# Threshold Lamport signatures can survive with much lower size by compromising on per-signature security level

Consider a Lamport signature scheme where there are 256 participants, and each participant is required to reveal one of eight values that they have committed to (ie. 128 bytes total for a Merkle branch). Given a value with hash H (assume a 768 bit hash), take 3 bits of H and assign them to each user. That user is required to provide the specific value that’s assigned to by the 3 hash bits. Any signature containing at least 170 valid preimages+Merkle branches passes as valid.

If you actually have 170 honest participants, then it’s easy to see how the scheme can succeed. What if one signature is made (with all 256 participants) and you want to forge it? Then, in expectation, only 32 fitting values will be available, so to make a valid signature for a given random specified H’ half the time you need ~158 participants to be colluding (of the ~98 non-colluding, ~12 will on average be available, getting you to 170). If you have 96 colluding participants, then you will only be able to make a signature for ~1 in 2^80 of possible values (ie. cryptographically infeasible, especially if a few rounds of PoW are done on H before using it as a source of query bits).

Hence, the size of a quantum-secure threshold signature is only ~96 bytes per participant for a committee of size 256, if we are willing to accept the ~1/3 slippage in safety (ie. safety reduces from 2/3 to 1/3); slippage reduces to ~1/5 if we increase from 96 bytes to 128 bytes, and to ~1/7 if we go up to 160 bytes per participant.

#### Edit (2018.05.28): Hash ladder sigs

We can trade off size for verification time by using a hash ladder sig instead of a regular sig. Each participant generates two values, x_1 and x_2, and commits to H^n(x_1) and H^n(x_2) (eg. n = 16). To make a signature, take log(n) bits from the message hash, call this value d, and compute s_1 = H^d(x_1) and s_2 = H^{n-d}(x_2), and output these values. Checking the signature involves taking H^{n-d}(s_1) and H^n(s_2) and checking that these values match the commitments. Sigs of this form are always 64 bytes; if we set n = 32 (leading to a sig verification time of 32 microseconds if we use blake2, which takes ~1 microsecond per round), then we get the same ~1/7 slippage.

A hybrid approach would be 96-byte signatures where we extract two values d_1 and d_2 from the message hash and put H^{d_1}(x_1), H^{d_2}(x_2), H^{2n - d_1 - d_2}(x_3) in the signature; this could allow us only 8% slippage with 32 hashes to verify a signature, or 6% slippage with 64 hashes to verify a signature.

Note that slippage becomes much lower for larger sets; for an infinite sized set, slippage approaches \frac{2}{3N} for 64-byte signatures and \frac{2}{3N^2} for 96-byte signatures.

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Hi! I believe this scheme is not secure and depends on the synchronicity of the signing parties.

Assume an adversary \mathcal{A}, who is a malicious block producer (BP), a hash function \mathcal{H} with an output of 256 bits, and 256 signers (1 per bit). For simplicity, we can assume all the signing parties are honest and online. Only the BP is rogue. Moreover, each signer has an ordering for each of the bits to be signed. Signer 1 signs the first bit, signer 2 signs the 2nd bit, and so on…

First, \mathcal{A} creates a malicious block and hashes it to obtain the digest to be signed. We can call this \mathcal{B}'. Second, \mathcal{A} produces i different (yet valid) blocks \mathcal{B}_{i} for each of the i signers, while ensuring that the i^{th} bit of the honest block matches the i^{th} bit of the malicious block.

\mathcal{A} only has to do this for 256 signers, that’s negligible work as for each bit, there’s a 50% probability that they’ll be able to succeed with a single attempt.

The adversary sends these different blocks individually to each of the i signers. The signers, check that the block is valid, release the Lamport signature for the corresponding bit to be signed, and return it to the BP.

The adversary \mathcal{A} now has a set of valid Lamport signatures for a malicious block \mathcal{B}'.

Note: This attack can be solved by having all the signers talk to each other to ensure they’re seeing the same hash… but this is expensive and implies that the security of the scheme depends on parties communicating with each other, thus making it an interactive scheme.