Binary Puzzle

Salmon is playing a game! He is given two integers $N$ and $K$. His goal is to output $K$ pairs $(x_i,y_i)$. He creates a function $f$ such that $f(x_i) = y_i$ and $f$ is undefined for all other values of $x$. His pairs must then meet the following conditions: - All $x_i$ are distinct. - All $y_i$ are distinct. - All $x_i, y_i$ are in the range $[0,2^N - 1]$ and are integers. - $f(f(x_i))