1:1 algorithmic shuffle of massive numerical space
There are three major options available to you, depending on the available preferences and guarantees, ordered in descending order of generality:

Format-Preserving Encryption is a class of encryption schemes that can accept arbitrary input spaces and then produce ciphertext from that input space as output. A classical example would be encryption of credit card numbers to encrypted ones that are still accepted by the database solution validating their format.
Wide Block Cipher Modes, e.g. EME* take a bit string of arbitrary length and then compute a keyed pseudo-random permutation on them, their original intended use was disk encryption where you can't store additional authentication data but want to maximize your chances of detecting modification by randomizing the data sector if the ciphertext sector is changed.
Traditional Symmetric Encryption, e.g. CBC mode with a fixed nonce / IV. Note that these suffer from the isuse that changes at bit position $n$ affect at most a constant amount of bits in front of $n$ and at most all bits following $n$ which seems to not be the avalanche effect you're looking for.

If you're wondering, the recommended baseline block cipher for the above constructions is AES.