Prime Numbers

88 Chapter 2 NUMBER-THEORETICAL TOOLS
Indeed, 62 modulo 3, 5, 7, respectively, gives the required residues 2, 2, 6.
Though ancient, the CRT algorithm still finds many applications. Some
of these are discussed in Chapter 8.8 and its exercises. For the moment,
we observe that the CRT affords a certain “parallelism.” A set of separate
machines can perform arithmetic, each machine doing this with respect to
a small modulus mi, whence some final value may be reconstructed. For
example, if each of x, y has fewer than 100 digits, then a set of prime moduli
{mi} whose product is M > 10 200 can be used for multiplication: The i-th
machine would find ((x mod mi) ∗ (y mod mi)) mod mi, and the final value
x ∗ y would be found via the CRT. Likewise, on one computer chip, separate
multipliers can perform the small-modulus arithmetic.
All of this means that the reconstruction problem is paramount; indeed,
the reconstruction of n tends to be the difficult phase of CRT computations.
Note, however, that if the small moduli are fixed over many computations, a
certain amount of one-time precomputation is called for. In Theorem 2.1.6,
one may compute the Mi and the inverses vi just once, expecting many future
computations with different residue sets {ni}. In fact, one may precompute
the products viMi. A computer with r parallel nodes can then reconstruct
niviMi in O(ln r) steps.
There are other ways to organize the CRT data, such as building up one
partial modulus at a time. One such method is the Garner algorithm [Menezes
et al. 1997], which can also be done with preconditioning.
Algorithm 2.1.7 (CRT reconstruction with preconditioning (Garner)).
Using the nomenclature of Theorem 2.1.6, we assume r ≥ 2 fixed, pairwise
coprime moduli m0,...,mr−1 whose product is M, and a set of given residues
{ni (mod mi)}. This algorithm returns the unique n ∈ [0,M− 1] with the given
residues. After the precomputation step, the algorithm may be reentered for future
evaluations of such n (with the {mi} remaining fixed).
1. [Precomputation]
for(1 ≤ i