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

1.2 Polynomial and Superpolynomial Time Bounds 5
resentation of n, which differs from log n by at most 1. Since (log n)/(log10 n)
is the constant (ln 10)/(ln 2) ≈ 3.322 ≈ 10
3 ,wehave�n�10 ≈ 3
10 log n. For
example, a number with 80 binary digits has about 24 decimal digits.) Similarly,
as an elementary operation we view the addition or the multiplication
of two bits. A rough estimate on the basis of the naive methods shows that
certainly c · ℓ bit operations are sufficient to add, subtract, or compare two
ℓ-bit numbers; for multiplication and division we are on the safe side if we
assume an upper bound of c · ℓ2 bit operations, for some constant c. Assume
now an algorithm A is given that performs TA(n) elementary operations on
input n. We consider possible bounds on TA(n) expressed as fi(log n), for
some functions fi : N → R; see Table 1.1. The table lists the bounds we
get for numbers with about 60, 150, and 450 decimal digits, and it gives the
binary length of numbers we can treat within 1012 and 1020 computational
steps.
i fi(x) fi(200) fi(500) fi(1500) si(10 12 ) si(10 20 )
1 c · x 200c 500c 1,500c 10 12 /c 10 20 2 c · x
/c
2
40,000c 250,000c 2.2c · 10 6
10 6 / √ c 10 10 / √ 3 c · x
c
3
8c · 10 6
1.25c · 10 8
3.4c · 10 9
10 4 / 3√ c 4.6 · 10 6 / 3√ 4 c · x
c
4
1.6c · 10 9
6.2c · 10 10
5.1c · 10 12
1,000/ 4√ c 100,000/ 4√ 5 c · x
c
6
6.4c · 10 13
1.6c · 10 16
1.1c · 10 19
100/ 6√ c 2150/ 6√ 6 c · x
c
9
5.1c · 10 20
2.0c · 10 24
3.8c · 10 28
22/ 9√ c 165/ 9√ c
7 x 2lnlnx
4.7 · 10 7
7.3 · 10 9
4.3 · 10 12
8 c · 2
1,170 22,000
√ x
18,000c 5.4c · 10 6
4.55c · 10 11
1,600 4,400
9 c · 2 x/2
1.3c · 10 30
1.6c · 10 60
2.6c · 10 120
80 132
Table 1.1. Growth functions for operation bounds. fi(200), fi(500), fi(1500) denote
the bounds obtained for 200-, 500-, and 1500-bit numbers; si(10 12 )andsi(10 20 )
are the maximal numbers of binary digits admissible so that an operation bound
of 10 12 resp. 10 20 is guaranteed
We may interpret the figures in this table in a variety of ways. Let us
(very optimistically) assume that we run our algorithm on a computer or
a computer network that carries out 1,000 bit operations in a nanosecond.
Then 10 12 steps take about 1 second (feasible), and 10 20 steps take a little
more than 3 years (usually unfeasible). Considering the rows for f1 and f2
we note that algorithms that take only a linear or quadratic number of operations
can be run for extremely large numbers within a reasonable time. If
the bounds are cubic (as for Algorithm 1.2.1, f3), numbers with thousands
of digits pose no particular problem; for polynomials of degree 4 (f4), we
begin to see a limit: numbers with 30,000 decimal digits are definitely out
of reach. Polynomial operation bounds with larger exponents (f5 or f6) lead