Calculating the Tate local pairing for any odd prime number

TL;DR

This paper extends the explicit computation of the Tate local pairing from the 3-torsion case to any odd prime p, providing a general formula for elliptic curves.

Contribution

It offers a new explicit formula for the Tate local pairing for any odd prime p, generalizing previous work limited to p=3.

Findings

Derived an explicit formula for the Tate local pairing for any odd prime p

Extended previous results from p=3 to all odd primes

Simplified calculations of Tate pairings in elliptic curve cryptography

Abstract

Fisher and Newton have given an explicit description of the Tate local pairing associated with the 3-torsion of an elliptic curve. The present paper summarizes the work from the author's master's thesis and gives an explicit formula for any odd prime $p$, thereby extending the work of Fisher and Newton.