Accelerating the CM method

Abstract

Given a prime q and a negative discriminant D, the CM method constructs an elliptic curve E/ F _q by obtaining a root of the Hilbert class polynomial H _D ( X) modulo q. We consider an approach based on a decomposition of the ring class field defined by H _D , which we adapt to a CRT setting. This yields two algorithms, each of which obtains a root of H _D mod q without necessarily computing any of its coefficients. Heuristically, our approach uses asymptotically less time and space than the standard CM method for almost all D. Under the GRH, and reasonable assumptions about the size of log q relative to ∣ D∣, we achieve a space complexity of O(( m+ n)log q) bits, where mn= h( D) , which may be as small as O(∣ D∣ ^1/4 log q) . The practical efficiency of the algorithms is demonstrated using ∣ D∣>10 ¹⁶ and q≈2 ²⁵⁶, and also ∣ D∣>10 ¹⁵ and q≈2 ³³²²⁰. These examples are both an order of magnitude larger than the best previous results obtained with the CM method.