[Resource Topic] 2008/202: Polynomials for Ate Pairing and $\mathbf{Ate}_{i}$ Pairing

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Polynomials for Ate Pairing and \mathbf{Ate}_{i} Pairing

Authors: Zhitu Su, Hui Li, JianFeng Ma


The irreducible factor r(x) of \mathrm{\Phi}_{k}(u(x)) and u(x) are often used in constructing pairing-friendly curves. u(x) and u_{c} \equiv u(x)^{c} \pmod{r(x)} are selected to be the Miller loop control polynomial in Ate pairing and \mathrm{Ate}_{i} 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 \mathrm{Ate}_{i} pairing to be efficient. We also show that the Miller loop length can not reach the bound \frac{\mathrm{log_{2}r}}{\varphi(k)} when we use the factorization of \mathrm{\Phi}_{k}(u(x)) to generate elliptic curves.

ePrint: https://eprint.iacr.org/2008/202

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