Unified and complete point addition formula for elliptic curves

As is well known, the unified point addition formula is useful for resisting side channel attacks in elliptic curve cryptography. Furthermore, if the unified formula is complete, which means it is valid for any two points, then there are no exceptional cases to be described particularly. This feature is especially preferable for elegant codes of elliptic crypto-algorithms. Therefore, the unified and also complete point addition formula can provide the advantage for good security and convenient implementation. In this paper, we exploit sufficient and necessary condition for the existence of unified and complete point addition formula on several known elliptic curve models such as Weierstrass cu-bics, Jacobi quartics, Edwards curves, etc.. Moreover, we study another form of elliptic curves called Selmer curves. For practical application, we finally give some numerical examples of cryptographic secure Selmer curves.