Miller-rabin algorithm Codevs 1702 prime number Determination 2

Reprinted from: http://www.dxmtb.com/blog/miller-rabbin/

We have an O (√n) test algorithm for ordinary prime numbers. In fact, we have an O (slog³n) algorithm.

Theorem One: If P is a prime number and (a,p) = 1, then a^ (p-1) ≡1 (mod p). That is, if p is a prime number and a,p coprime, then the remainder of a (p-1) divided by P is constant equal to 1. (Fermat theorem)

The inverse proposition of the theorem is not necessarily tenable, but it is gratifying that most cases are established.

So we get a direct application of the theorem, for the number of p to be verified, we continue to take a∈[1,p-1] and a∈z, verify that a^ (p-1) mod p is equal to 1, not p decisive is not prime, a total of s times. where a^ (p-1) mod p can be written by the p-1 binary, by (a*b) mod c= (a mod c) *b mod c, can be calculated in T=log (p-1) time, such as the complexity of integer multiplication, the total complexity of a calculation is log³ (p-1). This method is called fast power modulo.

In order to improve the accuracy of the algorithm, we have another theorem that can be used.
Theorem two: for 0<x<p,x^2 mod p =1 = x=1 or p-1.

We make p-1= (2^t) *u, that is, the p-1 is the U binary representation followed by the T 0. We first calculate the X[0]=a^u mod p, then the square T times and at each time modulo p, each time the results are recorded as X[i], and finally can be calculated a^ (p-1) mod p. If found X[i]=1 and x[i-1] is not equal to 1 is not equal to p-1, then found that P decisive is not prime.

It can be proved that after using the above two theorems, the probability of testing s error is at most 2^ (-s), so the algorithm is very reliable.

It is important to note that in order to prevent overflow (particularly large data), A*b mod C also uses a method similar to fast power modulo. Of course, the data is not very large can be waived

Codevs 1702 Prime number Determination 2time limit: 1 sspace limit: 128000 KBtitle level: Diamonds DiamondTitle Description Description

A number, is he a prime?

Set him up for P satisfaction (p<=2^63-1)

Enter a description Input Description

P

Output description Output Description

yes| No

Sample input Sample Input

2

Sample output Sample Output

Yes

Data range and Tips Data Size & Hint

An introduction to Algorithms--the section of number theory
Note Carmichael number

Category labels Tags Click here to expandNumber theory analysis: Because the data range is 2^63-1 large, the general method time complexity is very high, so using the stochastic algorithm--miller Rabin algorithm

1 /*The only thing I want to say is that the data on this topic is wrong.*/2 #defineS 153#include <iostream>4 using namespacestd;5#include <cstdio>6 #definell Long Long7#include <cstdlib>8#include <ctime>9 ll Quick_mod (ll x,ll u,ll N)Ten { Onex%=N; All ans=1; -      while(U) -     { the         if(u&1) -         { -u--; -ans= (ans*x)%N; +         } -u>>=1; +x= (x*x)%N; A     } at     returnans; - } - BOOLMiller_rabin (ll N) - { -     if(n==2)return true; -     if(n==1||! (n&1))return false; inll u=n-1; -ll t=0; to      while(! (u&1)) +     { -t++; theu>>=1; *     } $      for(intI=0; i<s;++i)Panax Notoginseng{/*Here you can randomly find x, or you can hand-hit a table, int ss[7]={2,3,5,7,11,13,17}; it turns out that the table is correct within a long long range .*/ -ll X=rand ()% (n1)+1; thex=Quick_mod (x,u,n); +          for(intI=1; i<=t;++i) A         { thell Y=quick_mod (x,2, n); +             if(y==1&&x!=1&&x!=n-1) -               return false; $x=y; $         } -         if(x!=1) -           return false; the     } -     return true;Wuyi } the intMain () - { WuSrand (Time (0)); - ll N; AboutCin>>N; $     if(Miller_rabin (n)) -printf"Yes"); -     Elseprintf"No"); -     return 0; A}

Miller-rabin algorithm Codevs 1702 prime number Determination 2