Depending on the implementation, there are two cryptographic functions that might be used (perhaps in conjunction):
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Cryptographic hash: An arbitrary amount of data (like a video file) is used to create a “hash”—a shorter, (effectively) unique text string. Anyone can run the file through the same function to see if it produces the same hash; if even a single bit of the file is changed, the hash will be completely different and you’ll know the data was altered.
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Public key cryptography: A pair of keys are created, one of which can only encrypt data (but can’t decrypt its own output), and the other, “public” key can only decrypt data that was encrypted by the first key. Users (like the White House) can post their public key on their website; then if a subsequent message purporting to come from that user can be decrypted using their public key, it proves it came from them.