Let $k<N$ and $P_1, ..., P_{2k+1}, \lambda_1, ..., \lambda_k$ be elements of a finite group $G$ of size $N$.

Find the minimum number of solution of the system

$$P_{2i} + P_{2i+1} = \lambda_i, \forall i\leq k$$

with the condition that the $P_1, ..., P_{2k+1}$ are pairwise distinct.

The law "+" on the $n-$bits strings is the "xor" ie we consider the n-bit strings as element of $(\mathbb{Z}/2\mathbb{Z})^n$.