On explicit constructions for solving the problem “A secret sharing”

“A secret sharing” problem was suggested to the participants of the second round competition in cryptography NSUCRYPTO-2015. The problem is to construct a subset $M\subset\mathbb F_2^n$ satisfying the following conditions: 1) any $u\in M$ can be represented as $u=x\oplus y\oplus z$, where $x,y,z$ are different elements of $\overline M=\mathbb F_2^n\setminus M$; 2) $x\oplus y\oplus z\in M$ for all different $x,y,z\in\overline M$. The paper presents some approaches to solving this problem. In particular, for even $n$, an explicit construction of the required set $M$ on the basis of a cubic parabola is proposed.