Prime Numbers

2.1 Modular arithmetic 87
and such operation counts depend on the binary expansion of the exponent n,
with typical operation counts being dramatically less than the value of n itself.
In fact, if x, n are integers the size of m, and we are to compute x n mod m
via naive multiply/add arithmetic and Algorithm 2.1.5, then O(ln 3 m)bit
operations suffice for the powering (see Exercise 2.17 and Section 9.3.1).
2.1.3 Chinese remainder theorem
The Chinese remainder theorem (CRT) is a clever, and very old, idea from
which one may infer an integer value on the basis of its residues modulo
an appropriate system of smaller moduli. The CRT was known to Sun-Zi in
the first century a.d. [Hardy and Wright 1979], [Ding et al. 1996]; in fact a
legendary ancient application is that of counting a troop of soldiers. If there
are n soldiers, and one has them line up in justified rows of 7 soldiers each,
one inspects the last row and infers n mod 7, while lining them up in rows of
11 will give n mod 11, and so on. If one does “enough” such small-modulus
operations, one can infer the exact value of n. In fact, one does not need the
small moduli to be primes; it is sufficient that the moduli be pairwise coprime.
Theorem 2.1.6 (Chinese remainder theorem (CRT)). Let m0,...,mr−1
be positive, pairwise coprime moduli with product M =Π r−1
i=0 mi. Letrre spective residues ni also be given. Then the system comprising the r relations
and inequality
n ≡ ni (mod mi), 0 ≤ n