Spectral Characterization of Superposition Leakage in Deterministic-LPN Authentication Protocols

Recent work showed that LPN-based authentication protocols can be vulnerable to quantum chosen-message (Q2) attacks because independently sampled Bernoulli noise becomes a global phase under a coherent superposition query. This paper studies a narrower and explicitly separated question: how does a known deterministic noise function affect the Q2 measurement distribution of an HB-style phase oracle? We prove an exact Walsh-spectral shift identity, derive the resulting single-query distribution, and formulate two information-theoretic consequences. First, for a uniformly random secret, the mutual information between the secret and the measured outcome is exactly I(X;W) = n − H(P_g), where P_g is the squared-normalized Walsh spectrum of the noise function. Second, repeated observations identify the secret only up to the translation stabilizer of P_g. Bent functions therefore give zero information in this oracle model, while affine functions reveal the secret completely in one query. We retain a finite-sample empirical-mode bound for the unique-maximum case and derive a noise-density–spectral-flatness trade-off that forces the noise density toward 1/2 as the spectrum becomes flat. We also analyze an idealized joint-input extension corresponding to the algebraic form of HB+, while explicitly avoiding a claim that this oracle is identical to the physical Q2 interface of the protocol. The paper does not claim classical hardness or quantum security for keyed deterministic-LPN families; those questions remain open. With a public noise function g, the construction studied here is classically insecure by linear algebra, and is used only as an exact setting for spectral analysis. The accompanying artifact, AegisGuard-Q v2.0, computes the spectrum, mutual information and translation stabilizer, and reproduces the numerical tables reported in the paper. It is available at https://github.com/jayant-kumar-dev/Aegisguard-Q (release v2.0, commit 44d392cf886bfc420e2cba53bf907bdd9b66ceb3). Results were verified on Linux (Python 3.12.3) and independently reproduced on Windows (Python 3.14.5).