Wikipedia’s definition of a digital signature is:
A digital signature is a mathematical scheme for verifying the authenticity of digital messages or documents. A valid digital signature on a message gives a recipient confidence that the message came from a sender known to the recipient.—Wikipedia
They also have a handy diagram of the process by which digital signatures are created and verified:
Alice signs a message using her private key and Bob can then verify that the message came from Alice, and hasn’t been tampered with, using her public key. This all seems straightforward and uncomplicated and is probably most developers’ view of what signatures are for and how they should be used. This has led to the widespread use of signatures for all kinds of things: validating software updates, authenticating SSL connections, and so on.
But cryptographers have a different way of looking at digital signatures that has some surprising aspects. This more advanced way of thinking about digital signatures can tell us a lot about what are appropriate, and inappropriate, use-cases.
Identification protocols
There are several ways to build secure signature schemes. Although you might immediately think of RSA, the scheme perhaps most beloved by cryptographers is Schnorr signatures. These form the basis of modern EdDSA signatures, and also (in heavily altered form) DSA/ECDSA.
The story of Schnorr signatures starts not with a signature scheme, but instead with an interactive identification protocol. An identification protocol is a way to prove who you are (the “prover”) to some verification service (the “verifier”). Think logging into a website. But note that the protocol is only concerned with proving who you are, not in establishing a secure session or anything like that.
There are a whole load of different ways to do this, like sending a username and password or something like WebAuthn/passkeys (an ironic mention that we’ll come back to later). One particularly elegant protocol is known as Schnorr’s protocol. It’s elegant because it is simple and only relies on basic security conjectures that are widely accepted, and it also has some nice properties that we’ll mention shortly.
The basic structure of the protocol involves three phases: Commit-Challenge-Response. If you are familiar with challenge-response authentication protocols this just adds an additional commitment message at the start.
Alice (for it is she!) wants to prove to Bob who she is. Alice already has a long-term private key, a, and Bob already has the corresponding public key, A. These keys are in a Diffie-Hellman-like finite field or elliptic curve group, so we can say A = g^a mod p where g is a generator and p is the prime modulus of the group. The protocol then works like this:
- Alice generates a random ephemeral key, r, and the corresponding public key R = g^r mod p. She sends R to Bob as the commitment.
- Bob stores R and generates a random challenge, c and sends that to Alice.
- Alice computes s = ac + r and sends that back to Bob as the response.
- Finally, Bob checks if g^s = A^c * R (mod p). If it is then Alice has successfully authenticated, otherwise it’s an imposter. The reason this works is that g^s = g^(ac + r) and A^c * R = (g^a)^c * g^r = g^(ac + r) too. Why it’s secure is another topic for another day.
Don’t worry if you don’t understand all this. I’ll probably do a blog post about Schnorr identification at some point, but there are plenty of explainers online if you want to understand it. For now, just accept that this is indeed a secure identification scheme. It has some nice properties too.
One is that it is a (honest-verifier) zero knowledge proof of knowledge (of the private key). That means that an observer watching Alice authenticate, and the verifier themselves, learn nothing at all about Alice’s private key from watching those runs, but the verifier is nonetheless convinced that Alice knows it.
This is because it is easy to create valid runs of the protocol for any private key by simply working backwards rather than forwards, starting with a response and calculating the challenge and commitment that fit that response. Anyone can do this without needing to know anything about the private key. That is, for any given challenge you can find a commitment for which it is easy to compute the correct response. (What they cannot do is correctly answer a random challenge after they’ve already sent a commitment). So they learn no information from observing a genuine interaction.
Fiat-Shamir
So what does this identification protocol have to do with digital signatures? The answer is that there is a process known as the Fiat-Shamir heuristic by which you can automatically transform certain interactive identification protocols into a non-interactive signature scheme. You can’t do this for every protocol, only ones that have a certain structure, but Schnorr identification meets the criteria. The resulting signature scheme is known, amazingly, as the Schnorr signature scheme.
You may be relieved to hear that the Fiat-Shamir transformation is incredibly simple. We basically just replace the challenge part of the protocol with a cryptographic hash function, computed over the message we want to sign and the commitment public key: c = H(R, m).
That’s it. The signature is then just the pair (R, s).
Note that Bob is now not needed in the process at all and Alice can compute this all herself. To validate the signature, Bob (or anyone else) recomputes c by hashing the message and R and then performs the verification step just as in the identification protocol.
