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RcppBigIntAlgos uses the C library GMP (GNU Multiple Precision Arithmetic) for efficiently factoring big integers. Links to RcppThread for factoring in parallel. For very large integers, prime factorization is carried out by a variant of the quadratic sieve algorithm that implements multiple polynomials. For smaller integers, a constrained version of the Pollard’s rho algorithm is used (original code from https://gmplib.org/… this is the same algorithm found in the R gmp package called by the function factorize). Finally, one can quickly obtain a complete factorization of a given number n via divisorsBig.
install.packages("RcppBigIntAlgos")
## Or install the development version
devtools::install_github("jwood000/RcppBigIntAlgos")First, we take a look at divisorsBig. It is vectorized and can also return a named list.
## Get all divisors of a given number:
divisorsBig(1000)
Big Integer ('bigz') object of length 16:
[1] 1 2 4 5 8 10 20 25 40 50 100 125 200 250 500 1000
## Or, get all divisors of a vector:
divisorsBig(urand.bigz(nb = 2, size = 100, seed = 42), namedList = TRUE)
Seed initialisation
$`153675943236425922379228498617`
Big Integer ('bigz') object of length 16:
[1] 1 3
[3] 7 9
[5] 21 27
[7] 63 189
[9] 813100228764158319466817453 2439300686292474958400452359
[11] 5691701601349108236267722171 7317902058877424875201357077
[13] 17075104804047324708803166513 21953706176632274625604071231
[15] 51225314412141974126409499539 153675943236425922379228498617
$`261352009818227569107309994396`
Big Integer ('bigz') object of length 12:
[1] 1 2
[3] 4 155861
[5] 311722 623444
[7] 419206873140534785974859 838413746281069571949718
[9] 1676827492562139143899436 65338002454556892276827498599
[11] 130676004909113784553654997198 261352009818227569107309994396It is very efficient as well. It is equipped with a modified merge sort algorithm that significantly outperforms the std::sort/bigvec (the class utilized in the R gmp package) combination.
hugeNumber <- pow.bigz(2, 100) * pow.bigz(3, 100) * pow.bigz(5, 100)
system.time(overOneMillion <- divisorsBig(hugeNumber))
user system elapsed
0.364 0.029 0.390
length(overOneMillion)
[1] 1030301
## Output is in ascending order
tail(overOneMillion)
Big Integer ('bigz') object of length 6:
[1] 858962534553352218394101882942702121170179203335000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
[2] 1030755041464022662072922259531242545404215044002000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
[3] 1288443801830028327591152824414053181755268805002500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
[4] 1717925069106704436788203765885404242340358406670000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
[5] 2576887603660056655182305648828106363510537610005000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
[6] 5153775207320113310364611297656212727021075220010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000Another benefit is that it will return correct orderings on extremely large numbers when compared to sorting large vectors in base R. Typically in base R you must execute the following: order(asNumeric(myVectorHere)). When the numbers get large enough, precision is lost which leads to incorrect orderings. Observe:
set.seed(101)
testBaseSort <- do.call(c, lapply(sample(100), function(x) add.bigz(pow.bigz(10,80), x)))
testBaseSort <- testBaseSort[order(asNumeric(testBaseSort))]
myDiff <- do.call(c, lapply(1:99, function(x) sub.bigz(testBaseSort[x+1], testBaseSort[x])))
## Should return integer(0) as the difference should always be positive
## NOTE that the result will be unpredictable because of lack of precision
which(myDiff < 0)
[1] 1 3 4 7 9 11 14 17 19 22 24 25 26 28 31 32 33 36 37 38 40 42 45 47 48
[26] 50 51 54 57 58 59 63 64 65 66 69 70 72 75 78 81 82 85 87 89 91 93 94 97 98
## N.B. The first and second elements are incorrect order (among others)
head(testBaseSort)
Big Integer ('bigz') object of length 6:
[1] 100000000000000000000000000000000000000000000000000000000000000000000000000000038
[2] 100000000000000000000000000000000000000000000000000000000000000000000000000000005
[3] 100000000000000000000000000000000000000000000000000000000000000000000000000000070
[4] 100000000000000000000000000000000000000000000000000000000000000000000000000000064
[5] 100000000000000000000000000000000000000000000000000000000000000000000000000000024
[6] 100000000000000000000000000000000000000000000000000000000000000000000000000000029The function quadraticSieve implements the multiple polynomial quadratic sieve algorithm. Currently, quadraticSieve can comfortably factor numbers with less than 70 digits (~230 bits) on most standard personal computers. If you have access to powerful computers with many cores, factoring 100+ digit semiprimes in less than a day is not out of the question.
