setting some of its bits to 0. This is a performancerelated optimization that some Poly1305 implementations can take advantage of when doing multiplication using 64bit registers. Clamping is performed by applying the following hexadecimal bitmask:
0ffffffc0ffffffc0ffffffc0fffffff
The message to authenticate is split into chunks of 128 bits each: $m_1$, $m_2$, $m_3$, … $m_n$. If the length of the message is not a multiple of 128 bits, then the last block may be shorter. The authentication tag is then calculated as follows:

Interpret $r$ and $s$ as two 128bit littleendian integers.

Initialize the Poly1305 state $a_0$ to the integer 0. As we shall see later, this state will need to hold at most 131 bits.

For each block $m_i$:

Interpret the block $m_i$ as a littleendian integer.

Compute $\overline{m}_i$ by appending a 1bit to the end of the block $m_i$. If $m_i$ is 128 bits long, then this is equivalent to computing $\overline{m}_i = 2^{128} + m_i$. In general, if the length of the block $m_i$ in bits is $\operatorname{len}(m_i)$, then this is equivalent to $\overline{m}_i = 2^{\operatorname{len}(m_i)} + m_i$.
This step ensures that the resulting block $\overline{m}_i$ is always nonzero, even if the original block $\overline{m}_i$ is zero. This is important for the security of the algorithm, as explained later.

Compute the new state $a_i = (a_{i1} + \overline{m}_i) \cdot r \pmod{2^{130}  5}$. Note that, because the operation is modulo $2^{130}  5$, the result will always fit in 130 bits.


Once each block has been processed, compute the final state $a_{n+1} = a_n + s$. Note that the state $a_n$ is at most 130 bits long, and $s$ is at most 128 bits long, hence the result will be at most 131 bits long.

Truncate the final state $a_{n+1}$ to 128 bits by removing the most significant bits.

Return the truncated final state $a_{n+1}$ as a littleendian byte string.
What this method is doing is computing the following polynomial in $r$ and $s$:
$$\begin{align*} tag & = ((((((\overline{m}_1 \cdot r) + \overline{m}_2) \cdot r) + \cdots + \overline{m}_n) \cdot r) \bmod{(2^{130}  5)}) + s \\ & = (\overline{m}_1 r^n + \overline{m}_2 r^{n1} + \cdots + \overline{m}_n r) \bmod{(2^{130}  5)} + s \end{align*}$$
$r$ and $s$ are secrets, and they come from the Poly1305 key. Note that if we didn’t add $s$ at the end, then the resulting polynomial would be a polynomial in $r$, and one could use polynomial rootfinding methods to figure out $r$ from the authentication tag, without knowledge of the key. Therefore it’s important that $s$ is nonzero.
In Python, this is what a Poly1305 implementation could look like (disclaimer: this is for learning purposes, and not necessarily secure or optimized for performance):
def iter_blocks(message: bytes):
"""
Splits a message in blocks of 16 bytes (128 bits) each, except for the last
block, which may be shorter.
"""
start = 0
while start < len(message):
yield message[start:start+16]
start += 16
def poly1305(key: bytes, message: bytes):
assert len(key) == 32 # 256 bits
# Prime for the evaluation of the polynomial
p = (1 << 130)  5
# Split the key into two parts r and s
r = int.from_bytes(key[:16], 'little') # 128 bits
s = int.from_bytes(key[16:], 'little') # 128 bits
# Clamp r
r = r & 0x0ffffffc0ffffffc0ffffffc0fffffff
# Initialize the state
a = 0
# Update the state with every block
for block in iter_blocks(message):
# Append a 1bit to the end of each block
block = block + b'\1'
# Convert the block to an integer
c = int.from_bytes(block, 'little')
# Update the state
a = ((a + c) * r) % p
# Add s to the state and truncate it to 128 bits, removing the most
# significant bits and keeping only the least significant 128 bits
a = (a + s) & ((1 << 128)  1)
# Convert the state from an integer to a 16byte string (128 bits)
return a.to_bytes(16, 'little')
And here is an example of how that code could be used:
key = bytes.fromhex('0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef')
msg = b'I had a very nice day today at the beach'
print(poly1305(key, msg).hex())
This returns b0c4cb74b3089e9a982e3baa90c1bb5f
, which is the same result that
we would get using OpenSSL:
openssl mac \
macopt hexkey:0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef \
in <(echo n 'I had a very nice day today at the beach') \
poly1305
A few things to note:

The same key cannot be reused to construct two distinct tags. In fact, suppose that we use the same hash key to compute
tag1 = Poly1305(key, msg1)
andtag2 = Poly1305(key, msg2)
. Then, because $s$ is the same for both, we could subtract the two tags (tag1  tag2
) to remove the $s$ part and obtain a polynomial in $r$. From there, we could use algebraic methods to figure out $r$. Once we have $r$, we can use either one of the tags and compute $s$, therefore recovering the full secret key.Similarly, if the keys were generated using a predictable algorithm (for example, incrementally:
key[i+1] = key[i] + 1
), it would still be possible to use a similar approach to figure out the secret key.For this reason, Poly1305 keys must be unique and unpredictable. Generating Poly1305 keys randomly or pseudorandomly is an acceptable approach. Authentication functions like Poly1305 are called onetime authenticators because they can be used only one time with the same key.

If we didn’t add the 1bits at the end of each block (in other words, if we used the $m_i$ blocks instead of $\overline{m}_i$), then encrypting a message full of zero bits would be the equivalent of encrypting an empty message. Adding the 1bits is a way to ensure that the length of the message always has an effect on the output.
Use of Poly1305 with ChaCha20 (ChaCha20Poly1305)
Let’s see how we can combine ChaCha20 and Poly1305 to construct an authenticated cipher. To recap:
 ChaCha20 is a stream cipher;
 Poly1305 is a onetime authenticator;
 ChaCha20, like most ciphers, requires the use of a unique nonce to allow key reuse.
Putting the two together gives birth to ChaCha20Poly1305. Here I’m going to describe how to implement it as standardized in RFC 8439.
The inputs to the ChaCha20Poly1305 encryption function are:
 a 256bit secret key;
 a 96bit nonce;
 a variablelength plaintext message.
The outputs from the ChaCha20Poly1305 encryption function are:
 a variablelength ciphertext (same length as the input plaintext);
 a 128bit authentication tag.
The ChaCha20Poly1305 decryption function will accept the same secret key, nonce, ciphertext, and authentication tag as the input, and produce either the plaintext or an error as the output. The error is returned in case the authentication fails.
ChaCha20Poly1305 works in the following way:

The ChaCha20 stream cipher is initialized with the 256bit secret key and the 96bit nonce.

