Undetected Error Bounds for Hybrid Integrity Protection Using Reed-Muller Codes, Algebraic Manipulation Detection, and Universal Hashing.

Ensuring information integrity requires not only reducing decoding errors but also reducing the probability that corrupted data are accepted as valid. This research presents a hybrid integrity protection system that incorporates seeded universal hash verification, algebraic manipulation detection (AMD), and a binary Reed-Muller outer code. Transmission over the binary symmetric channel BSC(p), outer encoding using RM(r, m), bounded-distance decoding, an εAMD-secure AMD layer, and a seeded 2-universal hash family with l-bit output define the model used in the analysis. Under explicitly stated freshness and conditional-independence assumptions, the system-level undetected error probability is upper-bounded by the residual decoder-miscorrection probability multiplied by the AMD acceptance bound and the seeded universal hash collision bound. A conservative alternative is also provided for settings in which the required conditional independence cannot be guaranteed. In this context, an explicit upper bound for the undetected error probability is derived. The outcome makes clear the different functions of outer coding and post-decoding verification and results in a direct dependency on the parameters r, m, p, and l. Finite-length Monte Carlo validation for a concrete instantiation based on RM(2, 5) complements the theoretical study and verifies that the hybrid construction offers a lower empirical undetected error probability compared to the comparable outer-only, AMD-only, and hash-only variations. The study does not propose new coding or verification primitives. Its contribution is a finite-length layered acceptance model and a Reed-Muller-specific undetected error analysis that incorporates the code weight distribution and bounded-distance decoding regions. The resulting spectrum-based bound distinguishes decoder miscorrection from the broader event of exceeding the guaranteed correction radius and is evaluated together with post-decoding verification and redundancy overhead. The model's formal manipulation detection and collision guarantees are provided by AMD and universal hash layers, while Reed-Muller code parameters and their standard distance formulas are conventional.