|This document is an attempt to define metrics quantifying the degree of privacy provided by a bitcoin transaction.|
|Definition of metrics measuring the resistance of a transaction to a set of attacks against users privacy.|
|Attacks considered in the scope of these metrics are:|
|- Merged Inputs Heuristic: methods identifying the inputs controlled by a same entity|
|- Coinjoin Sudoku: methods identifying the links existing between the inputs and outputs of a transaction|
|These metrics can be applied to all bitcoin transactions but are specifically useful for qualifying the "quality" of joint transactions (CoinJoin, SharedCoin, ...).|
|Entropy of a Transaction|
|The idea of this metric comes from a discussion with Greg Maxwell (1).|
|He was suggesting a notion of "Coinjoin Entropy" measuring how many possible mappings of inputs to outputs are possible given the values of the txos.|
|Considering that the number of combinations may be very high (b/o the combinatorial explosion created by coinjoin transactions) it's wise to define the notion of entropy:|
|E = log2(N)|
|E = entropy of the transaction|
|N = number of combinations (mappings of inputs to outputs)|
|Basically, it's the Shannon Entropy with the hypothesis that all events have the same probability (i.e. no additional information giving a "preference" to specific combinations).|
|Example 1: Basic Payment Transaction|
|* N = 1|
|* E = 0|
|* The unique interpretation is: [(I1,I2,I3,I4,I5,I6)=>(O1,O2)]|
|Example 2: Ambiguous Transaction|
|* N = 2|
|* E = 1|
|* Interpretations are:|
|- [(I1)=>(O2), (I2)=>(O1)]|
|Example 3: Basic Coinjoin Transaction (DarkWallet)|
|* N = 3|
|* E = 1.585|
|* Interpretations are:|
|- [(I1)=>(O1,O2), (I2)=>(O3,O4)]|
|- [(I1)=>(O3,O2), (I2)=>(O1,O4)]|
|O2 and O4 are the change outputs of the coinjoin transaction.|
|For all combinations we have I1=>O2 and I2=>O4.|
|We say that I1 is deterministically linked to O2 and I2 is deterministically linked to O4.|
|- Computation of the metric becomes quickly expensive when privacy tools like coinjoin are used to build the transaction (no surprise here).|
|- It's possible to decrease the computational cost thanks to side-channel information allowing to reduce the size of the problem space (e.g. inputs controlled by a same entity can be merged in a single input).|
|For example, these information can be provided by any heuristic allowing to cluster addresses (merged inputs, change address, ...).|
|- From this notion of Entropy, we can derive several metrics qualifying a transaction:|
|- Intrinsic Entropy: it's the value computed without any additional information, when the transaction is considered separately from the blockchain.|
|- Actual Entropy: it's the value taking into account additional information. On my hand I've worked on this one because I think it's the one which matters the most to users.|
|- Max Entropy: it's the value associated to a perfect coinjoin transaction having a structure similar or close to the evaluated transaction.|
|A perfect coinjoin is a coinjoin with all inputs having the same amount and all outputs having the same amount.|
|Here, we call structure the tuple (#inputs, #outputs)|
|Per se, Max Entropy isn't very interesting but it becomes much more useful when used to compute the following ratios:|
|Wallet efficiency = Intrinsic Entropy - Max Entropy (expressed in bits)|
|This metrics may be a proxy for qualifying the efficiency of a wallet when it builds a coinjoin transaction.|
|Wallet efficiency can also be expressed as a ratio: IntrinsicNumberOfCombinations(tx) / NumberOfCombinations(closest perfect coinjoin).|
|Blockchain efficiency = Actual Entropy - Max Entropy (expressed in bits)|
|This metrics may be a proxy for qualifying the efficiency of the whole ledger/ecosystem in term of protection of users privacy.|
|Blockchain efficiency can also be expressed as a ratio: ActualNumberOfCombinations(tx) / NumberOfCombinations(closest perfect coinjoin)|
|- Rule: Actual entropy of a bitcoin transaction is a dynamic value susceptible to decline over time. At best, it will stay constant. It will never increase.|
|- This metric is susceptible to be tricked by specific patterns of transactions like steganographic transactions.|
|A steganographic transaction aims to hide the real payment inside a "fake" payment.|
|Its fingerprint is similar to a classic payment but it differs from a classic payment by involving the payee in the payment process.|
|Example: UserA must pay 1.5btc to UserB. UserB (the payee) collaborates to the process by providing an input.|
|UserA 2btc -- -- 0.5btc UserA (change)|
|\ T1 /|
|UserB 1btc -- -- 2.5btc UserB|
|This transaction is indistinguishable from a classic payment (User1 provides 2 inputs. 1 output goes to the payee, 1 output is change).|
|According to our previous definition, we have E(T1)=0 instead of E(T1)=1 (i.e. classic payment + steganograhic transaction)|
|- This metric is a first good proxy for measuring privacy at transaction level. A transaction with a high entropy is likely to provide better privacy to the users.|
|But, as illustrated by the "Coinjoin Sudoku" attack (2), this metric fails to detect privacy leaks occuring at the lower level of specific inputs/outputs.|
|Therefore, this metric shall be completed by additional metrics (see part 2)|
|I've developed a python implementation of an algorithm computing this metric.|
|This implementation surely deserves the title of "worst possible solution": brute force algorithm, no parallelization, no optimization of the algorithm (memoization), ...|
|The unique optimization used by this implementation is a reduction of the problem space thanks to information provided by external heuristics.|
|As a direct consequence of this brute force approach, processing is limited to the most simple transactions (with #inputs <= 10 & #outputs <= 10).|
|Anyway, I was confident this implementation might crunch a good portion of the bitcoin ledger (60-70% of the transactions) so I've decided to keep it for a first round of tests.|
|Tests & Results|
|- Processing of transactions from block 1 to block 388602 (December 2015)|
|- #Txs (Total) = 97,977,912 (100,00%)|
|- #Txs with entropy computed = 96,597,663 ( 98,59%)|
|- #Txs not processed = 1,380,249 ( 1,41%)|
|TBH, I wasn't expecting such a large coverage ratio for this first round. I'm not sure it's great news for privacy.|
|Analysis of the unprocessed transactions|
|It's almost certain that a subset of these transactions is composed of true Coinjoin/SharedCoin transactions.|
|But it's also likely that another part is composed of large transactions with low entropy (classic payments with a large number of inputs and 2 outputs, ...).|
|A better/smarter implementation may allow the refinement of these results. To be done...|
|Analysis of the transactions processed|
|Cumulative distribution of txs per #combinations (loglog scale): http://i.imgur.com/jms8tpi.png|
|This distribution measures the probability of a transaction with #combinations >= value|
|Cumulative distribution of txs per entropy (linlog scale): http://i.imgur.com/NlLBL5W.png|
|This distribution measures the probability of a transaction with entropy >= value|
|Around 85.47% of the transactions processed have a null entropy (i.e. they have inputs & outputs deterministically linked)|
|Around 14.52% of the transactions processed have E(tx) >= 1 (i.e. they're as good or better than "ambiguous" transactions)|
|Around 1.89% of the transactions processed have E(tx) >= 1.585 (i.e. they're as good or better than the most basic coinjoin transactions)|
|In best case scenario, we may have around 3.3% of the transactions with an entropy greater or equal to the entropy of a basic coinjoin but it's likely that the real value is somewhere between 2% and 3%.|
|Note that it doesn't imply all these transactions have a coinjoin structure. They just produce higher entropies like coinjoin transactions.|
|(1): G.Maxwell - CoinJoin: Bitcoin privacy for the real world (https://bitcointalk.org/index.php?topic=279249.msg7117154#msg7117154)|
|(2): K.Atlas - Coinjoin Sudoku (http://www.coinjoinsudoku.com/)|
I have several questions about this methodology (I know this is very old, and I did briefly look at it earlier, but now someone is proposing to add this to JoinMarket I want to understand better).
(1) What is the definition of Actual Entropy? It seems you use E = log_2(N) for this, but I'm wondering about your choice of N based your examples - see the below points under (3).
