Welcome to the resource topic for
**2022/1104**

**Title:**

$\mu$Cash: Transparent Anonymous Transactions

**Authors:**
Liam Eagen

**Abstract:**

Zero Knowledge Set Membership Proofs (zkSMPs) allow efficiently, i.e. sublinearly in the size of the set, proving membership of a value in a set in zero knowledge with respect to the value. They have been used to construct anonymous cryptocurrencies such as ZCash, which uses a zero knowledge Merkle proof to show that the inputs of a transaction belong to the Transaction Output (TXO) set. Using a Merkle tree instantiated with a pair of Pedersen hash functions between an amicable cycle of elliptic curves, similarly to Curve Trees, and the Weil Elliptic Curve Inner Product (ECIPs) proofs, I design a set membership protocol with substantially smaller witness sizes than other Merkle zkSMPs. This protocol uses a pair of communicating Bulletproofs, one over each curve, whose total proof size I am able to reduce by proving portions of each verifier inside the other proof. Using these techniques, along with an adaptation of the Bulletproofs++ confidential transaction protocol, I design an anonymous transaction protocol for a decentralized cryptocurrency, whose security argument is reducible to the discrete log problem over a pair of elliptic curves and that does not require a trusted setup. Over a 256 bit field, these transactions are 1349 + 64n + 32 \lceil \log_2 c \rceil bytes for n inputs, m outputs, d depth, and c proof capacity, which is bounded by a linear function of n d, n, and m and is equal to 1 for up to m < 1000 or n < 37 when d = 48. Proving complexity is quasilinear and verifier complexity is linear in both n d and m, and in practice verification will be dominated by the cost of two Bulletproof verifications of length 1536 and 1744 for c=1. $\mu$Cash support efficient batch verification, user defined assets and multi-asset confidential transactions, privacy preserving multi-party proving, adaptor signatures, absolute and relative time locks, and a multiphase transaction structure to support scriptless scripts for private atomic swaps and payment channels. This protocol is likely compatible with the Halo accumulation scheme, although I do not investigate this.

**ePrint:**
https://eprint.iacr.org/2022/1104

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