Schnorr signatures built this way are secure (so long as you add some critical security checks!) and efficient. The EdDSA signature scheme is essentially just a modern incarnation of Schnorr with a few tweaks.
What does this tell us about appropriate uses of signatures
The way I’ve just presented Schnorr signatures and Fiat-Shamir is the way they are usually presented in cryptography textbooks. We start with an identification protocol, performed a simple transformation and ended with a secure signature scheme. Happy days! These textbooks then usually move on to all the ways you can use signatures and never mention identification protocols again. But the transformation isn’t an entirely positive process: a lot was lost in translation!
There are many useful aspects of interactive identification protocols that are lost by signature schemes:
- A protocol run is only meaningful for the two parties involved in the interaction (Alice and Bob). By contrast a signature is equally valid for everyone.
- A protocol run is specific to a given point in time. Alice’s response is to a specific challenge issued by Bob just prior. A signature can be verified at any time.
These points may sound like bonuses for signature schemes, but they are actually drawbacks in many cases. Signatures are often used for authentication, where we actually want things to be tied to a specific interaction. This lack of context in signatures is why standards like JWT have to add lots of explicit statements such as audience and issuer checks to ensure the JWT came from the expected source and arrived at the intended destination, and expiry information or unique identifiers (that have to be remembered) to prevent replay attacks. A significant proportion of JWT vulnerabilities in the wild are caused by developers forgetting to perform these checks.
WebAuthn is another example of this phenomenon. On paper it is a textbook case of an identification protocol. But because it is built on top of digital signatures it requires adding a whole load of “contextual bindings” for similar reasons to JWTs. Ironically, the most widely used WebAuthn signature algorithm, ECDSA, is itself a Schnorr-ish scheme.
TLS also uses signatures for what is essentially an identification protocol, and similarly has had a range of bugs due to insufficient context binding information being included in the signed data. (SSL also uses signatures for verifying certificates, which is IMO a perfectly good use of the technology. Certificates are exactly a case of where you want to convert an interactive protocol into a non-interactive one. But then again we also do an interactive protocol (DNS) in that case anyway 🤷).
In short, an awful lot of uses of digital signatures are actually identification schemes of one form or another and would be better off using an actual identification scheme. But that doesn’t mean using something like Schnorr’s protocol! There are actually better alternatives that I’ll come back to at the end.
Special Soundness: fragility by design
Before I look at alternatives, I want to point out that pretty much all in-use signature schemes are extremely fragile in practice. The zero-knowledge security of Schnorr identification is based on it having a property called special soundness. Special soundness essentially says that if Alice accidentally reuses the same commitment (R) for two runs of the protocol, then any observer can recover her private key.
This sounds like an incredibly fragile notion to build into your security protocol! If I accidentally reuse this random value then I leak my entire private key??! And in fact it is: such nonce-reuse bugs are extremely common in deployed signature systems, and have led to compromise of lots of private keys (eg Playstation 3, various Bitcoin wallets etc).
But despite its fragility, this notion of special soundness is crucial to the security of many signature systems. They are truly a cursed technology!
To solve this problem, some implementations and newer standards like EdDSA use deterministic commitments, which are based on a hash of the private key and the message. This ensures that the commitment will only ever be the same if the message is identical: preventing the private key from being recovered. Unfortunately, such schemes turned out to be more susceptible to fault injection attacks (a much less scalable or general attack vector), and so now there are “hedged” schemes that inject a bit of randomness back into the hash. It’s cursed turtles all the way down.
If your answer to this is to go back to good old RSA signatures, don’t be fooled. There are plenty of ways to blow your foot off using old faithful, but that’s for another post.
Did you want non-repudiation with that?
Another way that signatures cause issues is that they are too powerful for the job they are used for. You just wanted to authenticate that an email came from a legitimate server, but now you are providing irrefutable proof of the provenance of leaked private communications. Oops!
Signatures are very much the hammer of cryptographic primitives. As well as authenticating a message, they also provide third-party verifiability and (part of) non-repudiation.
You don’t need to explicitly want anonymity or deniability to understand that these strong security properties can have damaging and unforeseen side-effects. Non-repudiation should never be the default in open systems.
I could go on. From the fact that there are basically zero acceptable post-quantum signature schemes (all way too large or too risky), to issues with non-canonical signatures and cofactors and on and on. The problems of signature schemes never seem to end.
