Books / Crypto 101 / Chapter 7

Message authentication codes

A MAC is a small bit of information that can be used to check the authenticity and the integrity of a message. These codes are often called “tags”. A MAC algorithm takes a message of arbitrary length and a secret key of fixed length, and produces the tag. The MAC algorithm also comes with a verification algorithm that takes a message, the key and a tag, and tells you if the tag was valid or not. (It is not always sufficient to just recompute a tag and check if they are the same; many secure MAC algorithms are randomized, and will produce different tags every time you apply them.)

Note that we say “message” here instead of “plaintext” or “ciphertext”. This ambiguity is intentional. In this book we’re mostly interested in MACs as a way to achieve authenticated encryption, so the message will always be a ciphertext. That said, there’s nothing wrong with a MAC being applied to a plaintext message. In fact, we will be seeing examples of secure authenticated encryption schemes that explicitly allow for authenticated (but not encrypted) information to be sent along with the authenticated ciphertext.

Often, when you just want to talk about the authenticity and integrity of a particular message, it may be more practical to use a signature algorithm, which we’ll talk about in a later chapter. For now, all you need to know is that the term “signature” is normally reserved for asymmetric algorithms, whereas this chapter deals with symmetric algorithms.

Secure MACs

We haven’t quite defined yet exactly which properties we want from a secure MAC.

We will be defending against an active attacker. The attacker will be performing a chosen message attack. That means that an attacker will ask us the tag for any number of messages \(m_i\), and we’ll answer truthfully with the appropriate tag \(t_i\).

An attacker will then attempt to produce an existential forgery, a fancy way of saying that they will produce some new valid combination of \((m, t)\). The obvious target for the attacker is the ability to produce valid tags \(t^{\prime}\) for new messages \(m^{\prime}\) of their choosing. We will also consider the MAC insecure if an attacker can compute a new, different valid tag \(t^{\prime}\) for a message \(m_i\) that we previously gave them a valid tag for.

Why does a MAC take a secret key?

If you’ve had to deal with verifying the integrity of a message before, you may have used checksums (like CRC32 or Adler32) or even cryptographic hashes (like the SHA family) in order to compute a checksum for the message (depending on the algorithm and who you’re talking to, they may have called it “hash” or “digest”, too).

Let’s say that you’re distributing a software package. You have some tarballs with source code in them, and maybe some binary packages for popular operating systems. Then you put some (cryptographically secure!) hashes right next to them, so that anyone who downloads them can verify the hashes and be confident that they downloaded what they think they downloaded.

Of course, this scheme is actually totally broken. Computing those hashes is something everyone can do. You’re even relying on that fact for your user to be able to verify their download. That also means that an attacker that modified any of the downloads can just compute the hash again for the modified download and save that value. A user downloading the modified file will compute its hash and compare it against the modified hash, and conclude that the download worked. The scheme provided no help whatsoever against an attacker modifying the download, either as stored, or in transit.

In order to do this securely, you would either apply a signature algorithm to the binaries directly, or by signing the digests, as long as the hash function used to produce the digest is secure against second-preimage attacks. The important difference is that producing a signature (using either a pre-shared key with your users, or, preferably, a public-key signature algorithm) is not something that an attacker can do. Only someone who has the secret keys can do that.

Combining MAC and message

As we’ve mentioned before, unauthenticated encryption is bad. That’s why we introduced MACs. Of course, for a MAC to be useful, it has to make it to the recipient. Since we’re explicitly talking about authenticating encryption, now, we’ll stop using the word “message” and instead use the less ambiguous “plaintext” and “ciphertext”.

There are three common ways to combine a ciphertext with a MAC.

