explanation. Thank you.
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Please don't copy other answers author and provide detailed explanation. Thank you.
Please don't copy other answers author and provide detailed
explanation. Thank you.
Mix aggregation (1.5 marks): Suppose Bitcoin was changed such that transactions can have at most two inputs and two outputs (as is the case in Zcash, for example). Alice, Bob, and Carol each have a single UTXO worth 1 BTC. They want to mix their coins (without a trusted server) resulting in three output UTXOs worth 1 BTC each which they are perfectly shuffled: all six (3!) permutations are equally likely. They decide to do the following: Alice and Bob post a transaction with their two bitcoins as input and two fresh addresses, X and Y, as outputs, each holding 1 BTC. Alice controls one address and Bob controls the other, but an observer cannot tell which is which. Suppose X is listed as the first output in this transaction. Next, Carol and the owner of X post a similar transaction: Carol's coin and the coin X are the inputs, and two fresh addresses W and Z are the outputs, each holding 1 BTC. Carol controls one address and the owner of X controls the other. As before, an observer cannot tell which is which. They use Y, Z,W as the final shuffled coins. Pictorially, the mixing process looks as follows: A Y mix B Х Z mix С (W) a. What are the weighted anonymity sets for the addresses Y, Z, and W? Assume that each mix independently assigns equal probability to both outcomes. b. Suppose at a later time it is revealed that Bob controls W. What else does this reveal? C. Clearly this mix is inadequate as the six possible permutations are not equally likely. By adding one more 2-to-2 transaction, can you help Alice, Bob, and Carol design a better mix so that all six (3!) permutations are equally likely to an outside observer? Hint: Adding one extra mix transaction is sufficient. The three 2-to-2 mixes can function independently, but one or more of the mixes must choose the ordering of outputs non-uniformly. You should describe the probabilities for each mix's outcomes if they are not uniform.
explanation. Thank you.
Mix aggregation (1.5 marks): Suppose Bitcoin was changed such that transactions can have at most two inputs and two outputs (as is the case in Zcash, for example). Alice, Bob, and Carol each have a single UTXO worth 1 BTC. They want to mix their coins (without a trusted server) resulting in three output UTXOs worth 1 BTC each which they are perfectly shuffled: all six (3!) permutations are equally likely. They decide to do the following: Alice and Bob post a transaction with their two bitcoins as input and two fresh addresses, X and Y, as outputs, each holding 1 BTC. Alice controls one address and Bob controls the other, but an observer cannot tell which is which. Suppose X is listed as the first output in this transaction. Next, Carol and the owner of X post a similar transaction: Carol's coin and the coin X are the inputs, and two fresh addresses W and Z are the outputs, each holding 1 BTC. Carol controls one address and the owner of X controls the other. As before, an observer cannot tell which is which. They use Y, Z,W as the final shuffled coins. Pictorially, the mixing process looks as follows: A Y mix B Х Z mix С (W) a. What are the weighted anonymity sets for the addresses Y, Z, and W? Assume that each mix independently assigns equal probability to both outcomes. b. Suppose at a later time it is revealed that Bob controls W. What else does this reveal? C. Clearly this mix is inadequate as the six possible permutations are not equally likely. By adding one more 2-to-2 transaction, can you help Alice, Bob, and Carol design a better mix so that all six (3!) permutations are equally likely to an outside observer? Hint: Adding one extra mix transaction is sufficient. The three 2-to-2 mixes can function independently, but one or more of the mixes must choose the ordering of outputs non-uniformly. You should describe the probabilities for each mix's outcomes if they are not uniform.
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