Is there any theoretically safe notion of white-box?
I have been reading about white-box [1, 2, 3] recently. It seems white-box deals with code obfuscation and its not known whether a perfect white-box exists.
I am curios about theoretical notion behind this. For a stream cipher, the theoretical notion is PRF. For block cipher it is PRP. For a hash function, it is one-way function. Is there anything comparable for white-box?
I am looking for something which is not necessarily practical, but we can actually show this achieves perfect white-box, like a really scrambled code or something.
UPDATE. I vaguely remember reading something on the line of
White-box would be possible if a super-huge memory exists. The memory could be loaded as a look-up table with the plaintexts as the look-up-keys and the ciphertexts the look-up-values. Since the encryption-key is not involved in the look-up table (it has been used previously to create the look-up table), it cannot be recovered but encryption would be still possible.
But I could not retrieve the source. Does anybody know anything about it?