  1. Authenticate and encrypt. You authenticate and encrypt the plaintext separately. This is how SSH does it. In symbols: \(C = E(K_C, P)\), \(t = MAC(K_M, P)\), and you send both ciphertext \(C\) and tag \(t\).
  2. Authenticate, then encrypt. You authenticate the plaintext and then encrypt the combination of the plaintext and the authentication tag. This is how TLS usually does it. In symbols: \(t = MAC(K_M, P)\), \(C = E(K_C, P \| t)\), and you only send \(C\). (You don’t need to send \(t\), because it’s already an encrypted part of \(C\).)
  3. Encrypt, then authenticate. You encrypt the plaintext, compute the MAC of that ciphertext. This is how IPSec does it. In symbols: \(C = E(K_C, P)\), \(t = MAC(K_M, C)\), and you send both \(C\) and \(t\).

All of these options were studied and compared extensively.

We now know that out of all of these, encrypt-then-authenticate is unequivocally the best option. It’s so emphatically the best option that Moxie Marlinspike, a well-respected information security researcher, has a principle called “The Cryptographic Doom Principle” for any system that does not follow this pattern. Moxie claims that any system that does anything before checking the MAC is doomed. Both authenticate-and-encrypt and authenticate-then-encrypt require you to decrypt something before you can verify the authentication.

Authenticate-then-encrypt

Authenticate-then-encrypt is a poor choice, but it’s a subtle poor choice. It can still be provably secure, but only under certain conditions.

At first sight, this scheme appears to work. Sure, you have to decrypt before you can do anything, but to many cryptographers, including the designers of TLS, this did not appear to pose a problem.

In fact, prior to rigorous comparative study of different composition mechanisms, many preferred this setup. In a critique of IPSec, Schneier and Ferguson, two veteran cryptographers, considered IPSec’s use of encrypt-then-authenticate was a flaw, preferring TLS’s authenticate-then-encrypt. While they may have had a plausible (albeit mostly heuristic) argument for the time, this criticism is completely superseded by the provable security of encrypt-then-authenticate schemes.

Authenticate-and-encrypt

Authenticate-and-encrypt has some serious problems. Since the tag authenticates the plaintext and that tag is part of the transmitted message, an attacker will be able to recognize two plaintext messages are the same because their tags will also be the same. This essentially leads to the same problem we saw with ECB mode, where an attacker can identify identical blocks. That’s a serious problem, even if they can’t decrypt those blocks.

TODO: Explain how this works in SSH (see Moxie’s Doom article)

A naive attempt with hash functions

Many ways of constructing MACs involve hash functions. Perhaps one of the simplest ways you could imagine doing that is to just prefix the message with the secret key and hash the whole thing:

\[t = H(k \| m)\]

This scheme is most commonly called “Prefix-MAC”, because it is a MAC algorithm that works by using the secret key as a prefix.

The cryptographically secure hash function \(H\) guarantees a few things that are important to us here:

  • The tag \(t\) will be easy to compute; the hash function \(H\) itself is typically very fast. In many cases we can compute the common key part ahead of time, so we only have to hash the message itself.
  • Given any number of tags, there is no way for an attacker to “invert” the hash function to recover \(k\), which would allow them to forge arbitrary messages.
  • Given any number of tags, there is no way for an attacker to “rewind” the hash function to recover \(H(k)\), which may allow them to forge almost arbitrary messages.

One small caveat: we’re assuming that the secret key \(k\) has enough entropy. Otherwise, we have the same issue that we had for password storage using hash functions: an attacker could just try every single \(k\) until one of them matches. Once they’ve done that, they’ve almost certainly found the correct \(k\). That’s not really a failure of the MAC though: if your secret key contains so little entropy that it’s feasible for an attacker to try all of them, you’ve already lost, no matter which MAC algorithm you pick.

Breaking prefix-MAC

Despite being quite common, this MAC is actually completely insecure for most (cryptographically secure!) hash functions \(H\), including SHA-2.

As we saw in the chapter on hash functions, many hash functions, such as MD5, SHA-0, SHA-1 and SHA-2, pad the message with a predictable padding before producing the output digest. The output digest is the same thing as the internal state of the hash function. That’s a problem: the attacker can use those properties to forge messages.

