This repo contains various implementations of the evolution of pairings as explained in the book Pairings for Beginners by Craig Costello. The original scripts written in Magma can be found (with some minor fixes) in the magma_scripts folder. These scripts can be run for free at http://magma.maths.usyd.edu.au/calc/.

The content is the following:

• weil_tate_pairing_naive.sage implements Miller's algorithm for the Weil and the Tate pairings "as prescribed" (algorithm 5.1 in the book). That is, there are no optimizations over the initial definitions of both Weil and Tate pairings beyond the usage of Miller's double-and-add algorithm. The file divisors.sage contains the Divisors class useful for playing around with divisors.
• tate_pairing.sage contains the BKLS-GHS version of Miller's algorithm for the Tate pairing (algorithm 7.1 in the book). In this version, divisors are not needed anymore and the irrelevant factors technique has been used.
• ate_pairing.sage contains the BKLS-GHS version of Miller's algorithm for the ate pairing (algorithm 7.2 in the book). The ate pairing is the Tate pairing with a much shorter Miller's loop and domains reversed.

The file tools.sage contains various methods shared among all the previous files.

• Divisors.
• Weil pairing.
• Tate pairing.
• Ate pairing.
• Twists.
• Endomorphisms.
• Projective coordinates.
• Final exponentiation optimizations.

The following functionalities cannot be directly implemented in Sage (since we cannot extend finite fields of non-prime order), so I defer their implementation in Python to a later stage of this repo:

• Towered extension fields.
• Optimal pairings. In particular, the optimal ate pairing.