While your analysis of coinjoin is sound from the point of view of assuming it's a coinjoin without payment cross-entity (that's always how I've looked at it, too), i.e. you use subset-sum or "coinjoin sudoku" to list possibilities, it's also true that the above kind of "full listing", with the 6th Bell number, is the more full case right (so that'd be 203 possibilities).
It occurs to me that not only payjoin, fake coinjoin and to-self payments, and batched payments may lead to non-standard linkage possibilities, but also coinswap and similar.
Well tbh these are just introductory thoughts before figuring out something more concrete, but I'm particularly interested to know your answer to (1) above.
Q1 and Q2:
WRT Bell numbers, there are definitely a useful tool. I would also recommend to check Exponential Bell Polynomials which is some mind blowing stuff (https://en.wikipedia.org/wiki/Bell_polynomials#Combinatorial_meaning) and can be useful for specific cases. I've used them in Bolztmann to speed up the computation of coinjoins with denominated amounts (see https://github.com/Samourai-Wallet/boltzmann/blob/master/boltzmann/utils/tx_processor.py#L264)
You're absolutely right that the main goal of Boltzmann is to deal with ambiguous and coinjoin-like transactions. My first motivation for Boltzmann was to avoid overclustering on OXT.
My very personal conclusion about this subject of "privacy metrics":
For instance, Boltzmann served me well in order to avoid overclustering on OXT (with the consequence that OXT "suffers" from underclustering when compared to others analytics platforms but I'm fine with this tradeoff because I think it's easier to deal with it).
Its recent introduction into Samourai Wallet (with a visual indicator of deterministic links) follows the same principle. It's used as a way to build awareness about the issue of deterministic links but it follows the principle that it should be used as a negative indicator (a high entropy doesn't give you any guaranteee but a low entropy isn't a good sign for your privacy).
IMHO, the hardest part is all about the "education" of users. Almost all users expect tools providing a metrics telling them that their privacy is safe. The challenge for all of us is to change this mindset.
First, thanks for the detailed responses. I understand what you're doing better, although I don't think I fully understand why you're doing it; a couple more comments/questions below:
I'm still finding this slightly ambiguous. When you say "amounts of inputs and outputs" I think you mean btc amounts of inputs and outputs, but "amount" could mean number.
Moreover, I understand that this ("Intrinsic Entropy") really just means "this transaction taken in isolation, not considering inferred information from elsewhere, such as who owns which input/output", but I still find that that your base definition, of "entropy is number of interpretations" is ambiguous, to me, because in your given 3 examples, I don't understand why you came up with N=1,2,3 in examples 1,2,3 respectively. Or more precisely: I largely understand the reasoning, but it seems arbitrary, since other interpretations are possible. Well, this probably folds into the below comment anyway:
Right, so setting aside payjoin for a moment, what I find confusing about example 1 is that you assert only 1 interpretation, which you give as (all inputs) -> (o1, o2), but isn't it the case that the situation where o1 is change and o2 is payment is different from vice versa? Similarly, the case where o1 and o2 represent different recipients is different too. They represent a different financial flows; in one case the payer paid 0.01, in the second 300, in the third 300.01, to some external entit(ies).
If I'm reading you correctly, it seems like you were quite focused on coinjoin, i.e. you wanted specifically to distinguish cases where more than one entity is directing funds from A to B inside one tx. I guess I can imagine, that you find this useful, since you can combine this with techniques for tracing (e.g. change heuristics) and then apply the entropy concept in a "transitive" or compounded way across multiple transactions. So it will usually be a value of 0, unless you have some coinjoin behaviour (and as you note below, there are other transaction patterns it will just totally fail to see).
Thanks for those comments, helpful also, I have no response on them right now. As above ... I think I get it now (you are basically counting subset sum possibilities based on btc amounts), but I'm bothered that that isn't a real measure of how ambiguous/ "entropic" a transaction is.
I think the ambiguity comes from the fact that we use the term "interpretations" for the description of 2 very different things. On its side, Boltzmann is focused on the question of "deterministic links" while the model you're discussing seems focused on the question "who controls what?".
