https://crypto.stackexchange.com/questions/18091/is-there-any-true-anonymous-cryptocurrencies/18096#18096
"As has been mentioned, CryptoNote-based currencies use ring signatures (contrast group signatures, which involve a trusted party) rather than plain old ECDSA signatures in the proof-of-ownership part of their transaction signature scheme. Essentially, in Bitcoin each transaction output has a public key associated to it (identified, though not revealed, by the Bitcoin address on the output), and to spend that output you need to produce a small script including a digital signature using this key. When everyone sees that transaction, they see that the old output(s) are spent (so they can forget about it, except as far as keeping historical blockchain data for new users) and that new unspent output(s) have been created. Ring signatures, on the other hand, are associated to an arbitrary set of public keys, and knowledge of only one is required to make a signature. CryptoNote uses this by having each transaction input be a set of potential unspent outputs of the same value, signed by a ring signature using all those outputs' associated keys. It is impossible to determine which one is the "real" one that the signer is actually spending.
You might ask, if it's impossible to determine which outputs are being spent, how can double-spending by prevented? CryptoNote solves this using a special ring signature algorithm (which is a modified version of existing signatures, so this is original cryptography and their security proof should be scrutinized — fortunately it is not too complicated) in which the real signing key has a key image associated to it, which must be published alongside the signature. This key image cannot be reversed to get the original key and deanonymize the sender, but if a double-spend is attempted, users will see that the same key image is used twice and reject the second attempt."