I am looking for a robust answer to the best known algorithm for keeping cryptocurrency balances private.
The one implementation I know of has been created by Greg Maxwell of blockstreem. These algorithms are already easily implemented via the well known libsecp256k1 library.
Another approach I am aware of is known as Ring Confidential Transactions (pdf) by Shen Noether. I found an implementation that pulls code from Monero.
What other approaches are out there and what are the pro/cons of each approach? In particular what are the trade offs in space/time efficiency and privacy?