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 (2^64), 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\ p1600, 24 behaves as a pseudorandom permutation. We derive tight bounds on the maximum differential characteristic probability DP_ 2^-128 and the maximum linear characteristic correlation |LC_| 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) 2^{c/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 ₂ (1600/5) + 1 = 10 rounds, leaving a security margin of 14 additional rounds against structural attacks.