First, they use the digest as the internal state of the hash function. That state matches the state you get when you hash \(k \| m \| p\), where \(k\) is the secret key, \(m\) is the message, and \(p\) is that predictable padding. Now, the attacker gets the hash function to consume some new bytes: the attacker’s chosen message \(m^{\prime}\). The internal state of the hash function is now what you get when you feed it \(k \| m \| p \| m^{\prime}\). Then, the attacker tells the hash function to produce a digest. Again, the hash function appends a padding, so we’re now at \(k \| m \| p \| m^{\prime} \| p^{\prime}\). The attacker outputs that digest as the tag. That is exactly the same thing as what happens when you try to compute the tag for the message \(m \| p \| m^{\prime}\) under the secret key \(k\). So, the attacker has successfully forged a tag for a new message, and, by our definition, the MAC is insecure.

This attack is called a length extension attack, because you are extending a valid message. The padding in the middle \(p\), which started out as the padding for the original message but has become just some data in the middle, is called glue padding, because it glues the original message \(m\) and the attacker’s message \(m^{\prime}\) together.

This attack might sound a little academic, and far from a practical problem. We may have proven that the MAC is insecure by our definition, but the only tags the attacker can successfully forge are for very limited modifications of real messages. Specifically, the attacker can only forge tags for a message that consists of a message we sent, followed by some binary junk, followed by something the attacker chooses. However, it turns out that for many systems, this is plenty to result in real breaks. Consider the following Python code that parses a sequence of key-value pairs that look like k1=v1&k2=v2&...:1

def parse(s):
    pairs = s.split("&")
    parsed = {}
    for pair in pairs:
        key, value = pair.split("=")
        parsed[key] = value
    return parsed

The parsing function only remembers the last value for a given key: previous values in the dictionary are overwritten. As a result, an attacker mounting a length extension attack can effectively control the parsed dictionary entirely.

If you’re thinking that this code has many issues; sure, it does. For example, it doesn’t handle escaping correctly. But even if it did, that wouldn’t really fix the length extension attack problem. Most parsing functions will perfectly happily live with that binary junk in the middle. Hopefully it convinces you that there is in fact a pretty good chance that an attacker can produce messages with valid tags that say something entirely different from what you intended.

The prefix-MAC construction is actually secure with many current (SHA-3-era) hash functions, such as Keccak and BLAKE(2). The specifications for these hash functions even recommend it as a secure and fast MAC. They use various techniques to foil length extension attacks: for example, BLAKE keeps track of the number of bits that have been hashed so far, while BLAKE2 has a finalization flag that marks a specific block as the last.

Variants

Issues with prefix-MAC has tempted people to come up with all sorts of clever variations. For example, why not add the key to the end instead of the beginning (\(t = H(m \| k)\), or “suffix-MAC”, if you will)? Or maybe we should append the key to both ends for good measure (\(t = H(k \| m \| k)\), “sandwich-MAC” perhaps?)?

For what it’s worth, both of these are at least better than prefix-MAC, but both of these have serious issues. For example, a suffix-MAC system is more vulnerable to weaknesses in the underlying hash function; a successful collision attack breaks the MAC. Sandwich-MAC has other, more complex issues.

Cryptography has produced much stronger MACs, which we’ll see in the next few sections. There are no good reasons not to use them.

HMAC

HMAC is a standard to produce a MAC with a cryptographic hash function as a parameter. It was introduced in 1996 in a paper by Bellare, Canetti and Krawczyk. Many protocols at the time implemented their own attempt at message authentication using hash functions. Most of these attempts failed. The goal of that paper specifically was to produce a provably secure MAC that didn’t require anything beyond a secret key and a hash function.

One of the nice features of HMAC is that it has a fairly strong security proof. As long as the underlying hash function is a pseudorandom function, HMAC itself is also a pseudorandom function. The underlying hash function doesn’t even have to be collision resistant for HMAC to be a secure MAC. This proof was introduced after HMAC itself, and matched real-world observations: even though MD5 and to a lesser extent SHA-0 had serious collision attacks, HMAC constructions built from those hash functions still appeared to be entirely secure.