The stream cipher is used to encrypt a 256bit string of all zeros. The result is the Poly1305 subkey.
If you recall how a stream cipher works, you should know that encrypting using a stream cipher is equivalent to performing the XOR of a random bit stream with the plaintext. Here the plaintext is all zeros, so the process of generating the Poly1305 subkey is equivalent to grabbing the first 256 bits from the ChaCha20 bit stream.
We previously saw that the Poly1305 subkey must be unpredictable and unique in order for Poly1305 to be secure. The use of ChaCha20 with a unique nonce ensures that: because ChaCha20 is a stream cipher, its output will be random and unpredictable. Therefore, with this construction, the subkey will be unpredictable even if the nonce is predictable.

The stream cipher is used to encrypt another 256bit string. The result is discarded. This is equivalent to advancing the stream cipher state by 256 bits.
This step may seem weird, and in fact is not needed for security purposes, but it’s a mere implementation detail. This step is here because ChaCha20 has an internal state of 512 bits. In the previous step we obtained the first 256 bits of the state, and this next step is to discard the rest of the state to start with a fresh state. There is no particular reason for requiring a fresh state. The reason why RFC 8439 does that is because… spoiler alert: ChaCha20 is a block cipher under the hood. Its block size is 512 bits. If you read the RFC, you’ll see that it asks to call the ChaCha20 block encryption function once, grab the first 256 bits, and discard the rest. Here I’m using ChaCha20 as a stream cipher, so I have to include this extra step to discard the bits.

The plaintext is encrypted using the stream cipher.
Note that this is done without resetting the state of the cipher. We are continuing to use the same stream cipher instance that was used to generate the Poly1305 subkey.

The ciphertext is padded with zeros to make its length a multiple of 16 bytes (128 bits) and is authenticated using Poly1305, via the subkey generated in step 2.
This step may be done in parallel to the previous one, that is: every time we generate a chunk of ciphertext, we feed it to the Poly1305 authentication function.
Why pad the ciphertext before passing it to Poly1305? After all, ChaCha20 is a stream cipher, and Poly1305 can accept arbitrarysized messages. Again, this is an detail of RFC 8439 and padding does not serve any specific purpose.

The length of the ciphertext (in bytes) is fed into the Poly1305 authenticator. This length is represented as a 64bit littleendian integer padded with 64 zero bits.
The reason why the length is represented as 64 bits and padded (instead of representing it as 128 bits) will be clearer later: what I have given you so far is a simplified view of ChaCha20Poly1305 and authenticated encryption in general. I will give you the full picture when talking about associated data later on, and at that point this step will be slightly modified.

The ciphertext from ChaCha20 and the authentication tag from Poly1305 are returned.
The decryption algorithm works in a very similar way: ChaCha20 is initialized in the same way, the subkey is generated in the same way, the Poly1305 authentication tag is calculated from the ciphertext in the same way. The only difference is that ChaCha20 is used to decrypt the ciphertext (instead of encrypting the plaintext) and that the input authentication tag is compared to the calculated authentication tag before returning.
Here is a Python implementation of ChaCha20Poly1305, based on the implementations of ChaCha20 and Poly1305 from pycryptodome (usual disclaimer: this code is for educational purposes, and is not necessarily secure or optimized for performance):
from Crypto.Cipher import ChaCha20
from Crypto.Hash import Poly1305
def chacha20poly1305_encrypt(key, nonce, message):
assert len(key) == 32 # 256 bits
assert len(nonce) == 12 # 96 bits
# Initialize the ChaCha20 cipher with the key and nonce
cipher = ChaCha20.new(key=key, nonce=nonce)
# Derive the Poly1305 subkey using the ChaCha20 cipher
subkey = cipher.encrypt(b'\0' * 32) # 256 bits
subkey_r = subkey[:16]
subkey_s = subkey[16:]
# Initialize the Poly1305 authenticator with the subkey
authenticator = Poly1305.Poly1305_MAC(r=subkey_r, s=subkey_s, data=None)
# Discard the rest of the internal ChaCha20 state
cipher.encrypt(b'\0' * 32) # 256 bits
# Encrypt the message
ciphertext = cipher.encrypt(message)
# Authenticate the ciphertext
authenticator.update(ciphertext)
# Pad the ciphertext with zeros (to make it a multiple of 16 bytes)
if len(ciphertext) % 16 != 0:
authenticator.update(b'\0' * (16  len(ciphertext) % 16))
# Authenticate the length of the associated data (0 for simplicity)
authenticator.update((0).to_bytes(8, 'little')) # 64 bits
# Authenticate the length of the ciphertext
authenticator.update(len(ciphertext).to_bytes(8, 'little')) # 64 bits
# Generate the authentication tag
tag = authenticator.digest()
return (ciphertext, tag)
def chacha20poly1305_decrypt(key, nonce, ciphertext, tag):
assert len(key) == 32 # 256 bits
assert len(nonce) == 12 # 96 bits
assert len(tag) == 16 # 128 bits
# Initialize the ChaCha20 cipher and the Poly1305 authenticator, in the
# same exact way as it was done during encryption
cipher = ChaCha20.new(key=key, nonce=nonce)
subkey = cipher.encrypt(b'\0' * 32)
subkey_r = subkey[:16]
subkey_s = subkey[16:]
authenticator = Poly1305.Poly1305_MAC(r=subkey_r, s=subkey_s, data=None)
cipher.encrypt(b'\0' * 32)
# Generate the authentication tag, like during encryption
authenticator.update(ciphertext)
if len(ciphertext) % 16:
authenticator.update(b'\0' * (16  len(ciphertext) % 16))
authenticator.update((0).to_bytes(8, 'little'))
authenticator.update(len(ciphertext).to_bytes(8, 'little'))
expected_tag = authenticator.digest()
# Compare the input tag with the generated tag. If they're different, the
# plaintext must not be returned to the caller
if tag != expected_tag:
raise ValueError('authentication failed')
# The two tags match; decrypt the plaintext and return it to the caller
# Note that, because ChaCha20 is a symmetric cipher, there is no difference
# between the encrypt and decrypt method: here we are reusing the same
# exact code used during decryption
message = cipher.encrypt(ciphertext)
return message
And here is how it can be used:
key = bytes.fromhex('0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef')
nonce = bytes.fromhex('0123456789abcdef01234567')
message = b'I wanted to go to the beach, but now I changed my mind'
ciphertext, tag = chacha20poly1305_encrypt(key, nonce, message)
decrypted_message = chacha20poly1305_decrypt(key, nonce, ciphertext, tag)
assert message == decrypted_message
print(f'ciphertext: {ciphertext.hex()}')
print(f' tag: {tag.hex()}')
print(f' plaintext: {decrypted_message}')
Running it produces the following output:
ciphertext: 5d9b09cc5d90ca9ddff2d3470cfd6b563c5158e952bfae6acf1ebf9a3b968a488a41969567ef5ccfe05dcf9e548567028ff374a754af
tag: dac3c05d261920e278ceb22e2800aa95
plaintext: b'I wanted to go to the beach, but now I changed my mind'
This is the same output we would obtain by using the ChaCha20Poly1305 implementation from pycryptodome directly:
from Crypto.Cipher import ChaCha20_Poly1305
key = bytes.fromhex('0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef')
nonce = bytes.fromhex('0123456789abcdef01234567')
message = b'I wanted to go to the beach, but now I changed my mind'
cipher = ChaCha20_Poly1305.new(key=key, nonce=nonce)
ciphertext, tag = cipher.encrypt_and_digest(message)
print(f'ciphertext: {ciphertext.hex()}')
print(f' tag: {tag.hex()}')
As already stated, it is extremely important that the nonce passed to ChaCha20Poly1305 is unique. It may be predictable, but it must be unique. If the same nonce is reused twice or more, we can:

