July 22, 2019

nullprogram.com/blog/2019/07/22/

Over the last couple weeks I’ve spent a lot more time working with OpenPGP keys. It’s a consequence of polishing my passphrase-derived PGP key generator. I’ve tightened up the internals, and it’s enabled me to explore the corners of the format, try interesting things, and observe how various OpenPGP implementations respond to weird inputs.

For one particularly cool trick, take a look at these two (private) keys I generated yesterday. Here’s the first:

-----BEGIN PGP PRIVATE KEY BLOCK----- xVgEXTU3gxYJKwYBBAHaRw8BAQdAjJgvdh3N2pegXPEuMe25nJ3gI7k8gEgQvCor AExppm4AAQC0TNsuIRHkxaGjLNN6hQowRMxLXAMrkZfMcp1DTG8GBg1TzQ9udWxs cHJvZ3JhbS5jb23CXgQTFggAEAUCXTU3gwkQmSpe7h0QSfoAAGq0APwOtCFVCxpv d/gzKUg0SkdygmriV1UmrQ+KYx9dhzC6xwEAqwDGsSgSbCqPdkwqi/tOn+MwZ5N9 jYxy48PZGZ2V3ws= =bBGR -----END PGP PRIVATE KEY BLOCK-----

And the second:

-----BEGIN PGP PRIVATE KEY BLOCK----- xVgEXTU3gxYJKwYBBAHaRw8BAQdAzjSPKjpOuJoLP6G0z7pptx4sBNiqmgEI0xiH Z4Xb16kAAP0Qyon06UB2/gOeV/KjAjCi91MeoUd7lsA5yn82RR5bOxAkzQ9udWxs cHJvZ3JhbS5jb23CXgQTFggAEAUCXTU3gwkQmSpe7h0QSfoAAEv4AQDLRqx10v3M bwVnJ8BDASAOzrPw+Rz1tKbjG9r45iE7NQEAhm9QVtFd8SN337kIWcq8wXA6j1tY +UeEsjg+SHzkqA4= =QnLn -----END PGP PRIVATE KEY BLOCK-----

Concatenate these and then import them into GnuPG to have a look at them. To avoid littering in your actual keyring, especially with private keys, use the --homedir option to set up a temporary keyring. I’m going to omit that option in the examples.

$ gpg --import < keys.asc gpg: key 992A5EEE1D1049FA: public key "nullprogram.com" imported gpg: key 992A5EEE1D1049FA: secret key imported gpg: key 992A5EEE1D1049FA: public key "nullprogram.com" imported gpg: key 992A5EEE1D1049FA: secret key imported gpg: Total number processed: 2 gpg: imported: 2 gpg: secret keys read: 2 gpg: secret keys imported: 2

The user ID is “nullprogram.com” since I made these and that’s me taking credit. “992A5EEE1D1049FA” is called the long key ID: a 64-bit value that identifies the key. It’s the lowest 64 bits of the full key ID, a 160-bit SHA-1 hash. In the old days everyone used a short key ID to identify keys, which was the lowest 32 bits of the key. For these keys, that would be “1D1049FA”. However, this was deemed way too short, and everyone has since switched to long key IDs, or even the full 160-bit key ID.

The key ID is nothing more than a SHA-1 hash of the key creation date — unsigned 32-bit unix epoch seconds — and the public key material. So secret keys have the same key ID as their associated public key. This makes sense since they’re a key pair and they go together.

Look closely and you’ll notice that both keypairs have the same long key ID. If you hadn’t already guessed from the title of this article, these are two different keys with the same long key ID. In other words, I’ve created a long key ID collision. The GnuPG --list-keys command prints the full key ID since it’s so important:

$ gpg --list-keys --------------------- pub ed25519 2019-07-22 [SCA] A422F8B0E1BF89802521ECB2992A5EEE1D1049FA uid [ unknown] nullprogram.com pub ed25519 2019-07-22 [SCA] F43BC80C4FC2603904E7BE02992A5EEE1D1049FA uid [ unknown] nullprogram.com

I was only targeting the lower 64 bits, but I actually managed to collide the lowest 68 bits by chance. So a long key ID still isn’t enough to truly identify any particular key.

This isn’t news, of course. Nor am I the first person to create a long key ID collision. In 2013, David Leon Gil published a long key ID collision for two 4096-bit RSA public keys. However, that is the only other example I was able to find. He did not include the private keys and did not elaborate on how he did it. I know he did generate viable keys, not just garbage for the public key portions, since they’re both self-signed.

Creating these keys was trickier than I had anticipated, and there’s an old, clever trick that makes it work. Building atop the work I did for passphrase2pgp, I created a standalone tool that will create a long key ID collision and print the two keypairs to standard output:

Example usage:

$ go get -u github.com/skeeto/pgpcollider $ pgpcollider --verbose > keys.asc

This can take up to a day to complete when run like this. The tool can optionally coordinate many machines — see the --server / -S and --client / -C options — to work together, greatly reducing the total time. It took around 4 hours to create the keys above on a single machine, generating a around 1 billion extra keys in the process. As discussed below, I actually got lucky that it only took 1 billion. If you modify the program to do short key ID collisions, it only takes a few seconds.

The rest of this article is about how it works.

Birthday Attacks

An important detail is that this technique doesn’t target any specific key ID. Cloning someone’s long key ID is still very expensive. No, this is a birthday attack. To find a collision in a space of 2^64, on average I only need to generate 2^32 samples — the square root of that space. That’s perfectly feasible on a regular desktop computer. To collide long key IDs, I need only generate about 4 billion IDs and efficiently do membership tests on that set as I go.