The biggest difference between HMAC and prefix-MAC or its variants is that the message passes through a hash function twice, and is combined with the key before each pass. Visually, HMAC looks like this:

/img/books/crypto101/HMAC.svg

The only surprising thing here perhaps are the two constants \(p_{inner}\) (the inner padding, one hash function’s block length worth of 0x36 bytes) and \(p_{outer}\) (the outer padding, one block length worth of 0x5c bytes). These are necessary for the security proof of HMAC to work; their particular values aren’t very important, as long as the two constants are different.

The two pads are XORed with the key before use. The result is either prepended to the original message (for the inner padding \(p_{inner}\)) or to the intermediate hash output (for the outer padding \(p_{outer}\)). Because they’re prepended, the internal state of the hash function after processing the prefixes can be computed ahead of time, shaving a few cycles off the MAC computation time.

One-time MACs

So far, we’ve always assumed that MAC functions can be used with a single key to produce secure MACs for a very large number of messages. By contrast, one-time MACs are MAC functions that can only securely be used once with a single key. That might sound like a silly idea, since we’ve already talked about regular secure MACs. An algorithm that only works once just seems objectively worse. However, they have several big advantages:

  • They can be incredibly fast to evaluate, even for very large messages.
  • They have a compelling security proof based on the information content of the tag.
  • A construction exists to turn a one-time MAC{.interpreted-text role=”term”} into a secure multiple-use MAC, removing the principal problem.

A typical simple example of such one-time MACs consists of a simple multiplication and addition modulo some large prime \(p\). In this case, the secret key consists of two truly random numbers \(a\) and \(b\), both between 1 and \(p\).

\[t \equiv m \cdot a + b \pmod p\]

This simple example only works for one-block messages \(m\), and some prime \(p\) slightly bigger than the biggest \(m\). It can be extended to support bigger messages \(M\) consisting of blocks \(m_i\) by using a message-specific polynomial \(P\):

\[t \equiv \underbrace{(m_n \cdot a^n + \cdots + m_1 \cdot a)}_{P(M, a)} + b \pmod p\]

This might look like a lot of computation, but this polynomial can be efficiently evaluated by iteratively factoring out the common factor \(a\) (also known as Horner’s rule):

\[P(M, a) \equiv a \cdot (a \cdot (a \cdot (\cdots) + m_2) + m_1) + b \pmod p\]

By computing each multiplication modulo \(p\), the numbers will remain conveniently small.

In many ways, a one-time MAC is to authentication what a one-time pad is to encryption. The security argument is similar: as long as the key is only used once, an attacker learns no information about the key or the message, because they are being irreversibly mixed. This demonstrates that the MAC is secure against attackers trying to produce existential forgeries, even when that attacker has infinite computational power.

Also like a one-time pad, the security argument relies on two very important properties about the keys \(a, b\):

  • They have to be truly random.
  • They have to be used at most once.

Re-using \(a\) and \(b\)

We’ll illustrate that our example MAC is insecure if it is used to authenticate two messages \(m_1, m_2\) with the same key \((a, b)\):

\[\begin{aligned} t_1 &\equiv m_1 \cdot a + b \pmod p \\ t_2 &\equiv m_2 \cdot a + b \pmod p \end{aligned}\]

An attacker can reconstruct \(a, b\) with some simple modular arithmetic:2

\[\begin{aligned} t_1 - t_2 &\equiv (m_1 \cdot a + b) - (m_2 \cdot a + b) \pmod p \\ &\Downarrow \text{(remove parentheses)} \\ t_1 - t_2 &\equiv m_1 \cdot a + b - m_2 \cdot a - b \pmod p \\ &\Downarrow \text{($b$ and $-b$ cancel out)} \\ t_1 - t_2 &\equiv m_1 \cdot a - m_2 \cdot a \pmod p \\ &\Downarrow \text{(factor out $a$)} \\ t_1 - t_2 &\equiv a \cdot (m_1 - m_2) \pmod p \\ &\Downarrow \text{(flip sides, multiply by inverse of $(m_1 - m_2)$)} \\ a &\equiv (t_1 - t_2)(m_1 - m_2)^{-1} \pmod p \end{aligned}\]

