Resonant Cascade Complexity: A Constructive Approach to One-Way Function Candidates

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.