Dietzfelbinger M. Primality testing in polynomial time ... - tiera.ru

32 3. Fundamentals from Number Theory
and
bi = ai−1−qibi−1 = nxa,i−1+mya,i−1−qi(nxb,i−1+myb,i−1) =nxb,i+myb,i.
In particular, the coefficients xa,t and ya,t stored in xa and ya after the last
iteration of the loop satisfy gcd(n, m) =at = nxa,t + mya,t, as claimed. ⊓⊔
Concerning the running time of the Extended Euclidean Algorithm, we
note that the analysis in Lemma 3.2.3(a) carries over, so on input n, m
O(min{log(n), log(m)}) arithmetic operations are carried out. As for the cost
in terms of bit operations, we note without proof that the number of bit
operations is bounded by O(log(n) log(m)) just as in the case of the simple
Euclidean Algorithm.
3.3 Modular Arithmetic
We now turn to a different view on remainders: modular arithmetic. Let
m ≥ 2begiven(the“modulus”). We want to say that looking at an integer
a we are not really interested in a but only in the remainder a mod m. Thus,
all numbers that leave the same remainder when divided by m are considered
“similar”. We define a binary relation on Z.
Definition 3.3.1. Let m ≥ 2 be given. For arbitrary integers a and b we say
that a is congruent to b modulo m and write
a ≡ b (mod m)
if a mod m = b mod m.
The definition immediately implies the following properties of the binary
relation “congruence modulo m”.
Lemma 3.3.2. Congruence modulo m is an equivalence relation, i.e., we
have
Reflexivity: a ≡ a (mod m),
Symmetry: a ≡ b (mod m) implies b ≡ a (mod m), and
Transitivity: a ≡ b (mod m) and b ≡ c (mod m) implies a ≡ c (mod m). ⊓⊔
Further, it is almost immediate from the definitions of a mod m and of ≡
that
a ≡ b (mod m) if and only if m divides b − a. (3.3.5)
(Write a = mq + r and b = mq ′ + r ′ with 0 ≤ r, r ′