Zero-Knowledge cryptographic proofs
Bitcoin and preceding alternative cryptocurrencies have attempted to solve this problem through the use of transaction mixers or ring signatures. However, there are a number of drawbacks to these proposed solutions. For one, a malicious or compromised member of a mixer or ring signature can break privacy. Furthermore, the anonymity set is a key metric to understanding how private a cryptocurrency is. The anonymity set in formerly proposed solutions is limited by the size of the mixing cycle or ring signature. Each mixing cycle or ring signature is limited by the number of transactions per cycle, which is transitively limited by the the block size of the cryptocurrency. Thus, the anonymity set in previous attempts at privacy tends to only be a few hundred transactions.
With Zcoin, the anonymity set is on a dramatically higher magnitude. Instead of having an anonymity set limited to the few dozen, Zcoin has an anonymity set that encompasses all minted coins in a particular RSA accumulator that can scale to many thousands and unlike other solutions is not subject to transaction graph analysis.