Computing Hilbert class polynomials with the Chinese remainder theorem

Computing Hilbert class polynomials with the Chinese remainder theorem

We present a space-efficient algorithm to compute the Hilbert class polynomial HD(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+ log P ) space and has an expected running time of O(|D|1+ ). We describe practical optimizations that allow us to handle larger discriminants than othe...

Computing Igusa Class Polynomials via the Chinese Remainder Theorem

We present a new method for computing the Igusa class polynomials of a primitive quartic CM field. For a primitive quartic CM field, K, we compute the Igusa class polynomials modulo p for certain small primes p and then use the Chinese remainder theorem and a bound on the denominators to construct the class polynomials. We also provide an algorithm for determining endomorphism rings of Jacobian...

Computing Hilbert Class Polynomials

We present and analyze two algorithms for computing the Hilbert class polynomial HD. The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over finite fields. Our run time analysis gives tighter bounds for the complexity of all known algorithms for comput...

The Chinese Remainder Theorem

Computing Fibonacci Numbers Fast using the Chinese Remainder Theorem

The purpose of this paper is to investigate the calculation of Fibonacci numbers using the Chinese Remainder Theorem (CRT). This paper begins by laying down some general conclusions that can be made about the Fibonacci sequence. It will then go into specific cases of the CRT and how to calculate Fibonacci numbers with reduced forms of the CRT equations. For each of the cases, algorithms and ana...