Prime Numbers

508 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
Algorithm optimal B complexity
Basic FFT, fixed-base ... Oop(D ln D)
Basic FFT, variable-base O(ln n) O(n(C ln n)(C ln ln n) ...)
Schönhage O(n 1/2 ) O(n ln n ln ln n)
Nussbaumer O(n/ ln n) O(n ln n ln ln n)
Table 9.1 Complexities for fast multiplication algorithms. Operands to be
multiplied have n bits each, which during top recursion level are split into D = n/b
digits of b bits each, so the digit size (the base) is B =2 b . All bounds are for bit
complexity, except that Oop means operation complexity.
9.5.9 Application to the Chinese remainder theorem
We described the Chinese remainder theorem in Section 2.1.3, and there
gave a method, Algorithm 2.1.7, for reassembling CRT data given some
precomputation. We now describe a method that not only takes advantage
of preconditioning, but also fast multiplication methods.
Algorithm 9.5.26 (Fast CRT reconstruction with preconditioning).
Using the nomenclature of Theorem 2.1.6, we assume fixed moduli m0,...,mr−1
whose product is M, but with r =2 k for computational convenience. The goal
of the algorithm is to reconstruct n from its given residues (ni). Along the way,
tableaux (qij) of partial products and (nij) of partial residues are calculated. The
algorithm may be reentered with a new n if the mi remain fixed.
1. [Precomputation]
for(0 ≤ i