Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 354 — #362
354
Chapter 9
Number Theory
Problem 9.62.
Definition. Define the order of k over Z n to be
ord.k; n/ WWD minfm > 0 j k m D 1 .Z n /g:
If no positive power of k equals 1 in Z n , then ord.k; n/ WWD 1.
(a) Show that k 2 Z n iff k has finite order in Z n.
(b) Prove that for every k 2 Z n , the order of k over Z n divides .n/.
Hint: Let m D ord.k; n/. Consider the quotient and remainder of .n/ divided by
m.
Problem 9.63.
The general version of the Chinese Remainder Theorem (see Problem 9.58) extends
to more than two relatively prime moduli. Namely,
Theorem (General Chinese Remainder). Suppose a 1 ; : : : ; a k are integers greater
than 1 and each is relatively prime to the others. Let n WWD a 1 a 2 a k . Then for
any integers m 1 ; m 2 ; : : : ; m k , there is a unique x 2 Œ0::n/ such that
for 1 i k.
x m i .mod a i /;
The proof is a routine induction on k using a fact that follows immediately from
unique factorization: if a number is relatively prime to some other numbers, then it
is relatively prime to their product.
The General Chinese Remainder Theorem is the basis for an efficient approach
to performing a long series of additions and multiplications on “large” numbers.
Namely, suppose n was large, but each of the factors a i was small enough to be
handled by cheap and available arithmetic hardware units. Suppose a calculation
requiring many additions and multiplications needs to be performed. To do a single
multiplication or addition of two large numbers x and y in the usual way in
this setting would involve breaking up the x and y into pieces small enough to be
handled by the arithmetic units, using the arithmetic units to perform additions and
multiplications on (many) pairs of small pieces, and then reassembling the pieces
into an answer. Moreover, the order in which these operations on pieces can be
performed is contrained by dependence among the pieces—because of “carries,”