Whirlpool CoinJoin
Now let us look at a transaction that does privacy right: a Whirlpool CoinJoin.
This example demonstrates the Boltzmann entropy framework in action. If you have not yet read the Boltzmann Entropy section, we recommend starting there to understand the mathematical foundation.
Transaction ID: 323df21f0b0756f98336437aa3d2fb87e02b59f1946b714a7b09df04d429dec2
Structure: 5 inputs → 5 equal outputs of 5,000,000 sats each
What We Notice
This transaction has exactly 5 inputs and exactly 5 outputs, all of the same value (5 million sats each). This is the signature pattern of a Whirlpool CoinJoin round.
Let us look at the link probability matrix:
Why This Is Great
Unlike the consolidation example, this transaction creates real ambiguity:
- There are 1,496 valid interpretations of which input funded which output
- Entropy = 10.55 bits
- Each input has roughly a 34.2% chance of funding any given output (not 20% as you might expect)
What this means: An observer cannot determine which input funded which output. The Common Input Ownership Heuristic is broken.
Severity: This is a privacy win - positive signal
The Link Probability Matrix Explained
You might naturally think that with 5 inputs and 5 outputs, each input would have a 20% (1 in 5) chance of funding each output. But the matrix shows 34.2% for each cell. Why?
What Is a "Valid Interpretation"?
A valid interpretation (or "complete mapping") is a way of grouping ALL inputs and ALL outputs into groups where each group's input sum matches its output sum (within fee tolerance). This is a many-to-many mapping, not a one-to-one assignment.
For example, one valid interpretation might be: - Input 1 funds Output 1 AND Output 2 - Input 2 funds Output 3 - Input 3 funds Output 4 AND Output 5 - Inputs 4 and 5 fund nothing (they are combined with other inputs in the same groups)
Another valid interpretation: - Input 1 funds Output 1 - Input 2 funds Output 2 - Input 3 funds Output 3 - Input 4 funds Output 4 - Input 5 funds Output 5
And many more. For a 5-party equal-output CoinJoin like Whirlpool, there are exactly 1,496 valid interpretations.
Why Rows Sum to More Than 100%
This is the critical insight. In a single valid interpretation, one input can be linked to multiple outputs simultaneously. When we count how many interpretations contain each link, a single input can appear linked to several outputs in the same interpretation.
For the Whirlpool example: - There are 1,496 total interpretations - Each input-output pair appears together in 511 of those interpretations - 511 / 1,496 = 34.2%
This is why the link probability is 34.2%, not 20%. The 20% figure would only be correct if each input funded exactly one output per interpretation (a one-to-one assignment). But the Boltzmann algorithm counts many-to-many mappings, where one input can fund multiple outputs in the same interpretation.
The Math Behind 1,496 Interpretations
For equal-output CoinJoins, the number of valid interpretations can be computed using the integer partition formula. For 5 participants:
| Partition | Calculation | Term |
|---|---|---|
| [5] | 14400 / (14400 × 1) | 1 |
| [4,1] | 14400 / (576 × 1) | 25 |
| [3,2] | 14400 / (144 × 1) | 100 |
| [3,1,1] | 14400 / (36 × 2) | 200 |
| [2,2,1] | 14400 / (16 × 2) | 450 |
| [2,1,1,1] | 14400 / (4 × 6) | 600 |
| [1,1,1,1,1] | 14400 / (1 × 120) | 120 |
Total N = 1 + 25 + 100 + 200 + 450 + 600 + 120 = 1,496
Entropy = log2(1,496) = 10.55 bits
The classic permutation model (5! = 120) undercounts because it only considers one-to-one assignments. The partition model correctly accounts for the possibility that multiple outputs could be funded by the same input, yielding significantly more valid interpretations.
Perfect CoinJoin Entropy Table
| Participants | Interpretations (N) | Entropy E = log2(N) |
|---|---|---|
| 2 | 3 | 1.58 bits |
| 3 | 16 | 4.00 bits |
| 4 | 131 | 7.03 bits |
| 5 | 1,496 | 10.55 bits |
| 6 | 22,482 | 14.46 bits |
| 7 | 426,833 | 18.70 bits |
What This Means for Privacy
The 34.2% link probability means that for any given input-output pair, an analyst can only say "there is a 34.2% chance this input funded this output." This is far from the 100% certainty they get from a consolidation transaction.
With 5 participants, each output could have come from any of the 5 inputs. The analyst cannot narrow it down further. This is the power of CoinJoin.
Lesson
CoinJoin is the most powerful on-chain privacy tool available. It breaks the transaction graph by creating ambiguity. Use it regularly and do multiple rounds to increase your anonymity set.