Hello all,
I would like to discuss a way of computing a UTXO set hash that is
very efficient to update, but does not support any compact proofs of
existence or non-existence.
Much has been written on the topic of various data structures and
derived hashes for the UTXO/TXO set before (including Alan Reiner's
trust-free lite nodes [1], Peter Todd's TXO MMR commitments [2] [3],
or Bram Cohen's TXO bitfield [4]). They all provide interesting extra
functionality or tradeoffs, but require invasive changes to the P2P
protocol or how wallets work, or force nodes to maintain their
database in a normative fashion. Instead, here I focus on an efficient
hash that supports nothing but comparing two UTXO sets. However, it is
not incompatible with any of those other approaches, so we can gain
some of the advantages of a UTXO hash without adopting something that
may be incompatible with future protocol enhancements.
1. Incremental hashing
Computing a hash of the UTXO set is easy when it does not need
efficient updates, and when we can assume a fixed serialization with a
normative ordering for the data in it - just serialize the whole thing
and hash it. As different software or releases may use different
database models for the UTXO set, a solution that is order-independent
would seem preferable.
This brings us to the problem of computing a hash of unordered data.
Several approaches that accomplish this through incremental hashing
were suggested in [5], including XHASH, AdHash, and MuHash. XHASH
consists of first hashing all the set elements independently, and
XORing all those hashes together. This is insecure, as Gaussian
elimination can easily find a subset of random hashes that XOR to a
given value. AdHash/MuHash are similar, except addition/multiplication
modulo a large prime are used instead of XOR. Wagner [6] showed that
attacking XHASH or AdHash is an instance of a generalized birthday
problem (called the k-sum problem in his paper, with unrestricted k),
and gives a O(2^(2*sqrt(n)-1)) algorithm to attack it (for n-bit
hashes). As a result, AdHash with 256-bit hashes only has 31 bits of
security.
Thankfully, [6] also shows that the k-sum problem cannot be
efficiently solved in groups in which the discrete logarithm problem
is hard, as an efficient k-sum solver can be used to compute discrete
logarithms. As a result, MuHash modulo a sufficiently large safe prime
is provably secure under the DL assumption. Common guidelines on
security parameters [7] say that 3072-bit DL has about 128 bits of
security. A final 256-bit hash can be applied to the 3072-bit result
without loss of security to reduce the final size.
An alternative to multiplication modulo a prime is using an elliptic
curve group. Due to the ECDLP assumption, which the security of
Bitcoin signatures already relies on, this also results in security
against k-sum solving. This approach is used in the Elliptic Curve
Multiset Hash (ECMH) in [8]. For this to work, we must "hash onto a
curve point" in a way that results in points without known discrete
logarithm. The paper suggests using (controversial) binary elliptic
curves to make that operation efficient. If we only consider
secp256k1, one approach is just reading potential X coordinates from a
PRNG until one is found that has a corresponding Y coordinate
according to the curve equation. On average, 2 iterations are needed.
A constant time algorithm to hash onto the curve exists as well [9],
but it is only slightly faster and is much more complicated to
implement.
AdHash-like constructions with a sufficiently large intermediate hash
can be made secure against Wagner's algorithm, as suggested in [10].
4160-bit hashes would be needed for 128 bits of security. When
repetition is allowed, [8] gives a stronger attack against AdHash,
suggesting that as much as 400000 bits are needed. While repetition is
not directly an issue for our use case, it would be nice if
verification software would not be required to check for duplicated
entries.
2. Efficient addition and deletion
Interestingly, both ECMH and MuHash not only support adding set
elements in any order but also deleting in any order. As a result, we
can simply maintain a running sum for the UTXO set as a whole, and
add/subtract when creating/spending an output in it. In the case of
MuHash it is slightly more complicated, as computing an inverse is
relatively expensive. This can be solved by representing the running
value as a fraction, and multiplying created elements into the
numerator and spent elements into the denominator. Only when the final
hash is desired, a single modular inverse and multiplication is needed
to combine the two.
As the update operations are also associative, H(a)+H(b)+H(c)+H(d) can
in fact be computed as (H(a)+H(b)) + (H(c)+H(d)). This implies that
all of this is perfectly parallellizable: each thread can process an
arbitrary subset of the update operations, allowing them to be
efficiently combined later.
3. Comparison of approaches
Numbers below are based on preliminary benchmarks on a single thread
of a i7-6820HQ CPU running at 3.4GHz.
(1) (MuHash) Multiplying 3072-bit hashes mod 2^3072 - 1103717 (the
largest 3072-bit safe prime).
* Needs a fast modular multiplication/inverse implementation.
* Using SHA512 + ChaCha20 for generating the hashes takes 1.2us per element.
* Modular multiplication using GMP takes 1.5us per element (2.5us
with a 60-line C+asm implementation).
* 768 bytes for maintaining a running sum (384 for numerator, 384
for denominator)
* Very common security assumption. Even if the DL assumption would
be broken (but no k-sum algorithm faster than Wagner's is found), this
still maintains 110 bits of security.
(2) (ECMH) Adding secp256k1 EC points
* Much more complicated than the previous approaches when
implementing from scratch, but almost no extra complexity when ECDSA
secp256k1 signature validation is already implemented.
* Using SHA512 + libsecp256k1's point decompression for generating
the points takes 11us per element on average.
* Addition/subtracting of N points takes 5.25us + 0.25us*N.
* 64 bytes for a running sum.
* Identical security assumption as Bitcoin's signatures.
Using the numbers above, we find that:
* Computing the hash from just the UTXO set takes (1) 2m15s (2) 9m20s
* Processing all creations and spends in an average block takes (1)
24ms (2) 100ms
* Processing precomputed per-transaction aggregates in an average
block takes (1) 3ms (2) 0.5ms
Note that while (2) has higher CPU usage than (1) in general, it has
lower latency when using precomputed per-transaction aggregates. Using
such aggregates is also more feasible as they're only 64 bytes rather
than 768. Because of simplicity, (1) has my preference.
Overall, these numbers are sufficiently low (note that they can be
parallellized) that it would be reasonable for full nodes and/or other
software to always maintain one of them, and effectively have a
rolling cryptographical checksum of the UTXO set at all times.
4. Use cases
* Replacement for Bitcoin Core's gettxoutsetinfo RPC's hash
computation. This currently requires minutes of I/O and CPU, as it
serializes and hashes the entire UTXO set. A rolling set hash would
make this instant, making the whole RPC much more usable for sanity
checking.
* Assisting in implementation of fast sync methods with known good
blocks/UTXO sets.
* Database consistency checking: by remembering the UTXO set hash of
the past few blocks (computed on the fly), a consistency check can be
done that recomputes it based on the database.
[1] https://bitcointalk.org/index.php?topic=88208.0
[2]
https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2016-May/012715.html
[3]
https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2017-February/013591.html
[4]
https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2017-March/013928.html
[5] https://cseweb.ucsd.edu/~mihir/papers/inchash.pdf
[6] https://people.eecs.berkeley.edu/~daw/papers/genbday.html
[7] https://www.keylength.com/
[8] https://arxiv.org/pdf/1601.06502.pdf
[9] https://www.di.ens.fr/~fouque/pub/latincrypt12.pdf
[10]
http://csrc.nist.gov/groups/ST/hash/sha-3/Aug2014/documents/gligoroski_paper_sha3_2014_workshop.pdf
Cheers,
--
Pieter
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