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

6 1. Introduction: Efficient Primality Testing
to situations where the length of the numbers that can be treated is already
severely restricted — with (log n) 9 operations we may deal with one 7-digit
number in 1 second; treating a single 50-digit number takes years. Bounds
f7(log n) =(logn) 2lnlnn , f8(log n) =c · 2 √ log n ,andf9(log n) =c √ n exceed
any polynomial in log n for sufficiently large n. For numbers with small binary
length log n, however, some of these superpolynomial bounds may still be
smaller than high-degree polynomial bounds, as the comparison between f6,
f7, andf8 shows. In particular, note that for log n = 180,000 (corresponding
to a 60,000-digit number) we have 2 ln ln(log n) < 5, so f7(log n) < (log n) 5 .
The bound f9(log n) =c √ n, which belongs to the trial division method,
is extremely bad; only very short inputs can be treated.
Summing up, we see that algorithms with a polynomial bound with a truly
small exponent are useful even for larger numbers. Algorithms with polynomial
time bounds with larger exponents may become impossible to carry out
even for moderately large numbers. If the time bound is superpolynomial,
treating really large inputs is usually out of the question. From a theoretical
perspective, it has turned out to be useful to draw a line between computational
problems that admit algorithms with a polynomial operation bound
and problems that do not have such algorithms, since for large enough n,
every polynomial bound will be smaller than every superpolynomial bound.
This is why the class P, to be discussed next, is of such prominent importance
in computational complexity theory.
1.3 Is PRIMES in P?
In order to formulate what exactly the question “Is PRIMES in P?” means,
we must sketch some concepts from computational complexity theory. Traditionally,
the objects of study of complexity theory are “languages” and
“functions”. A nonempty finite set Σ is regarded as an alphabet, andone
considers the set Σ ∗ of all finite sequences or words over Σ.Themostimportant
alphabet is the binary alphabet {0, 1}, whereΣ ∗ comprises the words
ε (the empty word), 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101,... .
Note that natural numbers can be represented as binary words, e.g., by means
of the binary representation: bin(n) denotes the binary representation of n.
Now decision problems for numbers can be expressed as sets of words over
{0, 1}, e.g.
SQUARE = {bin(n) | n ≥ 0isasquare}
= {0, 1, 100, 1001, 10000, 11001, 100100, 110001, 1000000,...}
codes the problem “Given n, decide whether n is a square of some number”,
while