Decrypt arbitrary messages without using the secret key, if we can guess at least one message from its ciphertext.
This can be done using the techniques described at the beginning of this article: by recovering the random bit string from the XOR of the ciphertext with the guessed message.

Recover the Poly1305 subkey, and, at that point, tamper with ciphertexts and forge new, valid authentication tags.
This can be done by using algebraic methods on the polynomial of the authentication tag.
There is also a variant of ChaCha20Poly1305, called XChaCha20Poly1305, that features an extended 192bit nonce (the X stands for ‘extended’). This is described in an RFC draft but so far it hasn’t been accepted as a standard yet. I won’t cover XChaCha20 in detail here, because it’s slightly more complex and does not add much to the topic of this article, but XChaCha20Poly1305 has better security properties than ChaCha20Poly1305, so you should prefer it in your applications if you can use it. The reason why XChaCha20Poly1305 has better properties than ChaCha20Poly1305 is that, having a longer nonce, the probability of generating two random nonces with the same value are much lower.
Galois/Counter Mode (GCM)
Let’s now take a look at Galois/Counter Mode (GCM). This is commonly used with the Advanced Encryption Standard (AES), to construct the authenticated cipher AESGCM. One main difference between Poly1305 and GCM is that Poly1305 can work with any stream or block cipher, while GCM is designed to work with block ciphers with a block size of 128 bits.
GCM was proposed by David McGrew and John Viega in 2004 and is standardized in NIST Special Publication 80038D as well as RFC 5288. It takes its name from Galois fields, also known as finite fields, which in turn get their name from the French mathematician Évariste Galois, who introduced the concept of finite fields as we know them today.
As we did before with Poly1305, we are going to first see how the keyed hash function used by GCM works, and then we will see how to use it to construct an authenticated cipher like AESGCM on top of it. Before we can do that though, we need to understand what are finite fields, and what specific type of finite fields are used in GCM.
Finite Fields (Galois Fields)
What is a field? A field is a mathematical structure that contains a bunch of elements, and those elements can interact with each other using addition and multiplication. For both these operations there’s an identity element and an inverse element. Addition and multiplication in a field must obey the usual properties that we’re used to: commutativity, associativity, and distributivity.
A wellknown example of a field is the field of fractions. Here is why fractions form a field:
 the elements of the field are the fractions;
 addition is welldefined: if we add two fractions, we get a fraction out (example: $5/3 + 3/2 = 19/6$);
 multiplication is also welldefined: if we multiply two fractions, we get a fraction out (example: $1/2 \cdot 8/3 = 4/3$);
 the additive identity element is 0: if we add 0 to any fraction, we get the same fraction back;
 the additive inverse element is the negated fraction (example: $5/1$ is the additive inverse of $5/1$ because $5/1 + (5/1) = 0$);
 the multiplicative identity element is 1: multiplying any fraction by 1 yields the same fraction back;
 the multiplicative inverse element is what we get if we swap the numerator with the denominator (example: $3/2$ is the multiplicative inverse of $2/3$ because $3/2 \cdot 2/3 = 1$)—except for 0, which does not have a multiplicative inverse.
 and so on…
On top of addition, multiplication, and inverse elements, we can define derived operations like subtraction and division. Subtracting $a$ from $b$ is equivalent to adding $a$ to the additive inverse of $b$: $a  b = a + (b)$. Similarly, division can be defined in terms of multiplication with multiplicative inverses ($a / b = a b^{1}$).
Fields are a generalization of structures where addition, multiplication, subtraction, and division behave according to the rules that we’re used to. Field elements do not necessarily need to be numbers.
An example of something that is not a field is the integers. That’s because integers don’t have multiplicative inverses (for example, there’s no integer that multiplied by 5 makes the result equal to 1). However, there is a way to turn the integers into a field: if we take the integers and a prime number p, then we can construct the field of integers modulo p.
When we work in the integers modulo a prime p, whenever we see p appear in any of our expressions, we can replace it with 0. In other words, in such a field, p and 0 are two different ways to write the same element–they are two different representations of the same element.
Here is an example: in the field of integers modulo 7, the expression 5 + 3 equals 1 because
 5 + 3 evaluates to 8;
 8, by definition, is 7 + 1;
 if 7 and 0 are the same element, then 7 + 1 is equal to 0 + 1
 0 + 1 evaluates to 1
What we have just seen is that 8 is just a different representation of 1, just like 7 is a different representation of 0. Different symbols, same object. Just like, in programming languages, we can have multiple variables point to the same memory location: here the numbers are like variables, and what they point to is what really matters.
