We propose an improved implementation of modified Weil pairings. By reduction of operations in the extension field to those in the base field, we can save some operations in the extension field when computing a modified Weil pairing. In particular, computing e`(P, φ(P )) is the same as computing the Tate pairing without the final powering. So we can save about 50% of time for computing e`(P, φ(...

1. What is an elliptic curve? 2 2. Mordell-Weil Groups 5 2.1. The Group Law on a Smooth, Plane Cubic Curve 5 2.2. Reminders on Commutative Groups 8 2.3. Some Elementary Results on Mordell-Weil Groups 9 2.4. The Mordell-Weil Theorem 11 2.5. K-Analytic Lie Groups 13 3. Background on Algebraic Varieties 15 3.1. Affine Varieties 15 3.2. Projective Varieties 18 3.3. Homogeneous Nullstellensätze 20 3...

In this paper, we will implement a standard probabilistic method of computing bilinear pairings. We will compare its performance to a deterministic algorithm introduced in [5] to compute the squared Tate/Weil pairings which are claimed to be 20 percent faster than the standard method. All pairings will be evaluated over pairing-friendly ordinary elliptic curves of embedding degrees 8 and 10 and...