# [bitcoin-dev] Non-equal value CoinJoins. Opinions.

Yuval Kogman nothingmuch at woobling.org
Sun Dec 29 03:31:48 UTC 2019

```Hi,

On Sat, 28 Dec 2019 at 01:29, nopara73 via bitcoin-dev <
bitcoin-dev at lists.linuxfoundation.org> wrote:

I haven't read the whole thing in detail (and fwiw, I don't think I will by
this point), but I do want to respond to section about the combinatorics as
well as the proof, since both the premises and the implications don't seem
very solid to me, especially in light of the other replies in this thread.

It appears to be a step up from the Knapsack paper in terms of the
specificity of a concrete mixing protocol (which again, I did not
scrutinize, but see below), but a regression in terms of privacy (see other
replies), which even in the Knapsack paper's approach raises some concerns:

Now, there are 100!/(10!)^10 ~= 10^92 ways to partition the inputs into a
> list of 10 sets of 10 inputs, but only a tiny fraction of these partitions
> will produce the precise output list.
>
In the equal amount case, the search space of possible interpretations with
n = # inputs + # indistinguishable outputs is proportional to the nth Bell
number, i.e. it's exponential in the size of the transaction, which is an
inviting intuition. But this is an *upper* bound on the difficulty of

This quantitative framing is potentially misleading because:

1. attributing inputs/outputs (sub-transactions in the Knapsack paper's
terminology) is arguably not a search problem, but an optimization problem,
since approximate results are still partly useful to the adversary
2. there are many computational strategies, heuristics, etc that in
practice can make this more efficient than brute force[1], so framing it
that as a security parameter doesn't sit right with me
3. as individual sub-transactions are identified (for example using out of
band information), the computational challenge also *drops* exponentially
fast

not specific to unequal amounts, and in particular refers to ZmnSCPxj's
assertion of 0 linkability) I am very worried that perspectives that focus
on linkability information revealed by a single coinjoin transaction in
isolation. This problem was alluded in the document, to but I don't see
that it was addressed. Naively the post/pre mix transaction graph would
seem to present a computationally much harder problem when looking at the
combinatorics through the same lens, but reality it can also be used to
place many constraints on valid partitions/sub-transaction assignments for
a single transaction with equal amounts. The trivial example is post mix
linking of outputs, but there are many other ways to draw inferences or
eliminate possible interpretations of a single transaction based on its
wider context, which in turn may be used to attack other transactions.

> Based on the example above, we can see that not only are there a huge
> number of partitions, but that even with a fast algorithm that could find
> matching partitions, it would produce around 10^20 possible valid
> configurations. With 10^20 possibilities, there is essentially no linkage.
>
This is a better framing, but still doesn't address my third bullet, since
"Attacks always get better; they never get worse." In other words
"essentially no linkage" due to multiple possible interpretation is still
strictly more meaningful if you can add constraints out of band.

To be fair in equal amount CoinJoins this is also the case, but it's a much
simpler model to consider in the context of other privacy leak vectors
(e.g. transaction graph connectivity beyond a single coinjoin, wallet
fingerprinting, temporal patterns, network privacy leaks, etc etc), since
analyzing your level of exposure is *also* complicated by unequal amounts,
in other words higher chance of privacy leaks due to misuse, or ignorance
of some of the implications under intended use. Thinking through these
implications is much easier when the information content in the amounts is
minimized.

The Cash Fusion scheme actually extends this obfuscation even further. Not
> only can players bring many inputs, they can also have multiple outputs
>
And, quoting another section:

Unfortunately, the production of equal-amount coins is impractical for
> various reasons. Foremost, it has a "toxic waste"
>

I'm still cautiously optimistic about the potential of multiple
inputs/outputs per user (c.f. 3-phase chaumian CoinJoin ideas we've
previously discussed in the context of Wasabi, though I don't recall any
public discussion I can link to, sorry list), but with the additional
assumption of amounts with small popcounts/Hamming weights (e.g. only
amounts that are 2^n sat in size, or based on 1-2-5 series, and for a

Unfortunately this trades off that "toxic waste" problem for a very large
on chain footprint (e.g. if the popcount of the amount of a wallet is
limited to 1, the number of inputs and change outputs required in the worst
case is proportional to log of the payment amount) and significant UTXO
bloat (several mixed outputs per magnitude for transaction size to scale as
the popcount(payment amount) instead of the log(payment amount))

However, with OP_CHECKTEMPLATEVERIFY and Taproot, this overhead could
potentially be mitigated (something more like TumbleBit's privacy model,
but with an on chain footprint similar to multiparty payment channels as
described in the OP_CTV BIP draft) since the safety guaranteed by
CoinJoins' atomicity can be preserved without requiring atomicity of the
mixing itself, which can extend over multiple transactions and long time
intervals, while still providing the liquidity and finality and unilateral
exit option of payment channels. By moving such low hamming weight amount
outputs off chain, and allowing them to be mixed (with equal amounts),
split and merged off chain. The simpler analysis of equal amount outputs
could still be assumed which makes analysis easier and assumptions about
adversary weaker, and furthermore this approach would better align the
incentives for batching and privacy, which is why I think it's very
promising.

Finally, the proof as well as its applicability seems suspect to me, since
seems to involve trusting the server:
"Since the distinct list [...] [is] kept on the server and not shared with
the players"
"The server knows the linkages of the commitments but does not participate
as a verifier "
"If there is a problem [...] each component is assigned to another player
at random for verification"
these 3 statements together seems to suggest the server is trusted to not
use sybils in order the compromise privacy by participating in the
verification process?

(also, and although this is nitpicking, also does not seem to be
demonstrating "soundness" in any sense that I am familiar with - wouldn't
break in soundness only imply DoS vector?)

[1] btw, although I still owe you (nopara73) some consistently working
linear programming code to attribute change outputs in Wasabi (sorry! i
suck...), but anecdotally, in the special cases I did manage to solve even
transactions with tens of inputs do not present a challenge even to
supposedly slow/inefficient solvers compared to the state of the art. Even
though it's just anecdotal and in the context of the easier problem of
attributing change, which can still be ambiguous in Wasabi transactions, i
still think this puts into question the privacy of mixing based on a
dubious hardness assumption and a computationally bounded adversary, as
opposed to something which (in the scope of a single mixing transaction) is
perfectly hiding.
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