Assume I wanted to use RSA as the basis for a hash function or DRBG. Also assume that my construct would be insecure if someone were to have the private key. Is there any way generate the a usable public key to be used such that it is provably just as hard for me to generate the private key as it would be for anyone to do so?
I'm willing to take for granted:
- the existence of a suitable source of nothing-up-my-sleeve numbers
- the availability of significant computational resources.
The best idea I've come up with would depend on the ability to prove a lower bound the the size of the smallest factor in a number (e.g.: an $N$-bit number where all factors are at least of size $N*0.49$-bits) with significantly less work than it takes to actually factor it. (That and the ability to find such numbers.)