A p-adic Height Function Of Cryptanalytic Significance

It is noted that an efficient algorithm for calculating a p-adic height could have cryptanalytic applications. Elliptic curves and their generalizations are an active research topic with practical applications in cryptography [1], [2], [3]. If E is an elliptic curve defined over a finite field Fp, where p is prime, and if P and Q are points on the curve E such that Q = nP , then the elliptic curve discrete logarithm problem (ECDLP) is to find n given P and Q on E. The essential difficulty of the ECDLP arises from the curves being defined over finite fields and the addition law having a complicated nonlinear structure. Philosophically several papers have been written where efficient algorithms have been developed by taking advantage of the fact that finite fields sit very comfortably inside the p-adic fields. More precisely, objects defined over finite fields can be lifted to objects over p-adic fields using a simple or sophisticated versions of the Hensel’s lemma [4], [5]. Conversely, a p-adic object can be truncated to give an object in a finite field. Efficient algorithms using p-adic techniques include the attack on anomalous curves [6], [7] and [8], p-adic point counting [9], [10], constructing elliptic curves with the required number of points [11] etc. In this short note, we wish to bring to the attention of the cryptographic community some recent developments in the p-adic height function [12], [13], [14] which could be of cryptanalytic interest.