I'm working at understanding the Wesolowski and Pietrzak RSA group based VDFs (verifiable delay functions). These basically work by requiring the prover to do a bunch of repeated squaring within a semiprime group G of unknown order, and then computing a proof that a verifier can check without having to do the time consuming work of repeated squaring mod G. These can be used for proof of work, trustworthy randomness beacons, spam prevention, etc.
I'm curious as to why both rely on computing (g^(2^t) mod G) as the work to be done rather than just computing g to the power of some arbitrary huge exponent mod G. (The huge exponent could be generated by e.g. running a CSPRNG seeded by a known input.) In this case you could compute a proof of exponentiation for g^bignum mod G and supply that to the verifier.
Does the use of repeated squaring rather than some arbitrary exponent have any cryptographic significance? Is it just simpler? Just curious so I can understand why these choices were made.
Also curious about the generator (g). Any reason it needs to be a particular number, or should it just be something like 3? Maybe I didn't catch it but I didn't see that discussed.