In the field of integers modulo 7, the additive inverse for 5 is 2, because 5 + 2 = 7 = 0. If we manipulate the equation, we get that 5 = −2. In other words, 5 and −2 are two different representations of the same element, and for the same reason 2 and −5 are also two different representations of the same element. A similar story holds for multiplication: the multiplicative inverse for 5 is 3 because: 5 · 3 = 15 = 7 + 7 + 1 = 1, so we can write 5 = 3^{−1} as well as 3 = 5^{−1}.
What we have just seen is an example of a finite field. It’s different from a general field because it contains a finite number of elements (unlike fractions, which do not have a limit). In the case of the integers modulo 7, the number of elements is 7, and the list of elements is: {0, 1, 2, 3, 4, 5, 6}, or {−3, −2, −1, 0, 1, 2, 3}, or {0, 1, 2, 3, 2^{−1}, 3^{−1}, 6}, depending on what representation we like the most.
A few words about terminology, notation, and equivalences of finite fields
There can be many ways to construct a finite field (or even a general field). I have given an example using numbers, but a field does not necessarily need to be formed from numbers. We can also use vectors, matrices, polynomials, and anything you would like. As long as addition, multiplication, identity elements, and inverse elements are welldefined, you can get a field. Using programming terms, you can think of a field as an interface or a trait that can have arbitrary implementations.
An important result in algebra is that finite fields with the same number of elements are unique up to isomorphism. This means that if two finite fields the same number of elements, then there is an equivalence relation between the two. The number of elements of a field is therefore enough to define a field. It’s not enough to tell us what the elements of the field look like, or how they can be represented, but it’s enough to know how it behaves. To denote a field with $n$ elements, there are two major notations: $GF(n)$ and $\mathbb{F}_{n}$.
Another important result in algebra is that $n$ may be either a prime number, or a power of a prime. For example, we can have finite fields with 2 elements, or with 9 (= 3^{2}) elements, but we cannot have a field with 6 (= 2·3) elements. For this reason, you will often find finite fields denoted as $GF(p^k)$ or $\mathbb{F}_{p^k}$, where $p$ is a prime and $k$ is an integer greater than 0. The prime $p$ is called characteristic of the field, while $n = p^k$ is called order of the field.
Some common fields also have their own notation: in particular, the field of integers modulo a prime $p$ is denoted as $Z/pZ$. This notation encodes the “building instructions” to construct the field, in fact:
 $Z$ denotes the integers: $Z = \{\dots, 2, 1, 0, 1, 2, \dots\}$;
 $pZ$ denotes the integers multiplied by $p$: $pZ = \{\dots, 2p, p, 0, p, 2p\}$ (example: $2Z = \{\dots, 4, 2, 0, 2, 4, \dots\}$);
 $A/B$ is a quotient. This is a way to define an equivalence relation between elements, and its meaning is: within $A/B$, all the elements of $B$ are equivalent to 0. In the case of $Z/pZ$, all the multiples of $p$ are equivalent to 0, which is indeed what happens with the integers modulo $p$. The way I described this equivalence relation earlier is by saying that multiples of $p$ are different representations for 0.
Note that the integers modulo a power of a prime ($Z/p^kZ$, with $k$ greater than 1) do not form a field. The problem is that elements in $Z/p^kZ$ sometimes do not have a multiplicative inverse. For example, in $Z/4Z$, the number 2 does not have a multiplicative inverse (there is no element that multiplied by 2 gives 1). A field $GF(p^k)$ with $k$ greater than 1 needs to be constructed in a different way. One such way is to use polynomials, as described in the next section.
Polynomial fields
Let’s now move our attention from integers to polynomials, like this one:
$$x^7 + 5x^3  9x^2 + 2x + 1$$
Polynomials are a sum of coefficients multiplied by a variable (usually denoted by the letter x) raised to an integral power.
Let’s restrict our view to polynomials that have integer coefficients, like the one shown above. Something that is not a polynomial with integer coefficients is $1/2 x^2 + x$, because it has a fraction in it.
Integers and polynomials with integer coefficients are somewhat similar to each other. They kinda behave the same in many aspects. One important property of integers is the unique factorization theorem: if we have an integer, there’s a way to write it as a multiplication of some primary factors. For example, the integer 350 can be factored as the multiplication of 2, 5, 5, and 7.
$$350 = 7 \cdot 5 \cdot 5 \cdot 2$$
This factorization is unique: we can change the order of the factors, but it’s not possible to obtain a different set of factors (there’s no way to make the number 3 appear in the factorization of 350, or to make the number 7 disappear).
Polynomials with integer coefficients also have a unique factorization. In the case of integers, We call the unique factors “prime numbers”; in the case of polynomials we have “irreducible polynomials”. And just like we can have a field of integers modulo a prime, we can have a field of polynomials modulo an irreducible polynomial.
Integers  Polynomials (with integer coefficients) 

