Reducing the XOR's in Implementation Cost
Yes. I have a solution with 5 $\oplus$.

Hint 1: That would not work with $*$ (ordinary multiplication, or even multiplication in $\Bbb Z^*_p$) where there is $\oplus$, because we must use a property of $\oplus$ that $*$ does not have.

Hint 2: The right path is greedy.

Hint 3: It's key to compute $y_3$ late, with just one extra step.