Vectorized Factorization (Complete) with GMP

Description

Quickly generates the complete factorization for many (possibly large) numbers.

Usage

divisorsBig(v, namedList = FALSE, showStats = FALSE,
            skipPolRho = FALSE, skipECM = FALSE, nThreads = NULL)

Arguments

Details

Highly optimized algorithm to generate the complete factorization after first obtaining the prime factorization. It is built specifically for the data type that is used in the gmp library (i.e. mpz_t).

The main part of the algorithm that generates all divisors is essentially the same algorithm that is implemented in divisorsRcpp from the RcppAlgos package. A modified merge sort algorithm is implemented to better deal with the mpz_t data type. This algorithm avoids directly swapping elements of the main factor array of type mpz_t but instead generates a vector of indexing integers for ordering.

The prime factorization is obtained using primeFactorizeBig, which attempts trial division, Pollard's rho algorithm, Lentra's elliptic curve method, and finally the quadratic sieve.

See this stackoverflow post for examples and benchmarks : R Function for returning ALL factors.

See quadraticSieve for information regarding showStats.

Value

• Returns an unnamed vector of class bigz if length(v) == 1 regardless of the value of namedList.

• If length(v) > 1, a named/unnamed list of vectors of class bigz will be returned.

Author(s)

Joseph Wood

References

Divisor

See Also

divisorsRcpp, divisors

Examples

## Get the complete factorization of a single number
divisorsBig(100)

## Or get the complete factorization of many numbers
set.seed(29)
myVec <- sample(-1000000:1000000, 1000)
system.time(myFacs <- divisorsBig(myVec))

## Return named list
myFacsWithNames <- divisorsBig(myVec, namedList = TRUE)

## Get the complete factorization for a large semiprime
big = gmp::prod.bigz(gmp::nextprime(gmp::urand.bigz(2, size = 65, seed = 3)))
divisorsBig(big)

Vectorized Prime Factorization with GMP

Description

Quickly generates the prime factorization for many (possibly large) numbers, using trial division, Pollard's rho algorithm, Lenstra's Elliptic Curve method, and finally the Quadratic Sieve.

Usage

primeFactorizeBig(v, namedList = FALSE, showStats = FALSE,
                  skipPolRho = FALSE, skipECM = FALSE, nThreads = NULL)

Arguments

Details

This function should be preferred in most situations and is identical to quadraticSieve when both skipECM and skipPolRho are set to TRUE.

It is optimized for factoring big and small numbers by dynamically using different algorithms based off of the input. It takes cares to not spend too much time in any of the methods and avoids wastefully switching to the quadratic sieve when the number is very large.

See quadraticSieve for information regarding showStats.

Value

• Returns an unnamed vector of class bigz if length(v) == 1 regardless of the value of namedList.

• If length(v) > 1, a named/unnamed list of vectors of class bigz will be returned.

Note

Note, the function primeFactorizeBig(n, skipECM = T, skipPolRho = T) is the same as quadraticSieve(n)

Author(s)

Joseph Wood

References

• Gaj K. et al. (2006) Implementing the Elliptic Curve Method of Factoring in Reconfigurable Hardware. In: Goubin L., Matsui M. (eds) Cryptographic Hardware and Embedded Systems - CHES 2006. CHES 2006. Lecture Notes in Computer Science, vol 4249. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11894063_10

• Integer Factorization

See Also

primeFactorize, primeFactors, factorize, quadraticSieve

Examples

## Get the prime factorization of a single number
primeFactorizeBig(100)

## Or get the prime factorization of many numbers
set.seed(29)
myVec <- sample(-1000000:1000000, 1000)
system.time(myFacs <- primeFactorizeBig(myVec))

## Return named list
myFacsWithNames <- primeFactorizeBig(myVec, namedList = TRUE)

Prime Factorization with the Parallel Quadratic Sieve

Description

Get the prime factorization of a number, n, using the Quadratic Sieve.

Usage

quadraticSieve(n, showStats = FALSE, nThreads = NULL)

Arguments

Details

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 Time

The time measured for the multiple polynomial quadratic sieve section in hours h, minutes m, seconds s, and milliseconds ms.

Complete

The 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.

Polynomials

The number of polynomials sieved

Smooths

The number of Smooth numbers found

Partials

The 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.

Value

Vector of class bigz

Note

• primeFactorizeBig is preferred for general prime factorization.

• Both the extended Pollard's rho algorithm and the elliptic curve method are skipped. For general prime factorization, see primeFactorizeBig.

• Safely interrupt long executing commands by pressing Ctrl + c, Esc, or whatever interruption command offered by the user's GUI. If you are using multiple threads, you can still interrupt execution, however there will be a delay up to 30 seconds if the number is very large.

• Note, the function primeFactorizeBig(n, skipECM = T, skipPolRho = T) is the same as quadraticSieve(n)

Author(s)

Joseph Wood

References

• Pomerance, C. (2008). Smooth numbers and the quadratic sieve. In Algorithmic Number Theory Lattices, Number Fields, Curves and Cryptography (pp. 69-81). Cambridge: Cambridge University Press.

• Silverman, R. D. (1987). The Multiple Polynomial Quadratic Sieve. Mathematics of Computation, 48(177), 329-339. doi:10.2307/2007894

• Integer Factorization using the Quadratic Sieve

• From https://codegolf.stackexchange.com/ (Credit to user primo for answer) P., & Chowdhury, S. (2012, October 7). Fastest semiprime factorization. Retrieved October 06, 2017

See Also

primeFactorizeBig, factorize

Examples

mySemiPrime <- gmp::prod.bigz(gmp::nextprime(gmp::urand.bigz(2, 50, 17)))
quadraticSieve(mySemiPrime)

Max Number of Concurrent Threads

Description

Rcpp wrapper of std::thread::hardware_concurrency(). As stated by cppreference, the returned value should be considered only a hint.

Usage

stdThreadMax()

Value

An integer representing the number of concurrent threads supported by the user implementation. If the value cannot be determined, 1L is returned.

See Also

detectCores

Examples

stdThreadMax()