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modular tetration

the title is pretty clear.

we know alot about modular aritmetic.

but what about tetration ?

what do we know about e.g. a^^b mod p where p is a large odd prime and a and b are odd composites ?

or how do we solve for the smallest b satisifying

a^^b = c^^d mod p

how about cryptography ?

how about fixpoints ? or theorems ? conjectures ?

are there special kinds of primes p such that " equation mod p " is made easier ?

did i ask this before on this forum ?