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RSA Encryption asymptotic analysis

Now demonstrate an example of the review exercise 2: asymptotic analysis. This was how I answered the question 2 previous examination.

2. Find an asymptotic complexity function f (n) so the function g (n) defined experimentally by twelve points in the table below is $ \\ Omega f (n) $.

Justify this with a graph and discusses the quality of the elevation gained.






1.038

n	g (n)	n	g(n)	n	g(n)
3	15	130	3,800	3,982	190,000
12	170	203	7,500	5,002	240,000
30	600	603	22,000	10,393	550,000
57	1,500		42.000	40.039	2,600,000

First notice that the first 4 data, f (n) may be n ² because 3 ² approaches 15, 12 ² approaching 170, 30 ² approaching 600 and 57 ^{2 1.500} approaches but in calculating the past, no: 3.982

2 = 5.002
^{15,856,324 2 25,020,004}
= 10.393 ^{2 = 108,014,449
40.039 2 = 1,603,121,251}

As asymptotic complexity is more important to large bodies ² n grows too fast, we'll try to function but was obviously less than n.

^{3.982 1.5 5.002} = 251,276.5054
= 353,765.5438
1.5 10.393 1.5
= 1,059,525.4449 40.039 1.5 = 8,011,702.8514

too, now with power 1.4:

^{3.982 1.4 5.002} = 109,683.6155
^1.4 = 150,938.8936
^{10.393 1.4 40.039} = 420,181.8590
^1.4

= 2,776,364.6813 Enough for me, g (n) = n ^1.4 now seek a lower level.

know that a lower bound g (n) can be f (n) = log n, but not so close an 1.4, a more serious immediate ⁿ 1.35, so we'll take it and compare:

3.982 1.35
= 72,466.4013 5.002 98,592.5224 ^{= 1.35
10.393 1.35 40.039}
= 264,606.3437 1.35 = 1,634,377.4061

data are lower than 1.4 n, then f (n) = n 1.35 be our lower bound or $ \\ Omega f (n) $.

see the graph:

As you can see the red line represents our function g (n) approximately, the green lower bound f (n) and the blue the experimental data in the table. The lower bound is quite good because we did not go to log n to make it less than the linear n ^{1.4 1.35 but an} which is just below g (n).


Finally I wish to comment on how do this post because someone could serve in the future. The powers are like that and n, is made with a tool that is already included in the blogs is called superscript and subscript and become writing \u0026lt;sup> \u0026lt;/ sup> and \u0026lt;sub> \u0026lt;/ sub> respectively, while symbols such as $ \\ Omega $ are adding to the gadget blog (java code) to recognize LaTeX writing, but I already explained in this post .

other hand the graph I made with a program called gnuplot download and free use, when downloading a document I include tutorial or guide to learn and use. It is a program based on command line but is very powerful.