The Basics
Pedersen commitments are cryptographic algorithms that allow a prover to commit to a certain value without revealing it or being able to change it.
When you spend Monero, the value of the inputs you are spending and the value of the outputs you send are encrypted and opaque to everyone except the recipient. Pedersen commitments allow you to send Monero without revealing the value of the transactions. Pedersen’s commitments also make it possible for people to verify that transactions on the blockchain are valid and not create Monero out of thin air.
What It Means
As long as the encrypted output amounts created, which include an output for the recipient and a change output back to the sender, and the unencrypted transaction fee is equal to the sum of the inputs that are being spent, it is a legitimate transaction and can be confirmed not to be creating Monero out of thin air.
Pedersen’s commitments mean that the sums can be verified as being equal. Still, the Monero value of each sum and the Monero value of the inputs and outputs are undeterminable. Pedersen’s commitments also mean that even the ratio of one input to another or output to another is undeterminable.
It is unclear which inputs are spent as the ring signature lists both the real and decoy inputs. Therefore, you don’t know which input Pedersen commitments need to be summed. That’s okay because the RingCT ring signature only has to prove that the outputs are equal to the sum of the inputs for one combination of the inputs. For mathematical reasons, this is impossible to forge.
In-depth Information
See information in Ring Confidential Transactions paper by Shen Noether of the Monero Research Lab.