Unique factorization: $42 = 7 \cdot 3 \cdot 2$  Unique factorization: $x^3  1 = (x^2 + x + 1)(x  1)$ 
Prime numbers: 2, 3, 5, 7, 11, …  Irreducible polynomials: $x + 1$, $x^2  2$, $x^2 + x + 1$, … 
Integers modulo a prime number  Polynomials modulo an irreducible polynomial 
Let’s take a look at how arithmetic in polynomial fields work. Let’s take, for example, the field of polynomials with integer coefficients modulo $x^3 + x + 1$, and try to compute the result of $(x^2 + 1)(x^2 + 2)$. If we expand the expression, we get:
$$(x^2 + 1)(x^2 + 2) = x^4 + 3x^2 + 2$$
This expression can be reduced. Reducing a polynomial expression is the equivalent of what we were doing with integers modulo a prime, when we were saying that 8 = 7 + 1 = 1 (mod 7). That “conversion” from 8 to 1 is the equivalent of the reduction that we’re talking about here.
To reduce $x^4 + 3x^2 + 2$, first note that $x^4 = x \cdot x^3$. Also note that $x^3 = x^3 + x + 1  x  1$. Here we have just added and removed the term $x + 1$: the result hasn’t changed, but now the irreducible polynomial $x^3 + x + 1$ appears in the expression, and so we can substitute it with 0. Putting everything together, we get:
$$\begin{align*} (x^2 + 1)(x^2 + 2) & = x^4 + 3x^2 + 2 \\ & = x \cdot x^3 + 3x^2 + 2 \\ & = x \cdot (x^3 + x + 1  x  1) + 3x^2 + 2 \\ & = x \cdot (0  x  1) + 3x^2 + 2 \\ & = x^2  x + 3x^2 + 2 \\ & = 2x^2  x + 2 \end{align*}$$
It’s interesting to note that, if the polynomial field is formed by an irreducible polynomial with degree $n$, then all the polynomials in that field will all have degree less than $n$. That’s because if any $x^n$ (or higher) appears in a polynomial expression, then we can use the substitution trick I just showed to reduce its degree.
Binary fields
Let’s now look at polynomials where coefficients are from the field of integers modulo 2, meaning that they can be either 0 or 1. This is an example of such a polynomial:
$$x^7 + x^4 + x^2 + 1$$
or, in a more explicit form, where we can clearly see all the coefficients:
$$1 x^7 + 0 x^6 + 0 x^5 + 1 x^4 + 0 x^3 + 1 x^2 + 0 x^1 + 1 x^0$$
These are called binary polynomials. It’s interesting to note that if we ignore the variables and the powers, and keep only the coefficients, then what we get is a bit string:
$$(1 0 0 1 0 1 0 1)$$
This suggests that there’s an interesting duality between binary polynomials and bit strings. This means, in particular, that binary polynomials can be represented in a very compact and natural way on computers.
The duality between binary polynomials and bit strings also suggests that perhaps we can use bitwise operations to perform arithmetic on binary polynomials. And this turns out to be true, in fact:
 binary polynomial addition can be computed using the XOR on the two corresponding bit strings;
 binary polynomial multiplication can be computed using XOR, AND and bitshifting.
Computers are pretty fast at performing these bitwise operations, and this makes binary polynomials quite attractive for use in computer algorithms and cryptography.
Arithmetic with binary polynomials
The arithmetic of such polynomials is quite interesting: in fact, because $1 + 1 = 0$ (modulo 2), then also $x^k + x^k = 0$, in fact:
$$1 \cdot x^k + 1 \cdot x^k = (1 + 1) x^k = 0 \cdot x^k = 0$$
It’s easy to see that addition modulo 2 is equivalent to the XOR binary operator. And addition of two binary polynomials is equivalent to the bitwise XOR of their corresponding bit strings:
$$\begin{array}{ccccc} (x^3 + x^2 + 1) & + & (x^2 + x) & = & x^3 + x + 1 \\ \updownarrow & & \updownarrow & & \updownarrow \\ (1101) & \oplus & (0110) & = & (1011) \end{array}$$
Multiplication of binary polynomials can also be implemented as a bitwise operation on bit strings. First, note that multiplying a polynomial by a monomial is equivalent to bitshifting:
$$\begin{array}{ccccc} (x^3 + x + 1) & \cdot & x^2 & = & x^5 + x^3 + x^2 \\ \updownarrow & & \updownarrow & & \updownarrow \\ (1011) & \ll & 2 & = & (101100) \end{array}$$
Then note that multiplication of two polynomials can be expressed as the sum of multiplications by monomials:
$$(x^3 + 1)(x^2 + x + 1) = (x^3 + 1) \cdot x^2 + (x^3 + 1) \cdot x^1 + (x^3 + 1) \cdot x^0$$
Putting everything together, we have multiplications by monomials (equivalent to bitshifts) and sums (equivalent to bitwise XOR). This suggests that multiplication can be implemented on top of bitwise XOR and bitshifting.
Here is some Python code to implement binary polynomial multiplication, where
each polynomial is represented compactly as an int
:
def multiply(a, b):
"""
Compute a*b, where a and b are two integers representing binary
polynomials.
a and b are expected to have their most significant bit set to
the monomial with the highest power. For example, the polynomial
x^8 is represented as the integer 0b10000.
"""
assert a >= 0
assert b >= 0
result = 0
while b:
result ^= a * (b & 1)
a <<= 1
b >>= 1
return result
Other than XOR and bitshifting, this code also uses AND to “query” whether a certain monomial is present or not.
Here is an example of how to use the code:
a = 0b0101_0111 # x^6 + x^4 + x^2 + x + 1
b = 0b0001_1010 # x^4 + x^3 + x
c = multiply(a, b)
assert c == 0b0111_0110_0110 # x^10 + x^9 + x^8 + x^6 + x^5 + x^2 + x
Now that we have introduced binary polynomials, we can of course form binary polynomials modulo a binary irreducible polynomial. These form a finite field, which is more concisely called: binary field.
Note that in a binary field where the modulo is an irreducible polynomial of degree $n$, all polynomials in the field can be represented as $n$bit strings, and all $n$bit strings have a corresponding binary polynomial in the field.
Arithmetic in binary fields
If we have three integers $a$, $b$, and $p$, we can compute $(a + b) \bmod{p}$ or $a \cdot b \bmod{p}$ by performing the binary operation (addition or multiplication) and then taking the remainder of the division by $p$. This is a method that returns the results of addition or multiplication using a representation with the lowest number of digits possible.
What if instead of having 3 integers we have three binary polynomials $A$, $B$, and $P$ and we want to compute $(A + B) \bmod{P}$ or $A \cdot B \bmod{P}$? It turns out that these operations can be implemented with code that is even easier than the integer counterpart: no division needs to be involved!
