Beyond Quadratic: Unlocking Pseudorandomness with Quartic Character

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We study pseudorandomness arising from quartic Dirichlet characters and obtain results that connect analytic and cryptographic perspectives. From an analytic perspective, and following the framework of Mauduit and Sárközy, we define a Boolean function $\psi_\pi$ from the quartic character $\chi_\pi$ modulo a Gaussian prime $\pi$ and analyze sequence $E_{p-1}=(\psi_\pi(1),\ldots,\psi_\pi(p-1))$ for $p=\pi\bar\pi \equiv 1 \bmod{4}$. Using classical character-sum bounds (Pólya-Vinogradov and refine