## Generate large semi-primes
semiPrime120bits <- prod(nextprime(urand.bigz(2, 60, 42)))
semiPrime130bits <- prod(nextprime(urand.bigz(2, 65, 1)))
semiPrime140bits <- prod(nextprime(urand.bigz(2, 70, 42)))
## The 120 bit number is 36 digits
nchar(as.character(semiPrime120bits))
[1] 36
## The 130 bit number is 39 digits
nchar(as.character(semiPrime130bits))
[1] 39
## The 140 bit number is 42 digits
nchar(as.character(semiPrime140bits))
[1] 42
## Using factorize from gmp package which implements pollard's rho algorithm
##**************gmp::factorize*********************
system.time(print(factorize(semiPrime120bits)))
Big Integer ('bigz') object of length 2:
[1] 638300143449131711 1021796573707617139
user system elapsed
125.117 0.139 125.113
system.time(print(factorize(semiPrime130bits)))
Big Integer ('bigz') object of length 2:
[1] 14334377958732970351 29368224335577838231
user system elapsed
1437.246 0.309 1437.505
system.time(print(factorize(semiPrime140bits)))
Big Integer ('bigz') object of length 2:
[1] 143600566714698156857 1131320166687668315849
user system elapsed
2239.374 0.299 2239.641
##**************quadraticSieve*********************
## quadraticSieve is much faster and scales better
system.time(print(quadraticSieve(semiPrime120bits)))
Big Integer ('bigz') object of length 2:
[1] 638300143449131711 1021796573707617139
user system elapsed
0.086 0.001 0.085
system.time(print(quadraticSieve(semiPrime130bits)))
Big Integer ('bigz') object of length 2:
[1] 14334377958732970351 29368224335577838231
user system elapsed
0.102 0.000 0.103
system.time(print(quadraticSieve(semiPrime140bits)))
Big Integer ('bigz') object of length 2:
[1] 143600566714698156857 1131320166687668315849
user system elapsed
0.173 0.000 0.174As of version 0.3.0, we can utilize multiple threads with the help of RcppThread. For example, we factor the largest Cunnaningham Most Wanted number from the first edition released in 1983 in under a minute and RSA-79 can be factored in under 6 minutes on my machine. I obtained the best performance when nThreads = stdThreadMax() / 2. When the number of threads was maximized, there was a decrease in efficiency probably due to pollution of the cache.
quadraticSieve(mostWanted1983, nThreads=4, skipExtPolRho=TRUE, showStats=TRUE)
Summary Statistics for Factoring:
11111111111111111111111111111111111111111111111111111111111111111111111
| Pollard Rho Time |
|--------------------|
| 52ms |
| MPQS Time | Complete | Polynomials | Smooths | Partials |
|--------------------|----------|-------------|------------|------------|
| 34s 765ms | 100% | 15919 | 4511 | 4461 |
| Mat Algebra Time | Mat Dimension |
|--------------------|--------------------|
| 5s 123ms | 8836 x 8972 |
| Total Time |
|--------------------|
| 40s 133ms |
Big Integer ('bigz') object of length 2:
[1] 241573142393627673576957439049 45994811347886846310221728895223034301839
## ***************************************************************************
quadraticSieve(rsa79, showStats=TRUE, nThreads=4, skipExtPolRho=TRUE)
Summary Statistics for Factoring:
7293469445285646172092483905177589838606665884410340391954917800303813280275279
| Pollard Rho Time |
|--------------------|
| 66ms |
| MPQS Time | Complete | Polynomials | Smooths | Partials |
|--------------------|----------|-------------|------------|------------|
| 5m 19s 965ms | 100% | 100725 | 5581 | 7166 |
| Mat Algebra Time | Mat Dimension |
|--------------------|--------------------|
| 12s 694ms | 12605 x 12747 |
| Total Time |
|--------------------|
| 5m 33s 179ms |
Big Integer ('bigz') object of length 2:
[1] 848184382919488993608481009313734808977 8598919753958678882400042972133646037727If you encounter a number that is a product of multiple large primes, the algorithm will recursively factor the number into two numbers until every part is prime.
threePrime195bits <- prod(nextprime(urand.bigz(3, 65, 97)))
quadraticSieve(threePrime195bits, showStats = TRUE)
Summary Statistics for Factoring:
6634573213431810791169420577087478977215298519759798575509
| Pollard Rho Time |
|--------------------|
| 51ms |
| MPQS Time | Complete | Polynomials | Smooths | Partials |
|--------------------|----------|-------------|------------|------------|
| 11s 273ms | 100% | 2914 | 1704 | 2098 |
| Mat Algebra Time | Mat Dimension |
|--------------------|--------------------|
| 470ms | 3745 x 3802 |
Summary Statistics for Factoring:
369498233670465681342232176125551121921
| MPQS Time | Complete | Polynomials | Smooths | Partials |
|--------------------|----------|-------------|------------|------------|
| 115ms | 100% | 63 | 597 | 246 |
| Mat Algebra Time | Mat Dimension |
|--------------------|--------------------|
| 18ms | 801 x 843 |
| Total Time |
|--------------------|
| 11s 997ms |
Big Integer ('bigz') object of length 3:
[1] 11281626468262639417 17955629036507943829 32752213052784053513It can also be used as a general prime factoring function:
quadraticSieve(urand.bigz(1, 50, 1))
Seed initialisation
Big Integer ('bigz') object of length 5:
[1] 5 31 307 2441 4702723However gmp::factorize is more suitable for numbers smaller than 70 bits (about 22 decimal digits) and should be used in such cases.
quadraticSieveIf you want to interrupt a command which will take a long time, hit Ctrl + c, or esc if using RStudio, to stop execution.
## User hits Ctrl + c
## system.time(quadraticSieve(prod(nextprime(urand.bigz(2, 100, 42)))))
## Seed default initialisation
## Seed initialisation
##
## Error in QuadraticSieveContainer(n) : C++ call interrupted by the user.
##
## Timing stopped at: 1.623 0.102 1.726Credit to primo (Mike Tryczak) and his excellent answer to Fastest semiprime factorization.
Factoring large numbers with quadratic sieve on MSDN Archive.
A really nice concise example is given here: Factorization of n = 87463 with the Quadratic Sieve
Smooth numbers and the quadratic sieve by Carl Pomerance
Integer Factorization using the Quadratic Sieve by Chad Seibert
Currenlty, our main focus is on implementing our sieve in a parallel fashion.
I welcome any and all feedback. If you would like to report a bug, have a question, or have suggestions for possible improvements, please file an issue.