RcppBigIntAlgos uses the C library GMP (GNU Multiple Precision Arithmetic) for efficiently factoring big integers. 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 261352009818227569107309994396
It 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
0.557 0.063 0.622
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] 5153775207320113310364611297656212727021075220010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
Another 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] 100000000000000000000000000000000000000000000000000000000000000000000000000000029
The function quadraticSieve
implements the multiple polynomial quadratic sieve algorithm. Currently, quadraticSieve
can comfortably factor numbers with less than 60 digits (~200 bits).
## 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
## We did not test the 140 bit semi-prime as the 130 bit took a very long time
##**************gmp::factorize*********************
system.time(print(factorize(semiPrime120bits)))
Big Integer ('bigz') object of length 2:
[1] 638300143449131711 1021796573707617139
user system elapsed
126.603 0.052 126.694
system.time(print(factorize(semiPrime130bits)))
Big Integer ('bigz') object of length 2:
[1] 14334377958732970351 29368224335577838231
user system elapsed
1513.055 1.455 1517.524
##**************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
0.092 0.001 0.091
system.time(print(quadraticSieve(semiPrime130bits)))
Big Integer ('bigz') object of length 2:
[1] 14334377958732970351 29368224335577838231
user system elapsed
0.148 0.000 0.148
system.time(print(quadraticSieve(semiPrime140bits)))
Big Integer ('bigz') object of length 2:
[1] 143600566714698156857 1131320166687668315849
user system elapsed
0.265 0.000 0.265
Below, we factor a 50 digit semiprime in under 3 secs followed by a 60 digit semiprime factored in under 35 seconds. Also, we see can see summary statistics by setting showStats = TRUE
(N.B. This is will slow down execution slightly).
semiPrime164bits <- prod(nextprime(urand.bigz(2, 82, 42)))
## The 164 bit number is 50 digits
nchar(as.character(semiPrime164bits))
[1] 50
## We see this about 0.5 seconds faster than setting showStats = TRUE
system.time(quadraticSieve(semiPrime164bits))
user system elapsed
2.723 0.009 2.731
quadraticSieve(semiPrime164bits, showStats = TRUE)
Summary Statistics for Factoring:
10050120961360479179164300841596861740399588283187
| Time | Complete | Polynomials | Smooths | Partials |
|--------------------|----------|-------------|------------|------------|
| 3s 234ms | 100% | 1391 | 739 | 805 |
Big Integer ('bigz') object of length 2:
[1] 2128750292720207278230259 4721136619794898059404993
## And here is the 60 digit example
semiPrime200bits <- prod(nextprime(urand.bigz(2, 100, 1729)))
nchar(as.character(semiPrime200bits))
[1] 60
quadraticSieve(semiPrime200bits, showStats = TRUE)
Summary Statistics for Factoring:
394753378083444510740772455309612207212651808400888672450967
| Time | Complete | Polynomials | Smooths | Partials |
|--------------------|----------|-------------|------------|------------|
| 33s 235ms | 100% | 7952 | 1398 | 1671 |
Big Integer ('bigz') object of length 2:
[1] 514864663444011777835756770809 766712897798959945129214210063
Finally, we factor the largest Cunnaningham Most Wanted number from the first edition released in 1983 in under 9 minutes.
mostWanted1983 <- as.bigz(div.bigz(sub.bigz(pow.bigz(10, 71), 1), 9))
mostWanted1983
Big Integer ('bigz') :
[1] 11111111111111111111111111111111111111111111111111111111111111111111111
quadraticSieve(mostWanted1983, showStats = TRUE)
Summary Statistics for Factoring:
11111111111111111111111111111111111111111111111111111111111111111111111
| Time | Complete | Polynomials | Smooths | Partials |
|--------------------|----------|-------------|------------|------------|
| 8m 38s 680ms | 100% | 29484 | 2952 | 3382 |
Big Integer ('bigz') object of length 2:
[1] 241573142393627673576957439049 45994811347886846310221728895223034301839
If 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
| Time | Complete | Polynomials | Smooths | Partials |
|--------------------|----------|-------------|------------|------------|
| 21s 853ms | 100% | 5345 | 1243 | 1609 |
Summary Statistics for Factoring:
202568699792573213335520384055117307693
| Time | Complete | Polynomials | Smooths | Partials |
|--------------------|----------|-------------|------------|------------|
| 150ms | 100% | 117 | 409 | 211 |
Big Integer ('bigz') object of length 3:
[1] 11281626468262639417 17955629036507943829 32752213052784053513
It 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 4702723
However gmp::factorize
is more suitable for numbers smaller than 70 bits (about 22 decimal digits) and should be used in such cases.
quadraticSieve
If you want to interrupt a command which will take a long time, hit Ctrl + c, or esc if using RStudio, to stop execution. Underneath, we check for user interruption once every second.
## 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 1.726
Credit 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.