1811 Solution to POJ1811

Pseudocode of RMSPT (Sprache)
function powermod(a, s, n) { p := 1 b := a while s > 0 { if s & 1 == 1 then p := p * b % n b := b * b % n s := s >> 1 } }
Key Concepts (cont.)
Prime factorization algorithms: Algorithms devised for determining the prime factors of a given number Primality tests: Tests to determine whether or not a given number is prime, as opposed to actually decomposing the number into its constituent prime factors
Primality Tests
Deterministic: Adleman-PomeranceRumely Primality Test, Elliptic Curve Primality Proving… Probabilistic: Rabin-Miller Strong Pseudoprime Test…
Determine whether the given N is prime or not. If N is prime, print “Prime” and exit. Factorize N for its smallest prime factor.
The Brute-force Way Trial Division
Modified Brute-force
Construct a table that stores all prime numbers not greater than N1/2. Try dividing N only by prime numbers. Complexity: O(N1/2logN) for time, O(N1/2) for space using Sieve of Eratosthenes Estimation of space consumption: 226 bits = 223 bytes = 8,192 kilobytes Much time is used in the process of sieving
Brute-force with Trick
Start from N1/2 rather than 2. Do factorization recursively once a factor is found. Efficient in handling N = pq where p and q relatively close to each other. POJ accepts this! - westever’s solution
Rabin-Miller Strong Pseudoprime Test
Requires no more than (1 + o(1))logN multiplications (mod N). A number which passes the test is not necessarily prime. But a composite number passes the test for at most 1/4 of the possible bases a. If n multiple independent tests are performed on a composite number, the probability that it passes each test is 1/4n or less.
Key Concepts
Prime numbers: A prime number is a positive integer p > 1 that has no positive integer divisors other than 1 and p itself.
Framework of the Solution
Pseudocode of RMSPT (cont.)
function rabin-miller(n) { if n > 2 AND powermod(2, n - 1, n) != 1 then return FALSE if n > 3 AND powermod(3, n - 1, n) != 1 then return FALSE if n > 5 AND powermod(5, n - 1, n) != 1 then return FALSE if n > 7 AND powermod(7, n - 1, n) != 1 then return FALSE if n > 11 AND powermod(11, n - 1, n) != 1 then return FALSE if n > 13 AND powermod(13, n - 1, n) != 1 then return FALSE if n > 17 AND powermod(17, n - 1, n) != 1 then return FALSE if n > 19 AND powermod(19, n - 1, n) != 1 then return FALSE if n > 23 AND powermod(23, n - 1, n) != 1 then return FALSE return Tong Pseudoprime Test
Given an odd integer N, let N = 2rs + 1 with s odd. Then choose a random integer a between 1 and N - 1. If as = 1 (mod N) or a2^j s = -1 (mod N) for some j between 0 and r - 1, then N passes the test. A prime will pass the test for all a.
Rabin-Miller Strong Pseudoprime Test
Smallest composite numbers passing the RMSPT using the first k primes as bases: 2,047; 1,373,653; 25,326,001; 3,215,031,751; 2,152,302,898,747; 3,474,749,660,383; 341,550,071,728,321, 341,550,071,728,321; at most 41,234,316,135,705,689,041… 341,550,071,728,321 = 244.957…, 41,234,316,135,705,689,041 = 265.160… Tests show that randomized bases may fail sometimes.
If N is even, then 2 is its smallest prime factor. Try dividing N by every odd number k between 2 and N1/2. The smallest k by which N is divisible is the smallest prime factor of N. If such k does not exist, then N is prime. Complexity: O(N1/2) for time, O(1) for space
Prime Factorization Algorithms
Continued Fraction Algorithm, Lenstra Elliptic Curve Method, Number Field Sieve, Pollard Rho Method, Quadratic Sieve, Trial Division…
A Solution to POJ1811 Prime Test
STYC
Problem Description
Given an integer N which satisfies the relation 2 < N < 254, determine whether or not it is a prime number. If N is not a prime number, find out its smallest prime factor.
Pollard Rho Factorization Method
Iteration: Iterate the formula xn+1 = xn2 + a (mod N). Almost any polynomial formula (two exceptions being xn2 and xn2 - 2) for any initial value x0 will produce a sequence of numbers that eventually falls into a cycle.
Pollard Rho Factorization Method
Also known as Pollard Monte Carlo factorization method. Runs at O(p1/2) where p is the largest prime factor of the number to be factored. Two aspects to this method: iteration and cycle detection.