Here's an optimization used in Monero that optimizes the ledger size (initial confidential transaction proofs were too large).
You can have a public ledger but all of the data in it is encrypted and verifiable.
Each transaction, from the creation, to validation has its amounts and wallet ids hidden, the sender and the receiver cannot know anything about wallet contents of each other and receiver has no idea from where the sender is sourcing the coins.
OK, you and I exchanging coins do not know about each other's wallets or even the origin & destination of our transaction.
This anonymity depends on encryption to hide the wallets & transaction, and decryption to verify it, and for the receiver to be able to use the funds in the future.
So, who/what controls those keys? Seems they've just exchanged an open ledger problem for a key management problem - why is this not the case?
The key insight is that you design a cryptographic algorithm that will preserve addition between amounts, even when encrypted but you need to also provide some encrypted data that will allow an independent verifier to validate the whole transaction.
The problem is more nuanced because you need to guarantee that coins aren't produced from thin air.
The problem of the approach in the linked article is that proofs are large. To support encrypted verification (there is no need to decrypt anything in any step of the process) you need thousands of bytes for verifying a 32-bit amount.
Bulletproofs reduced the proof size significantly. There are then additional approaches like MimbleWimble where the proofs on the ledger can be discarded to make it even smaller.
To be honest, I don't see any maths. I see a document titled "investigation" which contains a bibliography consisting of a single forum post from 2013 by someone named Adam, and then the author goes on saying that he has invented some maths that allegedly would allow confidential transactions. Forgive me if I'm a bit sceptical... better wait until some experts review this material.
Here's Greg Maxwell (one of the Bitcoin core team members) explaining the confidential transactions.
https://crypto.stanford.edu/bulletproofs/
Here's an optimization used in Monero that optimizes the ledger size (initial confidential transaction proofs were too large).
You can have a public ledger but all of the data in it is encrypted and verifiable.
Each transaction, from the creation, to validation has its amounts and wallet ids hidden, the sender and the receiver cannot know anything about wallet contents of each other and receiver has no idea from where the sender is sourcing the coins.