Plugging \(a\) into either the equation for \(t_1\) or \(t_2\) gets \(b\):

\[\begin{aligned} \begin{aligned} t_1 &\equiv m_1 \cdot a + b \pmod p \\ &\Downarrow \text{(reorder terms)}\\ b &\equiv t_1 - m_1 \cdot a \pmod p \end{aligned} \end{aligned}\]

As you can see, as with one-time pads, re-using the key even once leads to a complete failure of the cryptosystem to preserve privacy or integrity, as the case may be. As a result, one-time MACs are a bit dangerous to use directly. Fortunately, this weakness can be solved with a construction called a Carter-Wegman MAC, which we’ll see in the next section.

Carter-Wegman MAC

As we’ve already stated, the obvious problem with one-time MACs is their limited practicality. Fortunately, it turns out that there is a construction, called a Carter-Wegman MAC, that turns any secure one-time MAC into a secure many-time MAC while preserving most of the performance benefit.

The idea behind a Carter-Wegman MAC is that you can use a one-time MAC \(O\) to produce a tag for the bulk of the data, and then encrypt a nonce \(n\) with a pseudorandom function \(F\), such as a block cipher, to protect that one-time tag:

\[CW((k_1, k_2), n, M) = F(k_1, n) ⊕ O(k_2, M)\]

As long as \(F\) is a secure pseudorandom function, the nonce’s encryption is totally unpredictable. In the eyes of an attacker, that means the XOR operation will randomly flip the bits of the one-time MAC tag \(O(k_2, M)\). Because this masks the real value of the one-time MAC tag, the attacker can not perform the algebraic tricks we saw for one-time MACs recovering the key when it is used more than once.

Keep in mind that while Carter-Wegman MACs take two distinct keys \(k_1\) and \(k_2\), and that Carter-Wegman MACs are related to one-time MACs, some of which also take two distinct keys \(a\) and \(b\), they are not the same two keys. The Carter-Wegman MAC’s \(k_2\) is the only key passed to the fast one-time MAC \(O\). If that fast one-time MAC is our earlier example that takes two keys \(a\) and \(b\), that \(k_2\) would have to get split up into those two keys. The Carter-Wegman MAC key would then be \((k_1, k_2) = (k_1, (a, b))\).

You can tell how a Carter-Wegman MAC exploits the benefits of both kinds of MACs by considering the two terms of the equation separately. In \(F(k_1, n)\), \(F\) is just a regular pseudorandom function, such as a block cipher. It is quite slow by comparison to the one-time MAC. However, its input, the nonce, is very small. The unpredictable output of the block cipher masks the output of the one-time MAC. In the second term, \(O(k_2, M)\), the large input message \(M\) is only handled by the very fast one-time MAC \(O\).

These constructions, in particular Poly1305-AES, currently represent some of the state of the art in MAC functions. An older, related MAC function called UMAC may also be good sources of extra background information, since they go into extensive details of the hows and whys of a practical Carter-Wegman MAC.

Authenticated encryption modes

So far, we’ve always clearly distinguished encryption from authentication, and explained the need for both. The majority of secure connections that are set up every day have that distinction as well: they treat encryption and authentication as fundamentally different steps.

Alternatively, we could make authentication a fundamental part of the mode of operation. After all, we’ve already seen that unauthenticated encryption is virtually never what you want; it is, at best, something you occasionally have to live with. It makes sense to use constructions that not only guarantee the privacy of an arbitrary stream, but also its integrity.