Let’s start with addition: we have already seen that addition with binary polynomials can be implemented with a simple XOR operation. This means that if the degree of $A$ and $B$ is lower than the degree of $P$, then the result of $A + B$ is also going to have degree less than $P$, hence no reduction is needed. We can use the result asis, without any transformation: adding two binary field elements can be implemented with a single XOR operation.
With multiplication the story is different: the product $A \cdot B$ may have degree equal to or higher than $P$. For example, if $A = B = x$ and $P = x^2 + 1$, the product $A \cdot B$ is equal to $x^2$, which has the same degree as $P$. We need to find a way to efficiently reduce the higherdegree terms of this product. To see one way to do that, note that we can write $P$ like this:
$$P = x^n + Q$$
where $n$ is the degree of $P$ (the maximum power of $P$) and $Q$ is another binary polynomial, with degree strictly lower than $n$. Rearranging the equation, we get:
$$x^n = P + Q$$
Note that subtraction and addition are the same operations in a binary field. Because $P$ equals 0, we can write:
$$x^n = Q$$
This equivalence gives us a way to eliminate higherlevel terms that appear during multiplication: whenever we see an $x^n$ appearing in the result, we can remove that term and add $Q$ instead. One way to do that, using binary strings, is to discard the highest bit (the one corresponding to $x^n$) and XOR with the binary string corresponding to $Q$.
Another way to do it is to just add $P$ (XOR by the binary string corresponding to $P$). This is equivalent to adding 0, results in the more compact representation that we’re interested in.
We could use similar tricks to eliminate terms like $x^{n+1}$, but these tricks are not necessary if we eliminate $x^n$ terms as soon as they appear in an iterative way.
Here is some Python code for multiplication in binary fields that uses the “add $P$” trick just described:
def multiply(a, b, p):
"""
Compute a*b modulo p, where a, b and c are three integers representing
binary polynomials.
a, b and p are expected to have their most significant bit set to the
highest power monomial. For example, the polynomial x^8 is represented as
0b10000.
"""
bit_length = p.bit_length()  1
assert a >= 0 and a < (1 << bit_length)
assert b >= 0 and b < (1 << bit_length)
result = 0
for i in range(bit_length):
result ^= a * (b & 1)
a <<= 1
a ^= p * ((a >> bit_length) & 1)
b >>= 1
return result
This code is essentially the same as the binary polynomial multiplication code we had before, except for this line in the for
loop:
a ^= p * ((a >> bit_length) & 1)
This line is what “adds $P$” whenever adding the shifted $A$ would result in a $x^n$ term to appear.
Again, we achieved implementing multiplication using only XOR, AND and bitshifting.
Note that the binary polynomial $P$ here does not necessarily need to be an irreducible polynomial for this algorithm to work. However, the resulting algebraic structure won’t be a field unless $P$ is irreducible. A similar story holds for integers: we can have integers modulo a nonprime number, but that’s not a field.
The GHASH keyed hash function
GCM uses a binary field. The irreducible binary polynomial that defines the binary field used by GCM is:
$$P = x^{128} + x^7 + x^2 + x + 1$$
We will call this field the GCM field. Note that this polynomial has degree 128, hence the GCM field elements can be represented as 128bit strings, and each 128bit string has a corresponding element in the GCM field.
The keyed hash function used by GCM is called GHASH and takes as input a 128bit key. We will call this key $H$. This key is interpreted as an element of the GCM field.
The message to authenticate is split into blocks of 128 bits each: $M_1$, $M_2$, $M_3$, … $M_n$. If the length of the message is not a multiple of 128 bits, then the last block is padded with zeros. Each block of message is also interpreted as an element of the GCM field.
Here is how the authentication tag is computed from $H$ and the padded message blocks $M_1$, …, $M_n$:
 The initial state (a GCM field element) is initialized to 0: $A_0 = 0$.
 For every block of message $M_i$, the next state $A_i$ is computed as $A_i = (A_{i1} + M_i) \cdot H \bmod{P}$.
 The final state $A_n$ is returned as a 128bit string.
What this function is doing is computing the following polynomial in $H$:
$$\begin{align*} Tag & = (((M_1 \cdot H + M_2) \cdot H + \cdots M_n) \cdot H) \bmod{P} \\ & = (M_1 H^n + M_2 H^{n1} + \cdots M_n H) \bmod{P} \end{align*}$$
This construction is somewhat similar to the one from Poly1305, although there are important differences:
 In Poly1305, the elements of the tag polynomial are integers modulo a prime, in GHASH they are elements of a binary field.
 GHASH does not perform any step to encode the length of the message, hence the tag for an empty message will be the same as the tag for a sequence of zero blocks. We will see later that GCM fixes this problem by appending the length of the message to the end of the input passed to GHASH.
 Most importantly, the final $Tag$ polynomial is a polynomial in one unknown, and as such $H$ may be easily recoverable using algebraic methods. For this reason, GHASH is not suitable as a secure onetime authenticator. We will see that GCM fixes this problem by encrypting the output of GHASH.
Use of GCM with AES (AESGCM)
GCM is the combination of a block cipher, Counter Mode (CTR), and the GHASH function that we have just seen. The block cipher is often AES. When we combine AES with GCM, the what we get is AESGCM, which is described below. However the block cipher does not necessarily need to be AES: what is important is that the block size of the cipher is 128 bits, and that’s because GHASH only works on 128bit blocks.
The inputs to the AESGCM encryption function are:
 a secret key (the length of the key depends on the variant of AES used: if AES128, this will be 128 bits);
 a 96bit nonce;
 a variablelength plaintext message.
The outputs of the AESGCM encryption function are:
 a variablelength ciphertext (same length as the input plaintext);
 a 128bit authentication tag.
The AESGCM decryption function will accept the same secret key, nonce, ciphertext, and authentication tag as the input, and produce either the plaintext or an error as the output. The error is returned in case the authentication fails.
AESGCM works in the following way:

