# How does CashFusion work?

CashFusion is a privacy-enhancing feature for Bitcoin transactions that combines two key ideas. The first idea involves enabling trustless, private, multi-input CoinJoins with blame capabilities through a commitment-based multiparty computation scheme. The second idea eliminates the equal-amount concept and focuses on increasing privacy by using a large number of inputs and outputs.

The primary goal of CashFusion is to make it computationally impractical to iterate through all possible partitions of inputs and outputs in a transaction. This is achieved by significantly increasing the numbers of inputs and outputs. The mathematical analysis involves Stirling numbers of the second kind, which determine the number of ways to partition a set of n objects into k non-empty subsets.

For example, if there are 8 players in a fusion round with a transaction containing 50 inputs and 50 outputs, the number of possible ways to partition the inputs into 8 non-empty subsets is approximately 3.5 x 10^40. The same applies to the outputs. Multiplying these two numbers gives an astronomical figure of approximately 10^81 possible combinations.

To secondary goal if CashFusion is to make it so that that even with infinite computing power, there would be a large number of valid partitions. Using a scenario with 10 players and conservative assumptions, the calculated ratio of valid partitions to possible values is 34 billion to 1 billion. This conservative estimate considers a uniform distribution of sum values within a range, acknowledging real-world variations. The argument emphasizes the wide distribution of possible combinations and the immense number of partitions, making finding valid ones highly probable.

In summary, CashFusion leverages mathematical principles, such as combinatorics and the concept of randomness, to create an extremely large and practically unsearchable space of possible transaction partitions, enhancing privacy and making it challenging for external parties to trace and analyze the path of ownership of funds.

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