D-HPPK KEM: A Defactorized Homomorphic Polynomial Public Key Encapsulation Mechanism for IND-CCA2 Security

We present the Defactorized Homomorphic Polynomial Public Key (D-HPPK) Encapsulation Mechanism, a post-quantum cryptosystem whose security reduces to searching over the ephemeral secret x in F_p. Classical secret recovery requires Θ(2^l) field-element trials, with O(L₂) bit operations per trial, yielding l+2log₂L bits of classical security and l/2+log₂L bits of quantum security. The scheme uses a three-layer architecture of additive random masking and hidden ring embeddings. Masking destroys exploitable factorisation structure, while hidden moduli S₁,S₂ and multipliers R₁,R₂ (all coprime to p) prevent adversaries from evaluating the cancellation condition in F_p. Key recovery requires O(p³) operations and is not the bottleneck. Under the Hidden Modulus Product Problem and uniform decryption assumptions, we prove the base PKE satisfies message-recovery security; a double-encryption consistency check yields IND-CCA2 security in the random oracle model. A unified parameter set (n=2,m=3,log₂p=λ) ensures efficient, uniquely invertible decryption at NIST security levels. At Level V (λ=256), D-HPPK achieves a 1234-byte public key and 392-byte CCA2 ciphertext — 21% smaller public keys and 87% smaller ciphertexts than Kyber-1024 — with 8× faster encapsulation and 4× faster decapsulation in reference implementations, all without requiring SIMD or NTT acceleration. A single implementation scales across levels. D-HPPK demonstrates that polynomial-based cryptography can deliver rigorous security, compact footprints, and practical efficiency.