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| This document is an attempt to define metrics quantifying the degree of privacy provided by a bitcoin transaction. | |
| Objectives | |
| 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) | |
| with: | |
| 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). | |
| Examples | |
| Example 1: Basic Payment Transaction | |
| * https://oxt.me/transaction/dcba20fdfe34fe240fa6eacccfb2e58468ba2feafcfff99706145800d09a09a6 | |
| * N = 1 | |
| * E = 0 | |
| * The unique interpretation is: [(I1,I2,I3,I4,I5,I6)=>(O1,O2)] | |
| Example 2: Ambiguous Transaction | |
| * https://oxt.me/transaction/8c5feb901f3983b0f28d996f9606d895d75136dbe8d77ed1d6c7340a403a73bf | |
| * N = 2 | |
| * E = 1 | |
| * Interpretations are: | |
| - [(I1)=>(O2), (I2)=>(O1)] | |
| - [(I1,I2)=>(O1,O2)] | |
| Example 3: Basic Coinjoin Transaction (DarkWallet) | |
| * https://oxt.me/transaction/8e56317360a548e8ef28ec475878ef70d1371bee3526c017ac22ad61ae5740b8 | |
| * N = 3 | |
| * E = 1.585 | |
| * Interpretations are: | |
| - [(I1)=>(O1,O2), (I2)=>(O3,O4)] | |
| - [(I1)=>(O3,O2), (I2)=>(O1,O4)] | |
| - [(I1,I2)=>(O1,O2,O3,O4)] | |
| * Note: | |
| 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. | |
| Observations | |
| - 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. | |
| Limitations | |
| - 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) | |
| Implementation | |
| 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. | |
| References | |
| (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/) |
On Entropy
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 (N), but notice what you want to examine is the strength of input - input, input-output and output-output relationships. With these, weighted by the values you can come up with a measure for entropy that does not rely on intuition. Also the work on top of this entropy can be mostly ported to the new entropy metric, so it's not like everything is lost by changing the intuition axiom with a mathematical axiom.
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 N + 1 (as @AdamISZ noted the possibility of the tx not being a CJ shall also not be ruled out, at least not by default.) Let's call it |M|, which is = N + 1. Next select a link and count how many valid sub-mappings the link is in the same partition, and let's call this number |L|. Finally you can get the probability to the link by dividing them: |L|/|M|. Then you get all the links and from there on getting your axiomatic entropy metric is obvious.
Feasibility Concern
Unless you optimized something since to estimate N, it is not computationally more expensive as you are relying on identifying the sub-mappings themselves, just like Maurer et al.
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.
Example
Take tx1 1, 1, 100 -> 1, 1, 100 and tx2 100, 100 -> 100, 100. Shannon entropy gives the same result for both transactions, while an entropy built on top of the Knapsack paper has more accurate score, because in Knapsack you must factors in that the 100 -> 100 partition of tx1 can be found in every valid sub-mapping, while in Shannon you're only concerned with the number of sub-mappings.
as in 2017, Maurer et al. in the paper titled Anonymous CoinJoin Transactions with Arbitrary Values worked out the mathematical basis of it.
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:
- The approach is very similar to the one used by Boltzmann and the paper provides a description far better than these gists (academic quality),
- Their model extends the analysis made by Boltzmann (links between inputs and outputs) to links between inputs and links between outputs.
Basing it on Shannon entropy is a creative intuition
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.
TL/DR:
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:
- an equal probability => formula proposed by Boltzmann (the scientist... not the algo),
- unequal probabilities => (generalized form proposed by Gibbs).
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:
- macrostate => the transaction that we can observe on the blockhain,
- microstates => all potential interpretations of the transaction,
- Boltzmann's (the scientist) entropy => Boltzmann's (the algo) entropy
In details, the algo enumerates all valid interpretations. From this, it then derives:
- the entropy,
- the link probabilities matrix (probabilities of links between inputs and outputs)
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.
Take tx1 1, 1, 100 -> 1, 1, 100 and tx2 100, 100 -> 100, 100. Shannon entropy gives the same result for both transactions, while an entropy built on top of the Knapsack paper has more accurate score, because in Knapsack you must factors in that the 100 -> 100 partition of tx1 can be found in every valid sub-mapping, while in Shannon you're only concerned with the number of sub-mappings.
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:
- tx1: Entropy = 3 (i.e. 8 interpretations)
- tx2: Entropy = 1.584963 (i.e. 3 interpretations)
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:
- How to do it trustlessly?
- Is it really more efficient than equal amount mixes across transaction chains?
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.
Regarding always creating unequal amounts to maximize Knapsack entropy in a single transaction, it makes sense but does it makes sense when we extend it to transaction chains and compare it with standard amount mixes? Remixes are frequent and creating standard output amounts facilitates mixes on the inputs of the remixes. Thus it may very well be the case where the Knapsack entropy of a tx chain would be ultimately greater with standard amounts than the Knapsack entropy maximizing method on single transaction.
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.
- Maybe the most obvious improvement would be that those links would be weighed by amounts as a 30% probable link between 1BTC and 100BTC isn't equivalent to a 30% link between 1BTC and 2BTC.
- Then maybe we could say that the total volume of a mix would be 1, instead of 47.5BTC so we can compare different mixes regardless the volume.
- Finally add them together?
Example. So for tx 1, 2 -> 1, 1, 1 we would get the following:
Sub mappings:
1,2 -> 1,1,1
1 -> 1 | 2 -> 1,1
1 -> 1 | 2 -> 1,1
1 -> 1 | 2 -> 1,1
Input match probabilities:
1 - inputs: 2(0.25) | outputs: 1(0.5) 1(0.5) 1(0.5)
2 - inputs: 1(0.25) | outputs: 1(0.75) 1(0.75) 1(0.75)
Output match probabilities:
1 - outputs: 1(0.5) 1(0.5) | inputs: 1(0.5) 2(0.75)
1 - outputs: 1(0.5) 1(0.5) | inputs: 1(0.5) 2(0.75)
1 - outputs: 1(0.5) 1(0.5) | inputs: 1(0.5) 2(0.75)
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 3BTC, so then the amounts would be 1/3, 2/3 -> 1/3, 1/3, 1/3.
And then maybe add the amounts together and multiply them with the link probabilities?
- 1BTC input links to other inputs and outputs:
(1/3 + 2/3) * 0.25 + (1/3 + 1/3) * 0.5 + (1/3 + 1/3) * 0.5 + (1/3 + 1/3) * 0.5 - 2BTC input links to other inputs and outputs:
(2/3 + 1/3) * 0.25 + (2/3 + 1/3) * 0.75 + (2/3 + 1/3) * 0.75 + (2/3 + 1/3) * 0.75 - 3 times 1BTC output links to other outputs:
3 * ((1/3 + 1/3) * 0.5 + (1/3 + 1/3) * 0.5)
The result of this would be 5.75 = (1/3 + 2/3) * 0.25 + (1/3 + 1/3) * 0.5 + (1/3 + 1/3) * 0.5 + (1/3 + 1/3) * 0.5 + (2/3 + 1/3) * 0.25 + (2/3 + 1/3) * 0.75 + (2/3 + 1/3) * 0.75 + (2/3 + 1/3) * 0.75 + 3 * ((1/3 + 1/3) * 0.5 + (1/3 + 1/3) * 0.5).
And the Boltzmann (not Shannon, I hope I name it properly now) entropy for 3 interpretations would be 1.58.
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?
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.