Let us — using parameters $ M, N$ and $ L$ —

create an ordered set of size $ M$ of $ N$ bit long vectors $ V$ and initialize them randomly: $ V_k[i] = b \sim Bin(n=1, p=0.5)\ \forall i \in \{0\ ..\ N1\}, \forall k \in \{0\ ..\ M1\}$ .

create an Nbit long vector $ A_0$ and initialize it to ones.

create an $ M$ bit long vector $ S$ and initialize it randomly in the same manner as any $ V_k$ .

for $ i$ from $ 0$ to $ M$ , if $ S[i] = 1$ , then $ A_i = A_{i1} \oplus V_k$ , otherwise $ A_i = A_{i1}$ , resulting in $ A_M$ after $ M$ steps

present an agent/player with $ A_M$ together with all vectors $ V_k$ and make him return $ S$ or the set of indices of vector $ S$ where $ S[i] = 1$ . It follows that $ A_M \oplus V_{i_0} \oplus V_{i_1} \oplus V_{i_2} \oplus\ …\ \oplus V_{i_{last}} = A_0$ .
A simple example with $ N=4$ and $ M=3$ :
$ A_M = [0, 1, 0, 1]$ , $ V_0 = [1, 1, 0, 1]$ , $ V_1 = [0, 0, 1, 1]$ , $ V_2 = [0, 1, 1, 1]$
solution $ \rightarrow S = [1, 0, 1],$ because $ A_M \oplus V_0 \oplus V_2 = [1, 1, 1, 1]$
This problem occurs in many games I have encountered and never really gave it much thought, until now. For the purpose of this question, I have called it the “binary toggling game”.
What I wonder is:
 what are “binary toggling games” really called
 what is the theory being them: algorithms, their complexity (classes), edge cases, etc.
Could you please provide links? I hope they exist.