A General Framework for p–adic Point Counting and Application to Elliptic Curves on Legendre Form

In 2000 T. Satoh gave the first p–adic point counting algorithm for elliptic curves over finite fields. Satoh’s algorithm was followed by the SST algorithm and furthermore by the AGM and MSST algorithms for characteristic two only. All four algorithms are important to Elliptic Curve Cryptography. In this paper we present a general framework for p–adic point counting and we apply it to elliptic curves on Legendre form. We show how the λ–modular polynomial can be used for lifting the curve and Frobenius isogeny to characteristic zero and we show how the associated multiplier gives the action of the lifted Frobenius isogeny on the invariant differential. The result is a point counting algorithm which is simpler and more practical than known algorithms for general elliptic curves. The algorithm extends the MSST algorithm to odd characteristics.