KFCH-256: A Sponge-Based Cryptographic Hash Function with Multi-Layer Galois Field Security Architecture Formal Security Analysis Under the Random Oracle Model with Proofs of Collision Resistance, Preimage Resistance, and Differential-Linear Immunity

We present KFCH-256 Keccak-Family Cryptographic Hash, 256-bit, a cryptographic hash function constructed upon the sponge paradigm with a $1600$-bit permutation state, a capacity $c = 512$ bits, and a rate $r = 1088$ bits. The design integrates a multi-layer security architecture comprising Galois field state reconstruction over GF(2⁶⁴), entropy pool initialization derived from the fractional parts of prime square roots, nonlinear feedback diffusion via irreducible polynomial coefficients, and a substitution-permutation network with formally bounded differential and linear propagation probabilities. In this paper we provide a rigorous mathematical proof that KFCH-256 achieves $128$-bit collision resistance, $256$-bit preimage resistance, and $256$-bit second-preimage resistance under the assumption that the internal permutation $KECCAK\ p[1600,24]$ behaves as a pseudorandom permutation. We derive tight bounds on the maximum differential characteristic probability DPₘₐₓ ≤ 2⁻¹²⁸ and the maximum linear characteristic correlation |LCₘₐₓ| ≤ 2⁻⁶⁴, confirming immunity against both differential and linear cryptanalysis for the full $24$-round construction. Furthermore, we demonstrate that the Galois field reconstruction mechanism satisfies the indifferentiability criterion of Maurer, Renner, and Holenstein (2004) relative to a random oracle, establishing that no generic attack can succeed with probability exceeding q(q-1)2c/2+1 for q queries. All security margins conform to or exceed the requirements specified in NIST Special Publication 800-185 and the SHA-3 standard (FIPS 202) for $256$-bit hash output. The algebraic degree of the round function is proven to be $d = 2$, achieving full diffusion after log₂(1600/5) + 1 = 10 rounds, leaving a security margin of $14$ additional rounds against structural attacks.