In Boltzmann, an "interpretation" is a specific configuration of the links between the inputs and the outputs of the transaction. In the case of a coinjoin transaction, Boltzmann tells us that there are N such configurations and for each configuration, it tells us the links between the inputs and outputs. That gives us a "link probability matrix" (as in "there are X configurations for which input1 in linked to output2") which can be expressed as a probability of an existing link between an input and an output (100% = deterministic link).
Actually, the approach used for Boltzmann is really similar to the concept of Entropy used in thermodynamics with a Boltzmann configuration being the equivalent of a microstate as defined in thermodynamics. From this, we derive the Entropy score by using Gibbs Entropy formula, with the assumption that the probability of each configuration is the same (i.e. we don't have additionnal information allowing to ponder a given microstate with a higher probability).
Boltzmann doesn't try to evaluate if an input and an output are controlled by a same entity. The point isn't that this information isn't useful, it's just that Boltzmann wasn't designed for that goal. It's a matter of tradeoff. As a consequence, Boltzmann is "useless" for Payjoin/P2EP transactions (and almost all transactions with payments made between the senders). So yes, we might say that this is an arbitrary choice but as I wrote in my previous conclusion, I truly believe that any "privacy metric" is limited in scope. There's no such thing as a perfect or a better "privacy metric". IMHO, the best we can expect is multiple metrics evaluating specific aspects and/or used to improve specific aspects during coin selection.
If my understanding is correct, in the model that you're describing, the term "interpretation" has a very different meaning. In this model, an interpretation is a specific configuration of "which entity controls which inputs/outputs".
Concerning the use of Bell numbers, it seems to make sense in the context of your goal for this metrics (if my understanding is correct).
Just one question: according to your metrics, what would be the number of interpretations for this transaction?
A few additional shower thoughts (in no specific order):
EDIT: Added a note about the scalability and composability of metrics.
This is a nice simple way to state it. "Link entropy" vs "Node entropy".
Sure, but I don't think this changes between the two models. What probably does change: with "node entropy" the probabilities of individual "microstates" are in practice wildly different, whereas in a coinjoin scenario (choosing between multiple subset sum linkages) that isn't the case, so the assumption of "all states equiprobable", while the simplest, would be a bit shitty for "node entropy".
What I described above would imply 15 possibilities for any 2-in 2-out, no matter the btc amounts (but of course I haven't proposed anything sophisticated like reasons to discount specific possibilities, etc.).
Re: the various metrics, sure, that looks reasonable, but let me state my principal concern/issue with the "Link Entropy" approach:
Consider the case of a simple tree of transactions, each of which is a payment from one of the two outputs of its parent. So, every transaction is a bog-standard bitcoin payment. Since all of them would fit Example 1 here, all would have zero entropy. This means that a normal bitcoin transaction graph has zero entropy.
I understand your reasoning - that there is no ambiguity in linkages, and it's assumed that each transaction has a deterministic flow, but I fear that this is massively misleading to consumers of this metric.
Since there is, in fact, massive ambiguity in the actual flow of funds. Change heuristics are usually dubious, and at best they're still probabilistic. So even in the absolutely least private case (well - I ignore address reuse), it concerns me that an entropy measure of 0 will be misused and will help parties who want to sell snake-oil "tracing" to people.
Thanks again, this is an interesting topic and a lot clearer to me now.
Agreed. It's just a matter of variables selected for the definition of a "microstate".
I tend to agree and this is also why I believe that the distinction between intrinsic and actual entropies is important. The intrinsic version is a kind of ideal case assuming the equiprobability. The computation of actual entropy is equivalent to removing the equiprobability hypothesis. A corollary is that in Bitcoin, the intrinsic entropy is the max possible value for the entropy score. In best case scenario, this value will stay the same as time passes but it will often decrease if new information are collected, allowing to associate a null probability to some microstates.
IMO, it depends on the purpose of the tool. As you noted, if you're an analytics company it may be misleading but if the goal is to alert users about potential privacy issues, displaying the "worst case scenario" seems to me like a good conservative approach.
