Let $\mathbb{F}_q$ be a finite field of large characteristic and $a_1, a_2, \cdots, a_n \in \mathbb{F}_q$ be some pairwise different elements. I assume that $\sqrt{-1} \in \mathbb{F}_q$. Consider the quadric intersection $I := \big\{ y_i^2 - y_0^2 = a_i \big\}_{i=1}^n$ in the space of dimension $n+1$ in the variables $y_0, y_1, \cdots, y_n$. I checked that $I$ is an absolutely irreducible curve (at least for the first values $n \in \mathbb{N}$), that is, $I$ is a complete intersection in the language of algebraic geometry. Therefore, $I$ possesses $q + O(\sqrt{q})$ solutions, that is, $\mathbb{F}_q$-points.
However, I would like to explicitely find any solution to this system in polynomial time in $\log_2(q)$ and $n$. In general, solving systems of quadratic equations is known to be a hard problem. This underlies so-called multivariate cryptography. Nevertheless, $I$ has a specific sparse form. Besides, many schemes of multivariate cryptography are successfully attacked. Thus, I hope that my system can also be efficiently solved. I am not a specialist in this research field, hence I need your help.
