Non-interactive zero-knowledge proofs are zero-knowledge proofs where information between a prover and a verifier can be authenticated by the prover, without revealing any of the specific information beyond the validity of the transaction itself. This function of encryption makes direct communication between the prover and verifier unnecessary, effectively removing any intermediaries. The core trustless cryptography "proofing" involves a hash function generation of a random number, constrained within mathematical parameters (primarily to modulate hashing difficulties) determined by the prover and verifier.
With this cryptographic engine, the prover must demonstrate the validity of the transaction, by solving the hash of a random number. Finally, proof of the answer is returned to the verifier, without revealing its value.
There are many different methods for establishing a cryptographic proof of hash validity. Perhaps the most notable method, proof of work, involves computing the proper hash value by means of brute force guessing using computational power. A far more scalable approach involves the more modern proof of stake, which leverages the pooled public-key authenticity of blockchain validator networks.