Criteria for testing the hypothesis on a noisy functional dependency between random binary vectors and bits

A sequence of random independent and uniformly distributed binary vectors $\vec x(t)$ and a sequence of random independent bits $y(t)$ are observed. Hypothesis $H_1\colon \{\vec x(t),y(t)$ are independent$\}$ is tested against $H_2\colon\{y(t)$ is corrupted value of $f(\vec x(t))\}$, the function $f$ essentially depends on an unknown part of the variables. We construct criteria based on sets of spectral statistics in situations of unknown $f$ or known $f$, and give asymptotic estimates of the volume of the size of the sample.