What to use instead?
Ok, so if signatures are so bad, what can I use instead?
Firstly, if you can get away with using a simple shared secret scheme like HMAC, then do so. In contrast to public key crypto, HMAC is possibly the most robust crypto primitive ever invented. You’d have to go really far out of your way to screw up HMAC. (I mean, there are timing attacks and that time that Bouncy Castle confused bits and bytes and used 16-bit HMAC keys, so still do pay attention a little bit…)
If you need public key crypto, then… still use HMAC. Use an authenticated KEM with X25519 to generate a shared secret and use that with HMAC to authenticate your message. This is essentially public key authenticated encryption without the actual encryption. (Some people mistakenly refer to such schemes as designated verifier signatures, but they are not).
Signatures are good for software/firmware updates and pretty terrible for everything else.
https://neilmadden.blog/2024/09/18/digital-signatures-and-how-to-avoid-them/
#authenticatedEncryption #cryptography #misuseResistance #signatures
In my previous post, I described the KEM/DEM paradigm for hybrid encryption. The key encapsulation mechanism is given the recipient’s public key and outputs a fresh AES key and an encapsulation of that key that the recipient can decapsulate to recover the AES key. In this post I want to talk about several ways that the KEM interface falls short and what to do about it:
- As I’ve discussed before, the standard definition of public key encryption lacks any authentication of the sender, and the KEM-DEM paradigm is no exception. You often want to have some idea of where a message came from before you process it, so how can we add sender authentication?
- If you want to send the same message to multiple recipients, a natural approach would be to encrypt the message once with a fresh AES key and then encrypt the AES key for each recipient. With the KEM approach though we’ll end up with a separate AES key for each recipient. How can we send the same message to multiple recipients without encrypting the whole thing separately for each one?
- Finally, the definition of public key encryption used in the KEM/DEM paradigm doesn’t provide forward secrecy. If an attacker ever compromises the recipient’s long-term private key, they can decrypt every message ever sent to that recipient. Can we prevent this?
In this article I’ll tackle the first two issues and show how the KEM/DEM abstractions can be adjusted to cope with each. In a follow-up post I’ll then show how to tackle forward secrecy, along with replay attacks and other issues. Warning: this post is longer and has more technical details than the previous post. It’s really meant for people who already have some experience with cryptographic algorithms.
Authenticated KEMs
Let’s look first at how to create authenticated encryption in the KEM/DEM paradigm. The DEM used to encrypt the body of a message in the KEM/DEM paradigm already provides authenticated encryption, so the only work that needs to be done is to authenticate the KEM so that the recipient can be confident the encapsulated DEM key came from a trusted source. Although we could try to manually combine a KEM with some kind of authentication mechanism, it’s almost always better to define a single abstraction that provides both properties. In this case, we can extend our KEM interface to become an authenticated KEM or AKEM. An AKEM is like a KEM, but the encapsulation function takes the sender’s private key as an additional argument, and the decapsulation function takes the sender’s public key:
function auth_encapsulate(senderPrivateKey, recipientPublicKey): // returns (aesKey, encapsulatedKey) as beforefunction auth_decapsulate(senderPublicKey, recipientPrivateKey, encapsulatedKey): // returns either the original aesKey or an error if // authentication fails
A straightforward implementation of this idea would be to combine an existing (unauthenticated) KEM with a signature scheme. The KEM produces an encapsulated key, which we then sign with our private key so that the recipient can be sure that it really came from us:
function auth_encapsulate_sig(senderPrivateKey, recipientPublicKey): var (aesKey, encapsulatedKey) = KEM.encapsulate(recipientPublicKey); var signature = sign(senderPrivateKey, encapsulatedKey); return (aesKey, encapsulatedKey || signature);function auth_decapsulate_sig(senderPublicKey, recipientPrivateKey, encapsulatedSignedKey): var (encapsulatedKey, signature) = parse(encapsulatedSignedKey); if !verify_signature(senderPublicKey, encapsulatedKey, signature): return error; return KEM.decapsulate(recipientPrivateKey, encapsulatedKey);