That last step is easier said than done. Naively, that might look like this (pseudo-code):

seen := map of long key IDs to keys loop forever { key := generateKey() longID := key.ID[12:20] if longID in seen { output seen[longID] output key break } else { seen[longID] = key } }

Consider the size of that map. Each long ID is 8 bytes, and we expect to store around 2^32 of them. That’s at minimum 32 GB of storage just to track all the long IDs. The map itself is going to have some overhead, too. Since these are literally random lookups, this all mostly needs to be in RAM or else lookups are going to be very slow and impractical.

And I haven’t even counted the keys yet. As a saving grace, these are Ed25519 keys, so that’s 32 bytes for the public key and 32 bytes for the private key, which I’ll need if I want to make a self-signature. (The signature itself will be larger than the secret key.) That’s around 256GB more storage, though at least this can be stored on the hard drive. However, to address these from the map I’d need at least 38 bits, plus some more in case it goes over. Just call it another 8 bytes.

So that’s, at a bare minimum, 64GB of RAM plus 256GB of other storage. Since nothing is ideal, we’ll need more than this. This is all still feasible, but will require expensive hardware. We can do a lot better.

Keys from seeds

The first thing you might notice is that we can jettison that 256GB of storage by being a little more clever about how we generate keys. Since we don’t actually care about the security of these keys, we can generate each key from a seed much smaller than the key itself. Instead of using 8 bytes to reference a key in storage, just use those 8 bytes to store the seed used to make the key.

counter := rand64() seen := map of long key IDs to 64-bit seeds loop forever { seed := counter counter++ key := generateKey(seed) longID := key.ID[12:20] if longID in seen { output generateKey(seen[longID]) output key break } else { seen[longID] = seed } }

I’m incrementing a counter to generate the seeds because I don’t want to experience the birthday paradox to apply to my seeds. Each really must be unique. I’m using SplitMix64 for the PRNG since I learned it’s the fastest for Go, so a simple increment to generate seeds is perfectly fine.

Ultimately, this still uses utterly excessive amounts of memory. Wouldn’t it be crazy if we could somehow get this 64GB map down to just a few MBs of RAM? Well, we can!

Rainbow tables

For decades, password crackers have faced a similar problem. They want to precompute the hashes for billions of popular passwords so that they can efficiently reverse those password hashes later. However, storing all those hashes would be unnecessarily expensive, or even infeasible.

So they don’t. Instead they use rainbow tables. Password hashes are chained together into a hash chain, where a password hash leads to a new password, then to a hash, and so on. Then only store the beginning and the end of each chain.

To lookup a hash in the rainbow table, run the hash chain algorithm starting from the target hash and, for each hash, check if it matches the end of one of the chains. If so, recompute that chain and note the step just before the target hash value. That’s the corresponding password.

For example, suppose the password “foo” hashes to 9bfe98eb , and we have a reduction function that maps a hash to some password. In this case, it maps 9bfe98eb to “bar”. A trivial reduction function could just be an index into a list of passwords. A hash chain starting from “foo” might look like this:

foo -> 9bfe98eb -> bar -> 27af0841 -> baz -> d9d4bbcb

In reality a chain would be a lot longer. Another chain starting from “apple” might look like this:

apple -> 7bbc06bc -> candle -> 82a46a63 -> dog -> 98c85d0a

We only store the tuples (foo, d9d4bbcb ) and (apple, 98c85d0a ) in our database. If the chains had been one million hashes long, we’d still only store those two tuples. That’s literally a 1:1000000 compression ratio!

Later on we’re faced with reversing the hash 27af0841 , which isn’t listed directly in the database. So we run the chain forward from that hash until either I hit the maximum chain length (i.e. password not in the table), or we recognize a hash:

27af0841 -> baz -> d9d4bbcb

That d9d4bbcb hash is listed as being in the “foo” hash chain. So I regenerate that hash chain to discover that “bar” leads to 27af0841 . Password cracked!

Collider rainbow table

My collider works very similarly. A hash chain works like this: Start with a 64-bit seed as before, generate a key, get the long key ID, then use the long key ID as the seed for the next key.

There’s one big difference. In the rainbow table the purpose is to run the hash function backwards by looking at the previous step in the chain. For the collider, I want to know if any of the hash chains collide. So long as each chain starts from a unique seed, it would mean we’ve found two different seeds that lead to the same long key ID.

Alternatively, it could be two different seeds that lead to the same key, which wouldn’t be useful, but that’s trivial to avoid.

A simple and efficient way to check if two chains contain the same sequence is to stop them at the same place in that sequence. Rather than run the hash chains for some fixed number of steps, they stop when they reach a distinguishing point. In my collider a distinguishing point is where the long key ID ends with at least N 0 bits, where N determines the average chain length. I chose 17 bits.

func computeChain(seed) { loop forever { key := generateKey(seed) longID := key.ID[12:20] if distinguished(longID) { return longID } seed = longID } }

If two different hash chains end on the same distinguishing point, they’re guaranteed to have collided somewhere in the middle.

To determine where two chains collided, regenerate each chain and find the first long key ID that they have in common. The step just before are the colliding keys.

counter := rand64() seen := map of long key IDs to 64-bit seeds loop forever { seed := counter counter++ longID := computeChain(seed) if longID in seen { output findCollision(seed, seen[longID]) break } else { seen[longID] = seed } }

Hash chains computation is embarrassingly parallel, so the load can be spread efficiently across CPU cores. With these rainbow(-like) tables, my tool can generate and track billions of keys in mere megabytes of memory. The additional computational cost is the time it takes to generate a couple more chains than otherwise necessary.