Get the prime factorization of a number, \(n\), using the Quadratic Sieve.
quadraticSieve(n, showStats = FALSE, nThreads = NULL)Vector of class bigz
An integer, numeric, string value, or an element of class bigz.
Logical flag. If TRUE, summary statistics will be displayed.
Number of threads to be used. The default is NULL.
Joseph Wood
First, trial division is carried out to remove small prime numbers, then a constrained version of Pollard's rho algorithm is called to quickly remove further prime numbers. Next, we check to make sure that we are not passing a perfect power to the main quadratic sieve algorithm. After removing any perfect powers, we finally call the quadratic sieve with multiple polynomials in a recursive fashion until we have completely factored our number.
When showStats = TRUE, summary statistics will be shown. The frequency of updates is dynamic as writing to stdout can be expensive. It is determined by how fast smooth numbers (including partially smooth numbers) are found along with the total number of smooth numbers required in order to find a non-trivial factorization. The statistics are:
MPQS TimeThe time measured for the multiple polynomial quadratic sieve section in hours h, minutes m, seconds s, and milliseconds ms.
CompleteThe percent of smooth numbers plus partially smooth numbers required to guarantee a non-trivial solution when Gaussian Elimination is performed on the matrix of powers of primes.
PolynomialsThe number of polynomials sieved
SmoothsThe number of Smooth numbers found
PartialsThe number of partially smooth numbers found. These numbers have one large factor, F, that is not reduced by the prime factor base determined in the algorithm. When we encounter another number that is almost smooth with the same large factor, F, we can combine them into one partially smooth number.
primeFactorizeBig, factorize
mySemiPrime <- gmp::prod.bigz(gmp::nextprime(gmp::urand.bigz(2, 50, 17)))
quadraticSieve(mySemiPrime)
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