This works, but it can produce very large encapsulated keys. For example, if you use RSA-KEM with an RSA signature scheme and 3072-bit keys, then the encapsulated key will end up being at least 768 bytes long. We can do better by removing the explicit signature and instead using implicit authentication, where we derive the DEM key in such a way that only the genuine sender could have calculated it. Any attempt to change the encapsulated key, without using the sender’s private key, will result in a different DEM key being calculated and the DEM authenticated decryption step will fail with an authentication error.One way to achieve this is to adapt the elliptic curve Diffie-Hellman (ECDH) KEM to include the sender’s private key in the key agreement calculation. Rather than performing a single ECDH calculation between an ephemeral private key and the recipient’s public key, we additionally perform one between the sender’s private key and the recipient’s public key. The two derived shared secrets are then concatenated before being fed into the Key Derivation Function to produce the AES key:
function auth_encapsulate_ecdh(senderPrivateKey, recipientPublicKey): var (ephemeralSecretKey, ephemeralPublicKey) = generateKeyPair(recipientPublicKey.curve); var ess = ecdh(ephemeralSecretKey, recipientPublicKey); var sss = ecdh(senderPrivateKey, recipientPublicKey); var aesKey = sha256(ess || sss || ephemeralPublicKey); return (aesKey, ephemeralPublicKey);function ecdh_decapsulate_ecdh(senderPublicKey, recipientPrivateKey, ephemeralPublicKey): var ess = ecdh(recipientPrivateKey, ephemeralPublicKey); var sss = ecdh(recipientPrivateKey, senderPublicKey); return sha256(ess || sss || ephemeralPublicKey);
(Note: as before we include the ephemeral public key in the KDF step to ensure non-malleability, and we’re using SHA-256 as a cheap and cheerful stand-in for a real KDF).The advantage of this approach is that the encapsulated key is still just an ephemeral public key—just 32 bytes for X25519. It’s also more deniable than the signature version. NIST call this scheme the One-Pass Unified Model, and it’s also broadly similar to the Noise “K” one-way handshake pattern and the AKEM defined for the upcoming HPKE standard. Although this scheme is more compact than the combination of a KEM plus a signature, it is still computationally quite expensive because of the two ECDH computations that are needed. More efficient options such as (one-pass) MQV can reduce this overhead, but these are not widely used in practice due to patent issues and potential weaknesses of some variants.
Both of these AKEM designs suffer from some security weaknesses:
- They are both vulnerable to something called Key Compromise Impersonation (KCI) attacks. What this means is that if an attacker compromises your long-term secret key, they can not only pretend to be you when sending messages to other people, but can pretend to be anyone else when sending messages to you! For the ECDH scheme they can immediately do this. The KEM-then-sign variant is a bit stronger, in that an attacker would have to capture at least one message sent to you by the party they want to impersonate, but in practice this is often possible. Preventing KCI generally involves either an interactive authentication protocol or signing the entire message, both of which I’ll leave out of this article.
- Neither design incorporates any notion of the identity of the users involved. In some cases this can lead to Unknown Key Share attacks, in which we think we’re communicating with Alice but actually are talking to Bob. You can solve this problem by including identity information and long-term public keys in the signature or key-derivation steps. It can be useful to add a general “context” argument to the encapsulate and decapsulate functions to allow additional data to be included in the KDF:
function auth_encapsulate_ecdh(senderPrivateKey, recipientPublicKey, context): var (ephemeralSecretKey, ephemeralPublicKey) = generateKeyPair(recipientPublicKey.curve); var ess = ecdh(ephemeralSecretKey, recipientPublicKey); var sss = ecdh(senderPrivateKey, recipientPublicKey); var aesKey = sha256(ess || sss || ephemeralPublicKey || context); return (aesKey, ephemeralPublicKey);function ecdh_decapsulate_ecdh(senderPublicKey, recipientPrivateKey, ephemeralPublicKey, context): var ess = ecdh(recipientPrivateKey, ephemeralPublicKey); var sss = ecdh(recipientPrivateKey, senderPublicKey); return sha256(ess || sss || ephemeralPublicKey || context);Sending to multiple recipients
The basic KEM/DEM paradigm runs into a problem if we want to send the same message to multiple recipients. In the traditional approach to hybrid encryption before KEMs, you’d simply generate a random AES key and then encrypt it for each recipient. But this doesn’t work for KEMs: we’d have to run the KEM separately for each recipient, obtaining a separate AES key for each one, and then encrypt the entire message with each AES key. This is very inefficient.We can solve this problem by introducing a notion called symmetric key-wrapping, where we use one AES key to encrypt (“wrap”) another AES key. In principle, any secure authenticated encryption scheme (such as a DEM) could be used, but there are some specialised algorithms that are more compact.
