论文标题
椭圆曲线和嵌入性学位的最佳食品配对$ 9,15 $和$ 27 $
Optimal Ate Pairing on Elliptic Curves with Embedding Degree $9,15$ and $27$
论文作者
论文摘要
自从基于配对的密码学出现以来，对椭圆曲线上的配对有效计算的有效计算都非常关注。在奇数嵌入学位的情况下，现有的几项作品需要一些改进。本文考虑了在嵌入度的椭圆曲线上的最佳ATE配对计算，$ k = 9 $，$ 15 $，$ 27 $，有订单三的曲折。我们的主要目标是在相应的扩展字段的塔式扩展字段中提供详细的算术和成本估算。与在这些情况下存在的前几种作品相比，使用基于晶状体的方法的最终指数可以提高米勒步骤的理论成本和最终的启用。特别是，对于$ k = 15 $，$ k = 27 $，我们就基地的运营而获得了改进，在计算最终指数的计算中分别高达25％和29％。我们还发现，嵌入度$ k = 15 $的椭圆曲线比在128位安全级别的BN12曲线更快。在每种情况下，我们提供岩浆实现，以确保本工作中使用的公式的正确性。
Much attention has been given to the efficient computation of pairings on elliptic curves with even embedding degree since the advent of pairing-based cryptography. The few existing works in the case of odd embedding degrees require some improvements. This paper considers the computation of optimal ate pairings on elliptic curves of embedding degrees $k=9$, $15$, $27$ which have twists of order three. Our main goal is to provide a detailed arithmetic and cost estimation of operations in the tower extensions field of the corresponding extension fields. A good selection of parameters enables us to improve the theoretical cost for the Miller step and the final exponentiation using the lattice-based method as compared to the previous few works that exist in these cases. In particular, for $k=15$, $k=27$, we obtain an improvement, in terms of operations in the base field, of up to 25% and 29% respectively in the computation of the final exponentiation. We also find that elliptic curves with embedding degree $k=15$ present faster results than BN12 curves at the 128-bit security level. We provide a MAGMA implementation in each case to ensure the correctness of the formulas used in this work.
