Deterministic Cryptographic Seed Generation via Cyclic Modular Inversion over $\mathbb{Z}\mathrm{/}{3}^p\mathbb{Z}$

We present a deterministic framework for cryptographic seed generation based on cyclic modular inversion over $\mathbb{Z}\mathrm{/}{3}^p\mathbb{Z}$. The method enforces algebraic admissibility on seed inputs via the identity $d_k\equiv -{\left(2^{k-1}\right)}^{-1}{3}^p$, thereby producing structured and invertible residue sequences. This mapping yields entropy-rich, cycle-complete seeds well-suited for cryptographic primitives such as DRBGs, KDFs, and post-quantum schemes. To assess the quality of randomness, we introduce the Entropy Confidence Score (ECS), a composite metric reflecting coverage, uniformity, and modular bias. Although not a cryptographic PRNG in itself, the framework serves as a deterministic entropy filter that conditions and validates seed inputs prior to their use by conventional generators. Empirical and hardware-based results confirm constant-time execution, minimal side-channel leakage, and lightweight feasibility for embedded applications. The framework complements existing cryptographic stacks by acting as an algebraically verifiable entropy filter, thereby enhancing structural soundness and auditability.