Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $K = \mathrm{Kum}(E_1\times E_2)$ in one of the two forms: $$y^2 = (x_1^3 + a_4x_1 + a_6)(x_2^3 + a_4x_2 + a_6),\quad {\rm where} \quad y = x_1x_2,$$ or $$ (x_1^3 + a_4x_1 + a_6)(y^\prime)^2 = (x_2^3 + a_4x_2 + a_6),\quad {\rm where} \quad y^\prime = x_2/x_1. $$ Katsura and Schütt proved in articles "Generalized Kummer surfaces and their unirationality in characteristic p (1987)" and "Zariski K3 surfaces (2017)" that the surface $K$ is a Zariski surface, i.e., there is a purely inseparable covering $\mathbb{A}^2 \to K$ over $\overline{\mathbb{F}_p}$, where $p \not\equiv 1$ (mod $12$). Is it still true over $\mathbb{F}_{p^2}$ at least for some supersingular elliptic curve and some $p$?

A positive answer to this question may have value for cryptography :)

Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $K = \mathrm{Kum}(A)$ for the superspecial abelian surface $A = E_1\times E_2$. Katsura and Schütt proved in the articles "Generalized Kummer surfaces and their unirationality in characteristic p (1987)" and "Zariski K3 surfaces (2017)" that the surface $K$ is a Zariski surface, i.e., there is a purely inseparable covering $\mathbb{A}^2 \to K$ over $\overline{\mathbb{F}_p}$, where $p \not\equiv 1$ (mod $12$). Is it still true over $\mathbb{F}_{p^2}$ at least for some supersingular elliptic curve and some $p$?

A positive answer to this question may have value for cryptography :) If $K$ is a Zariski surface over $\mathbb{F}_{p^2}$, then we have a method to compress a pair $(P_1, P_2) \in A(\mathbb{F}_{p^2})$ by computation of a map from $A$ to the affine plane $\mathbb{A}^2$ (through $K$) of separable degree 2. This is very compact and efficient method, because for decompression we need to solve only one quadratic equation. Computation of a preimage for a purely inseparable map is very fast. People usually take two projections on x-coordinates for $P_1$ and $P_2$ independently, hence they should solve two quadratic equations. The state of the art for this part of cryptography is represented, for example, in the article https://eprint.iacr.org/2017/1143.

Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite prime field $\mathbb{F}_p$, $p > 3$. Consider the Kummer surface $K = Kum(E_1\times E_2)$ in one of the two forms: $$y^2 = (x_1^3 + a_4x_1 + a_6)(x_2^3 + a_4x_2 + a_6),\quad {\rm where} \quad y = x_1x_2,$$ or $$ (x_1^3 + a_4x_1 + a_6)(y^\prime)^2 = (x_2^3 + a_4x_2 + a_6),\quad {\rm where} \quad y^\prime = x_2/x_1. $$ Katsura and Schütt proved in articles "Generalized Kummer surfaces and their unirationality in characteristic p (1987)" and "Zariski K3 surfaces (2017)" that the surface $K$ is a Zariski surface, i.e., there is a purely inseparable covering $\mathbb{A}^2 \to K$ over $\overline{\mathbb{F}_p}$, where $p \not\equiv 1$ (mod $12$). Is it still true over $\mathbb{F}_p$ at least for some supersingular elliptic curve and some $p$?

A positive answer to this question may have value for cryptography :)

Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $K = \mathrm{Kum}(E_1\times E_2)$ in one of the two forms: $$y^2 = (x_1^3 + a_4x_1 + a_6)(x_2^3 + a_4x_2 + a_6),\quad {\rm where} \quad y = x_1x_2,$$ or $$ (x_1^3 + a_4x_1 + a_6)(y^\prime)^2 = (x_2^3 + a_4x_2 + a_6),\quad {\rm where} \quad y^\prime = x_2/x_1. $$ Katsura and Schütt proved in articles "Generalized Kummer surfaces and their unirationality in characteristic p (1987)" and "Zariski K3 surfaces (2017)" that the surface $K$ is a Zariski surface, i.e., there is a purely inseparable covering $\mathbb{A}^2 \to K$ over $\overline{\mathbb{F}_p}$, where $p \not\equiv 1$ (mod $12$). Is it still true over $\mathbb{F}_{p^2}$ at least for some supersingular elliptic curve and some $p$?

A positive answer to this question may have value for cryptography :)