17 Tate pairing defined
This chapter covers
- A mathematical description of Tate pairing
- An implementation of Tate pairing
- A test with a tiny example to see how Tate pairings work
In this chapter, the Tate pairing is described. This will be used with the example in chapter 19. The Tate pairing has the properties of bilinearity and nondegeneracy but not the alternating property that the Weil pairing of chapter 16 possesses. I will first go over the mathematical description of the Tate pairing. In chapter 16, pairing test code utility routines were described, so this chapter will just show how the Tate pairing is computed with one listing. The tiny example introduced in chapter 13 will again be used to show how the number of points available for the Tate pairing is enlarged over the Weil pairing. Explaining the math with words is great, but seeing an example should help to get an intuitive feel for what the mathematics actually mean. Using the same tiny example, we will explore the Tate pairing in detail.
17.1 Tate pairing mathematics
In this section, the mathematics of the Tate pairing is explained. Similar to the general pairing description of a curve having two cyclic groups, the Tate pairing is described in section XI.9 in Silverman (2009) with the mathematical statement shown in equation 17.1: