Can we beat the square root bound for ECDLP over Fp2via representation?

Abstract

We give a 4-list algorithm for solving the Elliptic Curve Discrete Logarithm (ECDLP) over some quadratic field Fp2. Using the representation technique, we reduce ECDLP to a multivariate polynomial zero testing problem. Our solution of this problem using bivariate polynomial multi-evaluation yields a p1.314-algorithm for ECDLP. While this is inferior to Pollard's Rho algorithm with square root (in the field size) complexity O(p), it still has the potential to open a path to an o(p)-algorithm for ECDLP, since all involved lists are of size as small as p3/4, only their computation is yet too costly.