Paper 2026/964

Beyond Quadratic: Unlocking Pseudorandomness with Quartic Character

Abstract

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 refinements of Mauduit-Sárközy and Oon), we show that $E_{p-1}$ satisfies $W(E_{p-1}) =O(\sqrt{p}\log p)$ and $C_{\mu}(E_{p-1}) \leq 2^{\frac{\mu}{2}+1} \mu \sqrt{p}\log p,$ which imply strong pseudorandomness for small $\mu$. From the cryptographic side, we resolve an open question posed by Damgård by proving that quartic characters yield the secure pseudorandom generators and weak pseudorandom functions (wPRFs). Adapting similar techniques of Corrigan-Gibbs and Wu, we have shown that distinguishing quartic wPRF implies solving quadratic residuosity, via a chain of polynomial-time reductions. Our results show that under the Quadratic Residuosity Assumption, the quartic character also yields cryptographically secure wPRFs whose one-wayness was assumed in the construction of $\mathsf{Quartapus}$ signature scheme by Brier et al. and the post-quantum secure signature scheme $\mathsf{PorcRoast}_{4}$ by Beullens et al. that are more efficient and secure than legacy schemes based on the Legendre character.

Note: The authors would like to thank Prof. Santanu Sarkar of the Indian Institute of Technology (IIT), Madras, for pointing out a gap in Algorithm 1 of Section 4.3. In the earlier version, the output of Algorithm 1 mapped into a proper subset of the set of quadratic residues modulo N (QR_N) instead of the entire set QR_N. We have corrected these glitches and have accordingly modified Section 4.3 and Section 2.5.