$1000 TCR hashing competition

by Paul Crowley

In my day job, I do cryptography for Android. I have a problem where I need to make some cryptography faster, and I’m setting up a $1000 competition funded from my own pocket for work towards the solution.

On Android devices, key operating system components are stored in read-only partitions such as the /system partition. To prevent an attacker tampering with these partitions, we hash them using a Merkle tree, and sign the root. We don’t check all the hashes when the device boots; that would take too long. Instead, at boot time we check only the root of the tree, and then we check other sectors against their hashes as we load them using a Linux kernel module called dm-verity.

This likely works pretty well on phones sold in the US, which will have the ARM CE instructions that accelerate SHA2-256. But a lot of devices sold in poorer countries don’t have these instructions, and SHA2-256 can be pretty slow, and hurt overall system performance. For example, on the 900MHz Cortex-A7 Broadcom BCM2836, hashing takes 28.86 cpb [eBACS 2019-03-31], limiting reading speed to 31.2 MB/s. One partial fix is to switch to a hash function that is faster on such processors. BLAKE2b is nearly twice as fast on that processor, at 15.32 cpb. However, this is still a lot slower than I’m happy with.

Where sender and receiver have a shared secret, authentication can be very fast; a universal function like NH can run at around 1.5 cpb on such a processor. But this isn’t an option for verified boot, because it’s hard to keep the key out of the attacker’s hands, and given the key it’s trivial to forge messages.

Inbetween these two notions of security is the idea of a “target collision resistant” function, once known as a “universal one-way hash function”. With a TCR, hashing is randomized with a key chosen at signing time once the message to be signed is known. This makes the attacker’s job much harder since they cannot simply search for a pair of colliding messages. Instead, they must choose the first message to be hashed, and only then do they learn the key that will be used at hashing time, after which they must generate a second message that hashes with the first using this key; this problem is more akin to second preimage finding than collision finding. While collision attacks against hash functions are plentiful, second preimage attacks are far rarer.

Collision resistance Target collision resistance Universal function
K \xleftarrow{\$} \mathcal{K}
A \leftarrow K \\ A \rightarrow m_1 \\ A \rightarrow m_2 A \rightarrow m_1 \\ A \leftarrow K \\ A \rightarrow m_2 A \rightarrow m_1 \\ A \rightarrow m_2 \\ A \leftarrow K
Attacker succeeds if m_1 \neq m_2 and H(K, m_1) = H(K, m_2)

In principle, the vastly harder job facing the attacker of a TCR should mean that secure TCRs much faster than hash functions are possible. However, the main impetus to research on TCRs was a desire to bolster existing hash functions, when attacks on MD5 and SHA-1 were new and we didn’t know which if any of our existing hash functions would be left standing. As a result several ways to construct a TCR from a hash function with good provable properties were proposed, but none of these could be faster than their underlying hash functions. As far as I know, no-one has ever proposed a TCR as a primitive, designed to be faster than existing hash functions, and that’s what I need. I’m probably not the only one who’d find a primitive for much faster broadcast authentication useful, either!

To me this looks like an interesting, overlooked problem in symmetric cryptology, and I’d really like it to get some attention. So I’m offering a $1000 prize from my own pocket, to be awarded at Real World Crypto 2021, for the work that in my arbitrary opinion does the most to move the state of the art forwards or is just the most interesting.

I offered a similar prize at the rump session at FSE 2019, promising to award it at the end of the year, but I neglected to really tell anyone and didn’t get any entrants. Hopefully this will be a more successful launch for the prize, and see some of you at Real World Crypto 2020!

I’ve moved some technical notes into a subsequent blog post.