Computing Hilbert class polynomials with the Chinese remainder theorem

Andrew V. Sutherland
2010 Mathematics of Computation  
We present a space-efficient algorithm to compute the Hilbert class polynomial H_D(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D|^(1/2+o(1))log P) space and has an expected running time of O(|D|^(1+o(1)). We describe practical optimizations that allow us to handle larger discriminants than other methods, with |D| as large as 10^13 and h(D) up to 10^6. We apply these results to
more » ... t pairing-friendly elliptic curves of prime order, using the CM method.
doi:10.1090/s0025-5718-2010-02373-7 fatcat:bthdblwvvzgofhyrigrz46oowm