The GHASH subkey $H$ is generated by encrypting a zeroblock: $H = \operatorname{Encrypt}(key, \underbrace{000\dots0}_\text{128 bits})$.

The block cipher AES is initialized in Counter Mode (AESCTR) with the key, the nonce, and a 32bit, bigendian counter starting at 2.

The plaintext is encrypted using the instance of AESCTR just created.

The GHASH function is run with the following inputs:
 the subkey $H$, computed in step 1;
 the ciphertext padded with zeros to make its length a multiple of 16 bytes (128 bits), concatenated to the length (in bits) of the ciphertext represented as a 128bit bigendian integer.
The result is a 128bit block $S = \operatorname{GHASH}(H, ciphertext  padding  length)$.

The AESCTR counter is set to 1.

The block $S$ is then encrypted using AESCTR. The result of the encryption is the authentication tag.
Note that, because $S$ matches the block size of the cipher, this encryption won’t cause the counter value 2 to be reused.

The ciphertext and authentication tag are returned.
Here is how AESGCM and GHASH can be implemented in Python, using the AES implementation from pycryptodome (usual disclaimer: this code is for educational purposes, and it’s not necessarily secure or optimized for performance):
from Crypto.Cipher import AES
def multiply(a: int, b: int) > int:
"""
Compute a*b in the GCM field, where a and b are two integers representing
elements of the GCM field.
a and b are expected to have their least significant bit set to the highest
power monomial. For example, the polynomial x^125 is represented as 0b100.
"""
bit_length = 128
q = 0xe1000000000000000000000000000000
assert a >= 0 and a < (1 << bit_length)
assert b >= 0 and b < (1 << bit_length)
result = 0
for i in range(bit_length):
result ^= a * ((b >> 127) & 1)
a = (a >> 1) ^ (q * (a & 1))
b <<= 1
return result
def pad_block(data: bytes) > bytes:
"""
Pad data with zero bytes so that the resulting length is a multiple of 16
bytes (128 bits).
"""
if len(data) % 16 != 0:
data += b'\0' * (16  len(data) % 16)
return data
def iter_blocks_padded(data: bytes):
"""
Split the given data into blocks of 16 bytes (128 bits) each, padding the
last block with zeros if necessary.
"""
start = 0
while start < len(data):
yield pad_block(data[start:start+16])
start += 16
def ghash(subkey: bytes, message: bytes) > bytes:
subkey = int.from_bytes(subkey, 'big')
assert subkey < (1 << 128)
state = 0
for block in iter_blocks_padded(message):
block = int.from_bytes(block, 'big')
state = multiply(state ^ block, subkey)
return state.to_bytes(16, 'big')
def aes_gcm_encrypt(key: bytes, nonce: bytes, message: bytes):
assert len(key) in (16, 24, 32)
assert len(nonce) == 12
# Initialize a raw AES instance and encrypt a 16byte block of all zeros to
# derive the GHASH subkey H
cipher = AES.new(mode=AES.MODE_ECB, key=key)
h = cipher.encrypt(b'\0' * 16)
# Encrypt the message with AES in CTR mode, with the counter composed by
# the concatenation of the 12 byte (96 bits) nonce and a 4 byte (32 bits)
# integer, starting from 2
cipher = AES.new(mode=AES.MODE_CTR, key=key, nonce=nonce, initial_value=2)
ciphertext = cipher.encrypt(message)
# Compute the GHASH of the ciphertext plus the ciphertext length in bits
s = ghash(h, pad_block(ciphertext) + (len(ciphertext) * 8).to_bytes(16, 'big'))
# Encrypt the GHASH value using AES in CTR mode, with the counter composed
# by the concatenation of the 12 byte (96 bits) nonce and a 4 byte (32
# bits) integer set at 1. The GHASH value fits in one block, so the counter
# won't be increased during this round of encryption
cipher = AES.new(mode=AES.MODE_CTR, key=key, nonce=nonce, initial_value=1)
tag = cipher.encrypt(s)
return (ciphertext, tag)
def aes_gcm_decrypt(key: bytes, nonce: bytes, ciphertext: bytes, tag: bytes):
assert len(key) in (16, 24, 32)
assert len(nonce) == 12
assert len(tag) == 16
# Compute the GHASH subkey, the GHASH value, and the authentication tag, in
# the same exact way as it was done during encryption
cipher = AES.new(mode=AES.MODE_ECB, key=key)
h = cipher.encrypt(b'\0' * 16)
s = ghash(h, pad_block(ciphertext) + (len(ciphertext) * 8).to_bytes(16, 'big'))
cipher = AES.new(mode=AES.MODE_CTR, key=key, nonce=nonce, initial_value=1)
expected_tag = cipher.encrypt(s)
# Compare the input tag with the generated tag. If they're different, the
# plaintext must not be returned to the caller
if tag != expected_tag:
raise ValueError('authentication failed')
# The two tags match; decrypt the plaintext and return it to the caller.
# Note that, because AESCTR is a symmetric cipher, there is no difference
# between the encrypt and decrypt method: here we are reusing the same
# exact code used during decryption
cipher = AES.new(mode=AES.MODE_CTR, key=key, nonce=nonce, initial_value=2)
message = cipher.encrypt(ciphertext)
return message
And here is how the code can be used:
key = bytes.fromhex('0123456789abcdef0123456789abcdef0123456789abcdef0123456789abcdef')
nonce = bytes.fromhex('0123456789abcdef01234567')
message = b'I went to the zoo yesterday but not today'
ciphertext, tag = aes_gcm_encrypt(key, nonce, message)
print(f'ciphertext: {ciphertext.hex()}')
print(f' tag: {tag.hex()}')
decrypted_message = aes_gcm_decrypt(key, nonce, ciphertext, tag)
print(f' plaintext: {decrypted_message}')
assert message == decrypted_message
This snippet produces the following output:
ciphertext: e0c32db2962f9b729c69028d9a1fdfb2c93839fc1188f314c58ee97fd6a242404953bb208df609a33c
tag: 9fa6fe2f77a0c98282868924ace0e4ec
plaintext: b'I went to the zoo yesterday but not today'
This is the same output we would obtain by using the AESGCM implementation from pycryptodome directly:
from Crypto.Cipher import AES
cipher = AES.new(mode=AES.MODE_GCM, key=key, nonce=nonce)
ciphertext, tag = cipher.encrypt_and_digest(message)
print(f'ciphertext: {ciphertext.hex()}')
print(f' tag: {tag.hex()}')
Nonce reuse is catastrophic for AESGCM in two ways:

Because the ciphertext produced by AESGCM is just a variant of AESCTR, nonce reuse with GCM can have the same consequences as nonce reuse with AESCTR, or any other stream cipher: if someone is able to guess the plaintext, they can recover the random stream, and use that to decrypt other messages (or portions of them).