Basing it on Shannon entropy is a creative intuition, but there is no need to resort to any intuition at all, as in 2017, Maurer et al. in the paper titled Anonymous CoinJoin Transactions with Arbitrary Values worked out the mathematical basis of it.
Here the entropy depends on solely the number of valid non-complete sub-mappings (
You may be wondering how the strength of the links of input-input, input-output and output-output are determined: take all the valid sub-mappings and count them. This would be with your notation
Unless you optimized something since to estimate
Math Heavy Explanation
User Friendly Exmplanation
You can find a more user friendly introduction and some suggestions on how to build other metrics out of it here.
It's been a while since I've read this paper but my souvenir it that it was indeed an interesting paper. For at least 2 reasons:
Actually, the use of the concept of Entropy wasn't based on an intuition but on the certainty that it was the right tool for the task that motivated the design of the algo.
Sometimes, in statistical physics or in thermodynamics, entropy is interpretated as a measure of uncertainty (as in "a measure of our ignorance about the system"). For instance, we can observe a macrostate and we know that this macrostate may correspond to N different microstates. We aren't able to know the exact microstate but at least, we can evaluate our lack of knowledge, measured as the Entropy.
The specific form of the formula used to compute the entropy will depend on the fact that microstates have:
In the case of bitcoin transactions, the algo was initially designed under the angle of blockchain analysis. Thus, having (among others things) a metrics that measures our lack of knowledge about the right interpretation was a win '"Knowledge of our lack of knowledge is still useful knowledge).
In a nutshell we have these correspondences:
In details, the algo enumerates all valid interpretations. From this, it then derives:
At this point, tyou may wonder why we use the Boltzmann's form and not the Gibb's form to compute the entropy.
Well, The reason is pretty simple. When we analyze a transaction in isolation (without additional info that we may collect in the blockchain or by others means), all interpretations are equiprobable. Thus the choice of this form. This is what I called the "Intrinsic Entropy" in this gist.
But here's the most interesting part, IMO. If we can collect additionnal info (e.g: 2 inputs are controlled by the same entiy), then we can rule out some of the interpretations , which is roughly equivalent to setting a null probability for these interpretations in the Gibb's forms while considering that others interpretations have the same probability. This is what I called the "Actual Entropy" in this gist.
Under the hood, this is how the Boltzmann algo is used by OXT. At first, OXT computes the "Intrinsic" entropy of the tx but if new information are gathered later from new transactions, then OXT recomputes the entropy with these new information. This is what is displayed by OXT.
I'm not sure to understand what you mean by Shannon's entropy in this context. All I can say for sure is that Boltzmann (with the hypothesis of equiprobability of the interpretations) returns 2 different results:
Wrong example and probably wrong about Shannon naming of the entropy, it's just AFAIK Shannon's intuition was using the logarithm and ended up working in practice, but as I'm trying to look it up, I can see there's much more to this, anyway I won't spend much time on this, as it's less relevant to the conversation. Rather I'd get back to my tailor made example that was wrong and can't help but wondering if it's not that simple to construct such example, then maybe maximizing the subsets could give roughly the same result in practice as the Knapsack way where probabilities between i/o, i/i, o/o and come up with entropy from that.
Anyway, as a side note, the Knapsack paper's goal was to create unequal output mixing and only as a side effect they had to work out a mathematical model on how to score mixes. IMO this spinoff research is more applicable than the original goal itself, as the original goal suffers from lack of further research:
Regarding the trustlessness aspect I'm not sure if it's even possible as mixing networks rely on standard package sizes in one way or another.
Side note aside, what would be the Knapsack entropy anyway?
With Knapsack we can calculate the probabilities of links, but from there we have to get to entropy somehow.
Example. So for tx
This is how far Knapsack goes. Next thing could be to normalize the amounts to 1 based on the total volume for comparisions between transactions, which is
And then maybe add the amounts together and multiply them with the link probabilities?
The result of this would be
And the Boltzmann (not Shannon, I hope I name it properly now) entropy for 3 interpretations would be
I'm far from sure my calculations are correct, but it communicates the idea. Is this how entropy based on link probabilities should be calculated? Or there's a better way to do it?