The idea is a bit like the original version of hybrid encryption, but instead of directly encrypting the AES key for each recipient, we will instead use the KEM to derive a key-encryption key (KEK) for each recipient, and then use that to encrypt the random AES key (which we’ll call the Data Encryption Key, or DEK):
- Generate a random DEK,
dek = DEM.keygen()- Encrypt the message with dk:
ciphertext = DEM.encrypt(dek, message);- For each recipient, r, calculate the Key Encryption Key:
kek[r], encapsulatedKey[r] = KEM.encapsulate(r.publicKey);- For each KEK, encrypt the DEK:
wrappedKey[r] = wrap(kek[r], dk);- Send the encrypted ciphertext, the wrapped keys, and the encapsulated keys to the recipients.
Each recipient then runs the KEM to decapsulate their encapsulated key and uses the result to decrypt (“unwrap”) the DEK. They then use the DEK to decrypt the rest of the message.
There is still an inefficiency here though, in that we’ll have a separate encapsulated key for each recipient. For the ECDH-KEM the encapsulated key is an ephemeral public key, and we can safely reuse the same ephemeral key pair for every recipient. This saves a costly key-pair generation step for each recipient and reduces the size of the final ciphertext, but we can’t achieve this with the existing KEM API.
Once again, we can adapt the KEM interface to natively support this. The encapsulation function is changed to take a set of public keys for the recipients rather than just one. The result is known as an mKEM, and you can find more details in this article by Nigel Smart (*cough* over here). For example, here’s an unauthenticated mKEM based on ECDH:
function mkem_encapsulate(recipientPublicKeys): var aesKey = DEM.keygen(); var (ephemeralSecretKey, ephemeralPublicKey) = generateKeyPair(); var wrappedKeys = []; for pk in recipientPublicKeys: var ss = ecdh(ephemeralSecretKey, pk); var kek = sha256(ss || ephemeralPublicKey); var wrappedKey = wrap(kek, aesKey); wrappedKeys += wrappedKey; end return (aesKey, ephemeralPublicKey || wrappedKeys);Authenticated mKEMs
It’s easy to see how we could adapt the generic mKEM approach of the previous section to provide authenticated encryption (AmKEM? mAKEM?): simply add the sender’s private key to the interface and then perform an authenticated key derivation for each recipient, just as we did for the single-recipient AKEM:
function auth_mkem_encapsulate(senderPrivateKey, recipientPublicKeys): var aesKey = DEM.keygen(); var (ephemeralSecretKey, ephemeralPublicKey) = generateKeyPair(); var wrappedKeys = []; for pk in recipientPublicKeys: var ess = ecdh(ephemeralSecretKey, pk); var sss = ecdh(senderPrivateKey, pk); var kek = sha256(ess || sss || ephemeralPublicKey); var wrappedKey = wrap(kek, aesKey); wrappedKeys += wrappedKey; end return (aesKey, ephemeralPublicKey || wrappedKeys);
This does provide authenticated encryption, but only if all the recipients are honest. If any of the recipients is malicious, or have had their private key compromised, they can create a new message that appears to come from the original sender:
- First they decapsulate the genuine message to obtain the DEM key,
dek.- They then create a new message and encrypt it using the DEM and the same DEK:
var c2 = DEM.encrypt(dek, "My new message");- Finally, they re-attach the wrapped keys and ephemeral public key from the original message and send the whole thing to the same recipients (or a subset of them).
When the recipients decrypt the message it will appear that it is a genuine message from the original sender. This can happen regardless of whether you use signatures or implicit authentication in the AKEM. The fundamental reason is that neither the encapsulated keys nor the wrapped keys are in any way tied to the actual message. The KEM ensures the authenticity of the data encryption key, and the DEM ensures that a message could only have been created by someone who had access to that DEK. The problem is that the set of people who have access to the DEK is the original sender and all of the recipients.
Protecting against this kind of attack is known as insider security, and to solve it we’ll need to give up some measure of deniability. We could just sign the entire message, in which case we give up all deniability: any third party can verify that we sent that message, not just our intended recipients. Let’s call this outsider non-repudiation. What we really want is a slightly weaker guarantee: the recipients can prove to each other that the message came from the original sender (or didn’t in the case of a forgery), but they can’t prove that to anyone outside the group. We’ll call this insider non-repudiation.