If the same nonce is used twice or more, the GHASH subkey $H$ will always be the same. Even if the output of GHASH is encrypted in step 7, we can use the XOR of two authentication tags to “cancel” the encryption and obtain a polynomial in $H$. From there, we can use algebraic methods to recover $H$. This gives us the ability to forge new, valid authentication tags.
It’s worth mentioning that there’s a variant of AESGCM, called AESGCMSIV, (Synthetic Initialization Vector) specified in RFC 8452. This differs from AESGCM in that it uses a littleendian version of GHASH called POLYVAL (which is faster on modern CPUs), and in that it allows nonce reuse without the two catastrophic consequences that I mentioned above.
(Nonce reuse with AESGCMSIV however still presents a problem, just not as serious as the two ones above: specifically, it breaks ciphertext indistinguishability.)
Authenticated Encryption with Associated Data (AEAD)
The way I have described authenticated encryption, and in particular the constructions ChaCha20Poly1305 and AESGCM, is accurate, but incomplete. What I have told you is that when you use an authenticated encryption cipher, the ciphertext is checked for integrity and authenticity. But we can use the same technique to authenticate anything, not just ciphertexts: we can, for example, authenticate some plaintext data, or authenticate a piece of plaintext data and a piece of ciphertext altogether.
When we use a method to authenticate a plaintext message only, what we get is a Message Authentication Code (MAC). We don’t use the word “encryption” in this context, because the confidentiality of the message is not ensured (only its authenticity).
When we use a method to authenticate both a ciphertext and a plaintext message, what we get is Authenticated Encryption with Associated Data (AEAD). In this construction, there are two messages involved: one to be encrypted (resulting in a ciphertext), and one to be kept in plaintext. The plaintext message is called “associated data” (AD) or “additional authenticated data” (AAD). Both the ciphertext and the associated data are authenticated at encryption time, so their integrity and authenticity will be enforced.
The inputs to the encryption function of an AEAD cipher are, generally speaking:
 a key;
 a nonce;
 the additional data;
 the message to encrypt.
The outputs of the encryption are:
 the ciphertext;
 the authentication tag.
Note that there’s only one authentication tag that covers both the additional data and the ciphertext.
The inputs to the decryption function are:
 the key used for encryption;
 the nonce used for encryption;
 the additional data used for encryption;
 the ciphertext.
And the output of the decryption is either an error or the decrypted message.
It’s important to note that the associated data must be both at encryption time and decryption time. Changing a single bit of it will make the entire decryption operation fail.
Both ChaCha20Poly1305 and AESGCM (and their variants, XChaCha20Poly1305 and AESGCMSIV) are AEAD ciphers. Here’s how they implement AEAD:
 When the Poly1305 or GHASH authenticator is first initialized, they are fed the additional data, padded with zeros to make its size a multiple of 16 bytes (128 bits).
 Then the padded ciphertext is fed into the authenticator.
 The length of the additional data and the length of the ciphertext are represented as two 64bit integers, concatenated, and fed into the authenticator.
If the additional data is empty, then what you get are exactly the constructions that I described earlier in this article.
Authenticated Encryption with Associated Data is useful in situations where you want to encode some metadata along with your encrypted data. For example: an identifier for the resource that is encrypted, or the type of data encrypted (text, image, video, …), or some information that indicates what key and algorithm was used to encrypt the resource, or maybe the expiration of the data. The associated data is in plaintext so systems that do not have access to the secret key can gather some properties about the encrypted resource. It must however be understood that the associated data cannot be trusted until it’s verified using the secret key. Systems that analyze the associated data must be designed in such a way that, if the associated data is tampered, nothing bad will happen, and such tampering attempt will be detected sooner or later.
A word of caution
Something very important to understand is that when using authenticated encryption ciphers like ChaCha20Poly1305 or AESGCM, decryption can in theory succeed even if the verification of the authentication tag fails.
For example, we can decrypt a ciphertext encrypted with ChaCha20Poly1305 by using ChaCha20 and ignoring the authentication tag. Similarly, we can decrypt a ciphertext encrypted with AESGCM by using AESCTR and, again, ignoring the authentication tag. This possibility opens the doors to all the nasty scenarios that we have seen at the beginning of this article, removing all the benefits of authenticated encryption.
Perhaps the most important thing to remember when using authenticated encryption is: never use decrypted data until you have verified its authenticity.
Why am I emphasizing this? Because some AE or AEAD implementations do return plaintext bytes before verifying their authenticity.
The code samples that I have provided do the following: they first calculate the authentication tag, compare it to the input tag, and only if the comparison succeeds they perform the decryption. This is a simple approach, but it may be expensive when encrypting large amounts of data (for example: several gigabytes). The reason why this approach is expensive is that, if the ciphertext is too large, it may not fit all in memory, and the ciphertext would have to be read from the storage device twice: once for calculating the tag, and once for decrypting the ciphertext. Also, chances are that by the time the application has computed the tag, the underlying ciphertext may have changed without detection.
What some authenticated encryption implementations do when dealing with large amounts of data is that they calculate the tag and perform the decryption in parallel. They read the ciphertext chunkbychunk, and pass each chunk to both the authenticator and the decryption function, returning a chunk of decrypted bytes to the caller at each iteration. Only at the end, when the full ciphertext has been read, the authenticity is checked, and the application may return an error only at that point. With such implementations, it is imperative that the exit status of the application is checked before using any of the decrypted bytes.
An implementation that works like that (returning decrypted bytes before authentication is complete) is GPG. Here is an example of the output that GPG produces when decrypting a tampered message:
gpg: AES256.CFB encrypted data
gpg: encrypted with 1 passphrase
This is a very long message.
gpg: WARNING: encrypted message has been manipulated!
The decrypted message (“This is a very long message”) got printed, together with a warning, and the exit status is 2, indicating that an error occurred. It is important in this case that the decrypted message is not used in any way.
Other implementations avoid this problem by simply not encrypting large amounts of data. If given a large file to encrypt, the file is first split into multiple chunks of a few KiB, then each chunk is encrypted independently, with its own nonce and authentication tag. Because each chunk is small, authentication and decryption can happen in memory, one before the other. If a chunk was tampered, decryption would stop, returning truncated output, but never tampered output. It’s still important to check the exit status of such an implementation, but the consequences are less catastrophic than before. The drawback of this approach is that the total size of the ciphertext increases, because each chunk requires a nonce, an authentication tag, and some information about the position of the chunk (to prevent the chunks from being reordered). Storing the nonces or the positions can be avoided by using an algorithm to generate them on the fly, but storing the tag cannot be avoided.
The method of splitting that I have just described (of splitting long messages into chunks that are individually encrypted and authenticated) is used for example in TLS, as well as the command line tool AGE.
Summary and final considerations
At the beginning of this article we have seen some risks of using bare encryption ciphers: one of them in particular was malleability, that is: the property that ciphertexts may be modified without detection.
This problem was addressed by using Authenticated Encryption (AE) or Authenticated Encryption with Associated Data (AEAD), which are methods to provide integrity and authenticity in addition to confidentiality when encrypting data.
We have seen the details of the two most popular authenticated encryption ciphers and briefly mentioned some of their variants. Their features are summarized here:
Cipher  Cipher Type  Key Size  Nonce Size  Nonce Reuse  Tag Size 

ChaCha20Poly1305  Stream, AEAD  256 bits  96 bits  Catastrophic  128 bits 
XChaCha20Poly1305  Stream, AEAD  256 bits  192 bits  Catastrophic  128 bits 
AESGCM  Stream, AEAD  128, 192, 256 bits  96 bits  Catastrophic  128 bits 
AESGCMSIV  Stream, AEAD  128 or 256 bits  96 bits  Reduced risk  128 bits 
Authenticated encryption is used in most of our modern protocols, including TLS, S/MIME, PGP/GPG, and many more. Failure to implement authenticated encryption correctly has lead to some serious issues in the past.
Whenever you’re using encryption, ask yourself: how is integrity and authentication verified? And remember: it’s essential to verify the authenticity of data before using it.
I hope you enjoyed this article! As usual, if you have any suggestions or spotted some mistakes, let me know in the comments or by contacting me!
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