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Here are some popular coins that hold privacy in high regard:
First we have monero:
Monero is a secure, private, untraceable currency. It is open-source and freely available to all.
It uses a cryptographically sound system that allows you to send and receive funds without your transactions being publicly visible on the blockchain.
Next I would be remiss if I didn’t talk about zcash.
So Zcash is the first open, permissionless cryptocurrency that can fully protect the privacy of transactions using zero-knowledge cryptography. If Bitcoin is like http for money, Zcash is https—a secure transport layer
Zcash brings fungibility to cryptocurrency by unlinking shielded coins from their history on the blockchain.
There is also a coin called zclassic which is a fork of zcash that allocates all mining rewards directly to the miners, while with zcash, 20% of the rewards go to the founders for the first 4 years. However, zclassic does use the same trusted parameters as zcash. Zclassic believes in equity, no censorship and no governing body.
Let’s take a better look
Monero takes advantage of ring signatures, now according to their website, A ring signature is a type of group signature that makes use of your account keys and a number of public keys (also known as outputs) pulled from the blockchain using a triangular distribution method. Over the course of time, past outputs could be used multiple times to form possible signer participants. In a “ring” of possible signers, all ring members are equal and valid. There is no way an outside observer can tell which of the possible signers in a signature group belongs to your account. So, ring signatures ensure that transaction outputs are untraceable.
Zcash is pioneering the use of zero-knowledge proofs, applying cutting-edge cryptography to blockchain technology. Zero-knowledge proofs allow fully encrypted transactions to be confirmed as valid.
With Zcash, users can enjoy the advantages of using a public blockchain, while still being sure that their private information is protected.
Ethereum can take advantage of zero knowledge proofs or zk-SNARKs because Ethereum’s flexible smart contract interface enables a large variety of applications, many of which have probably not yet been conceived. The possibilities grow considerably when adding the capacity for privacy. Imagine, for example, an election or auction conducted on the blockchain via a smart contract such that the results can be verified by any observer of the blockchain, but the individual votes or bids are not revealed. Another possible scenario may involve selective disclosure where users would have the ability to prove they are in a certain city without disclosing their exact location. The key to adding such capabilities to Ethereum is zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs) – precisely the cryptographic engine underlying Zcash.
Ring signatures https://getmonero.org/knowledge-base/moneropedia/ringsignatures
Ethereum & zk-SNARKs https://z.cash/blog/zksnarks-in-ethereum.html
More on Ethereum https://blog.ethereum.org/2017/01/19/update-integrating-zcash-ethereum/