For example, if Alice sends a message to Bob and Charlie, then Bob can prove to Charlie that Alice really did send that message (even if Charlie failed to receive it), but Bob can’t prove anything about the origin of the message to Delilah, who wasn’t in the original set of recipients. As far as Delilah is concerned, Bob could have fabricated the message.
It turns out that we can quite easily solve this problem, but first we need a little detour.
Tag-KEM: KEMs with associated data
The original formulation of the KEM/DEM paradigm required both the KEM and the DEM to be secure against chosen ciphertext attacks (CCA). This is a strong notion of security, but was found to be too strong in some cases. There are efficient hybrid encryption schemes in which either the KEM or the DEM is not CCA-secure, but when combined in the right way can achieve CCA-secure public key encryption. Some of these schemes are more efficient than they would be otherwise. To accommodate these schemes into the KEM/DEM paradigm, a new variant known as Tag-KEM was introduced. A Tag-KEM (TKEM) splits the encapsulation of a symmetric key into two phases:
- First, a function is called with the recipient’s public key and it outputs an AES key to use with the DEM and a private state value:
var (aesKey, state) = TKEM.key(recipientPublicKey);- Then, after the message has been encrypted with the DEM, a second function of the TKEM is called to generate an encapsulated key from the internal state and a tag. The tag is some arbitrary associated data that will be linked to the encapsulated key. You can think of it a bit like the context argument we added earlier to allow user identities to be bound to the encapsulated key.
var encapsulatedKey = TKEM.encapsulate(state, tag);- To decapsulate the key, the recipient has to provide the same tag. If they provide a different tag then an error is returned.
In the original paper, this is used to allow the DEM to be a simple CPA-secure (unauthenticated) encryption algorithm. The entire encrypted ciphertext from the DEM is passed in as the tag to the TKEM. The TKEM then applies a MAC to the tag as part of the encapsulation process, effectively transforming it into a secure Encrypt-then-MAC scheme. We can use the same mechanism to ensure that the key encapsulation is tied to the message to prevent insider attacks against authenticated KEMs, by ensuring that the tag is included in the authentication step. The same idea has been used to construct secure signcryption schemes from Tag-KEM (signcryption is a strong form of public key authenticated encryption that provides the same authenticity guarantees as a signature, and so provides the stronger outsider non-repudiation we want to avoid).
There are two ways that we could accomplish this:
- We can add a tag argument to the AKEM mechanism, so that the tag is directly included in the authentication provided by the AKEM. For example, using the context argument we described earlier. If the AKEM is based on signatures, then this effectively becomes a signcryption scheme, which has the advantage of also preventing KCI attacks, but destroys deniability.
- If we are using key-wrapping, and the key-wrapping function allows associated data (SIV-AES does, AES-KW doesn’t) then we can include the tag there instead. This better preserves deniability, but remains vulnerable to KCI.
If we’re using a KEM based on ECDH with a strong KDF like HKDF, then these two options are pretty similar: the tag ends up getting included in a strong MAC (PRF) protected by a key known only to the sender and an individual recipient. If the tag is the whole message, then this ensures that a new message can’t be created with the same encapsulated DEK without the other recipients being able to detect this.
mATKEMs
Putting it all together, we’d get a multi-recipient authenticated Tag-KEM, or mATKEM for short! A mATKEM has the following functions:
keygen()– generates public and private keys for a sender/recipient and returns them.key(senderPrivateKey, recipientPublicKeys)– returns a fresh DEM key and a private state object that is tied to the sender, recipients, and the DEM key.auth_encapsulate(state, tag)– encapsulates the DEM key for each of the recipients in a way that they can authenticate the sender, with the tag also authenticated (but not encapsulated). Returns the encapsulated key.auth_decapsulate(senderPublicKey, recipientPrivateKey, encapsulatedKey, tag)– authenticates that the encapsulated key was produced by the given sender with the given tag, and then decapsulates and returns the DEM key. If authentication fails then an error is returned instead.A sketch of an implementation of a secure mATKEM based on ECDH with key-wrapping is shown below. This is just a sketch of how such a thing could be implemented, I haven’t rigorously analysed it or attempted to prove it correct.