As we’ve already seen, many of the methods of composing authentication and encryption are inherently insecure. By doing that in a fixed, secure way such as a properly designed authenticated encryption mode, an application developer no longer has to make that choice, which means they also can’t inadvertently make the wrong choice.

AEAD

AEAD is a feature of certain modes of authenticated encryption. Such modes of operation are called AEAD modes. It starts with the premise that many messages actually consist of two parts:

  • The actual content itself
  • Metadata: data about the content

In many cases the metadata should be plaintext, but the content itself should be encrypted. The entire message should be authenticated: it should not be possible for an attacker to mess with the metadata and have the resulting message still be considered valid.

Consider an e-mail alternative as an example cryptosystem. The metadata about the content might contain the intended recipient. We definitely want to encrypt and authenticate the content itself, so that only the recipient can read it. The metadata, however, has to be in plaintext: the e-mail servers performing the message delivery have to know which recipient to send the message to.

Many systems would leave this metadata unauthenticated, allowing attackers to modify it. In our case, that looks like it may just lead to messages being delivered to the wrong inbox. That also means that an attacker can force e-mail to be delivered to the wrong person, or not delivered at all.

AEAD modes address this issue by providing a specified way to add metadata to encrypted content, so that the whole of the encrypted content and the metadata is authenticated, and not the two pieces separately:

/img/books/crypto101/AEAD.svg

OCB mode

Usually, you will want to use a much more high level cryptosystem, such as OpenPGP, NaCl or TLS.

OCB mode is an AEAD mode of operation. It is one of the earliest developed AEAD modes.

/img/books/crypto101/Encryption1.svg

As you can see, most of this scheme looks quite similar to ECB mode. The name OCB is quite similar to electronic codebook, as well. OCB does not share the security issues ECB mode has, however, as there are several important differences, such as the offsets \(\Delta_i\) introduced in each individual block encryption.

Being an AEAD mode, OCB mode provides a cryptographically secure authentication tag \(t\), which is built from \(X\), a very simple (not cryptographically secure by itself) checksum of the plaintext. There is also another, separate tag \(t_a\), which authenticates the AEAD associated data. That associated data tag \(t_a\) is computed as follows:

/img/books/crypto101/Auth.svg

This design has a number of interesting properties. For example, it is very fast: only requiring roughly one block cipher operation per encrypted or associate data block, as well as one additional block cipher operation for the final tag. The offsets (\(\Delta_i\)) are also extremely easy to compute. The checksum block \(X\) is just all of the plaintext blocks \(P_i\) XORed together. Finally, OCB mode is easy to compute in parallel; only the final authentication tag is dependent on all the preceding information.

OCB mode also comes with a built-in padding scheme: it behaves slightly differently when the plaintexts or authentication text is not exactly a multiple of the block size. This means that, unlike with PKCS#5/PKCS#7 padding, there isn’t an entire block of “wasted” padding if the plaintext happens to be a multiple of the block size.

Despite having several interesting properties going for it, OCB mode has not received as much attention as some of the alternatives; one of the main reasons being that it is patent encumbered. Even though a number of patent licenses are available, including a free-of-charge one for open source software, this does not appear to have significantly impacted how much OCB mode is used in the field.

GCM mode

Usually, you will want to use a much more high level cryptosystem, such as OpenPGP, NaCl or TLS.

GCM mode is an AEAD mode with an unfortunate case of RAS (redundant acronym syndrome) syndrome: GCM itself stands for “Galois Counter Mode”. It is formalized in a NIST Special Publication and roughly boils down to a combination of classical CTR mode with a Carter-Wegman MAC. That MAC can be used by itself as well, which is called GMAC.

Authentication

GCM mode (and by extension GMAC)

  1. I realize there are briefer ways to write that function. I am trying to make it comprehensible to most programmers; not pleasing to advanced Pythonistas. 

  2. For a refresher on modular arithmetic, including an explanation of the modular inverse, please refer to the appendix <modular-arithmetic>{.interpreted-text role=”ref”}. 


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