Polynomials for Ate Pairing and Atei Pairing

The irreducible factor r(x) of Φk(u(x)) and u(x) are often used in constructing pairing-friendly curves. u(x) and uc ≡ u(x) (mod r(x)) are selected to be the Miller loop control polynomial in Ate pairing and Atei pairing. In this paper we show that when 4|k or the minimal prime which divides k is larger than 2, some u(x) and r(x) can not be used as curve generation parameters if we want Atei pairing to be efficient. We also show that the Miller loop length can not reach the bound log2r φ(k) when we use the factorization of Φk(u(x)) to generate elliptic curves.