function key(senderPrivateKey, recipientPublicKeys): var dek = DEM.keygen(); var state = (dek, senderPrivateKey, recipientPublicKeys); return (dek, state);function auth_encapsulate(state, tag): var (dek, senderPrivateKey, recipientPublicKeys) = state; var (ephemeralSecretKey, ephemeralPublicKey) = generateKeyPair(); var wrappedKeys = []; for pk in recipientPublicKeys: var ess = ecdh(ephemeralSecretKey, pk); var sss = ecdh(senderPrivateKey, pk); var kek = KDF(ess || sss || ephemeralPublicKey || tag); var wrappedKey = wrap(kek, dek); wrappedKeys += wrappedKey; end return ephemeralPublicKey || wrappedKeys;function auth_decapsulate(senderPublicKey, recipientPrivateKey, encapsulatedKey, tag): var ephemeralPublicKey || wrappedKeys = encapsulatedKey; var ess = ecdh(recipientPrivateKey, ephemeralPublicKey); var sss = ecdh(recipientPrivateKey, senderPublicKey); var kek = KDF(ess || sss || ephemeralPublicKey || tag); for wrappedKey in wrappedKeys: var dek = unwrap(kek, wrappedKey); if dek != error: return dek end return error;Compactly committing DEMs
In the original formulation of a Tag-KEM, the tag input is the full ciphertext output of the (unauthenticated) DEM. But in the mATKEM implementation this entire ciphertext gets passed into the KDF or MAC for every recipient, which is very inefficient for long messages. This can be optimised in the multi-recipient case by hashing the tag once with a collision-resistant hash function rather than including the whole ciphertext in the KEM or key-wrap calculation for every recipient.An alternative approach would be to use an authenticated DEM that is compactly committing (which I’ll refer to as ccDEM) and then use the MAC tag from the DEM as the tag for the authenticated TKEM. The advantage of reusing the authentication tag from the DEM rather than calculating it in the KEM, is that it allows us to use DEM schemes that are not based on Encrypt-then-MAC, for example misuse-resistant modes like SIV-AES.
Not all authenticated encryption algorithms are compactly committing, and several popular algorithms such as AES-GCM and ChaCha20-Poly1305 are definitely not. Although those algorithms prevent an attacker who doesn’t know the DEK from altering or forging new ciphertexts, if the attacker does know the key (as in insider security) then they can actually find other messages that have the same tag. The same is true for CMAC used by SIV and CCM. The definition of a compactly committing encryption algorithm requires that the MAC is a collision-resistant pseudorandom function (PRF) such as HMAC. If we assume that Alice (the sender) is honest then we should be able to get away with just second-preimage resistance instead for this application, which allows us to use shorter tags.
A ccDEM would be a DEM that takes a key, message, and (optional) associated data and returns a ciphertext and compactly committing authentication tag. An insider-secure multi-recipient authenticated public key encryption scheme (phew!) can then be built as the following generic composition:
function encrypt(senderPrivateKey, recipientPublicKeys, plaintext): var (dek, state) = mATKEM.key(senderPrivateKey, recipientPublicKeys); var (ciphertext, tag) = ccDEM.encrypt(dek, plaintext); var encapsulatedKey = mATKEM.auth_encapsulate(state, tag); return (encapsulatedKey, ciphertext, tag);function decrypt(senderPublicKey, recipientPrivateKey, receivedMessage): var (encapsulatedKey, ciphertext, tag) = receivedMessage; var dek = mATKEM.auth_decapsulate(senderPublicKey, recipientPrivateKey, encapsulatedKey, tag); if dek = error: return error; return DEM.decrypt(dek, ciphertext, tag);
The combination of a compactly committing DEM and an authenticated Tag-KEM seems like a powerful way of thinking about secure public key encryption schemes. This was first suggested to me by Paul Grubbs on the CFRG mailing list, but as he says it has not been formally analysed so please don’t rush to deploy such a scheme in production. Hopefully a real cryptographer will find this problem interesting and decide to fill in the gaps.That concludes this look at building authenticated KEMs for multiple recipients. We’ve seen how the original KEM interface can be adapted to accommodate stronger security goals. As with the original KEM proposal, finding the right API is an important part of building secure software, and the wrong API can leave you constantly fighting subtle security issues or inefficiencies. In the next article, I look at forward secrecy and how that tantalising state argument in the Tag-KEM API can be a bridge towards secure interactive protocols like Signal or Noise.
https://neilmadden.blog/2021/02/16/when-a-kem-is-not-enough/
#authenticatedEncryption #cryptography #kemDemParadigm #publicKeyEncryption