You have a bit vector $\langle a,b \rangle$, and I have another bit vector $\langle c,d \rangle$. We want to compute the scalar product of the two. Typically, we would need at least two multiplications and one addition ($\ell$ multiplications and $\ell-1$ additions for bit vectors of length $\ell$).
Using a certain algebraic property of some algebraic objects, we can do it using two additions and one multiplication ($2(\ell-1)$ additions and one multiplication).
I have got correctly working examples for vectors up to 6 items till now. I am working on increasing that number and getting to know the upper limit or any other limitations. So far it works for vectors over $\mathbb{R}$ not just bit vectors.