Due to the publicly known and deterministic characteristic of pilot tones, pilot authentication (PA) in multi-user multi-antenna orthogonal frequency-division multiplexing systems is very susceptible to the jamming/nulling/spoofing behaviors. To solve this, in this paper, we develop a hierarchical 2-D feature (H2DF) coding theory that exploits the hidden pilot signal features, i.e., the energy feature and independence feature, to secure pilot information coding which is applied between legitimate parties through a well-designed five-layer hierarchical coding model to achieve secure multiuser PA (SMPA). The reliability of SMPA is characterized using the identification error probability (IEP) of pilot encoding and decoding with the exact closed-form upper and lower bounds. However, this phenomenon of non-tight bounds brings about the risk of long-term instability in SMPA. Therefore, a reliability bound contraction theory is developed to shrink the bound interval, and practically, this is done by an easy-to-implement technique, namely, codebook partition within the H2DF code. In this process, a tradeoff between the upper and lower bounds of IEP is identified and a problem of optimal upper and lower bound tradeoff is formulated, with the objective of optimizing the cardinality of sub-codebooks such that the upper and lower bounds coincide. Solving this, we finally derive an exact closed-form expression for IEP, which realizes a stable and highly reliable SMPA. Numerical results validate the stability and resilience of H2DF coding in SMPA.
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