We present a constructive approach to creating function cascades with provably high inversion complexity. Our main contribution is the discovery of a resonance condition: for the function f (x) = x^ (kx) mod p composed with modular inversion, the parameter k dividing (p-1) creates maximum collision density, resulting in inversion complexity Θ (p). We analyze the security of this construction against known attacks, discovering that smooth primes are vulnerable to subgroup attacks. We resolve this by demonstrating that safe primes (p = 2q+1) maintain resonance while limiting attack speedup to ~2x. The optimal construction uses safe primes with k = q = (p-1) /2, achieving effective security O (p/2). We discuss the relationship to the P vs NP problem, noting that while we construct a candidate one-way function, proving its one-wayness would itself resolve P vs NP. PRELIMINARY VERSION - Subject to revision pending formal proofs.
Aliaksei Naboko (Sun,) studied this question.