```
Revision history for Perl module Math::Prime::Util::GMP
0.43 2017-?
- random_strong_prime(nbits) random strong prob prime of nbits bits
- random_maurer_prime(nbits) random nbits-bits proven prime
- random_shawe_taylor_prime(nbits) random nbits-bits proven prime
- random_maurer_prime_with_cert(nbits)
- random_shawe_taylor_prime_with_cert(nbits)
- csprng_urandomb(n) random number between 0 and 2^n-1
- csprng_urandomr(lo,hi) random number in [lo,hi], inclusive.
[PERFORMANCE]
- sieve_primes with small widths should perform much better.
0.42 2017-02-27
[ADDED]
- lambertw(x[,digits]) Lambert W function
- random_prime(a,b) random prob prime in range [a,b]
- random_nbit_prime(nbits) random prob prime of exactly nbits bits
- random_ndigit_prime(ndigs) random prob prime of exactly ndigs digits
- seed_csprng(bytes,data) supply a seed to the internal CSPRNG
- is_csprng_well_seeded() returns 1 if the CSPRNG has a proper seed
- is_semiprime(n) does n have exactly two prime factors
[FIXES]
- is_power behaviour for 1 and -1.
- is_nminus1_prime could assert on some inputs. Fix.
- chinese([3,0],[2,3]) made GMP go belly up. Found by Trizen.
- divisors(1) in list context would segfault. Found by Trizen.
[PERFORMANCE]
- Adjust zeta algorithm crossover for large precision. Makes a huge
difference for bern{frac/real} with values > 53000.
Thanks to Trizen for pointing this out.
- zeta works for all real values, returns undef for 1. It has issues
below -10 or so that need to be dealt with in a later release.
- is_primitive_root a bit faster with large inputs.
- stirling about 40% faster. Thanks to Karim Belabas.
[OTHER]
- The ISAAC CSPRNG has been added internally and all functions call it.
While it is quite fast it is slower than GMP's Mersenne Twister.
- On startup, we attempt to seed ISAAC with 256 bytes from /dev/urandom.
Callers can call is_csprng_well_seeded() to check if this succeeded,
and call as needed seed_csprng() to seed or reseed.
0.41 2016-10-09
[API CHANGES]
- bernreal and harmreal will use the second argument to mean the digits
of precision to use, rather than the number of digits past the decimal
place.
- is_pseudoprime and is_strong_pseudoprime act like Math::Prime::Util.
[ADDED]
- todigits(n[,base[,len]]) Convert number to digit array
- zeta(s[,digits]) Riemann Zeta of integer or FP s
- riemannr(s[,digits]) Riemann R function of integer or FP s
- divisors(n) Returns list of divisors
- is_euler_pseudoprime(n,@a) Euler-Jacobi primality test
[OTHER]
- With verbose >= 3, prints factors found in partial sieve.
- factor(1) returns empty list, just like non-GMP code.
- factor() went through a Perl layer for obsolete reasons. Removed.
- bernreal and bernfrac will use the Zeta/Pi method for large values,
making it orders of magnitude faster for large sizes.
- Added internal FP log, exp, pow functions, which are not in GMP.
- is_prime will do one extra M-R test for probable primes, down from 1-5.
Also, if is_provable_prime adds two Frobenius tests if returning a 1.
- Removed Perl layer from is_strong_pseudoprime.
- is_pseudoprime and is_strong_pseudoprime take a list of bases, and
there is no default base.
- sieve_primes with small n and large range (e.g. 10^20 to 10^20+8e9) is
much faster. This tunes the full vs. partial sieve crossover.
0.40 2016-08-01
[ADDED]
- sqrtint(n) Integer square root of n
- rootint(n,k) Integer k-th root of n
- is_prime_power(n) Returns k if n=p^k for p a prime.
[OTHER]
- is_perrin_pseudoprime 2x faster. Takes optional
second argument for additional restrictions.
0.39 2016-07-24
[ADDED]
- bernreal returns float value of Bernoulli number
- is_euler_plumb_pseudoprime Colin Plumb's Euler Criterion test
- surround_primes returns offsets of prev and next primes
[PERFORMANCE]
- prev/next/surround prime sieve is slightly deeper
- Add very simple composite filter for is_perrin_pseudoprime.
- Internal refactor of Miller-Rabin code to remove one mpz variable.
[OTHER]
- Add option for restricted Perrin pseudoprimes.
0.38 2016-06-18
[FIXES]
- Minor updates for Kwalitee.
- Rewrite of BLS75 internals, rewrite BLS75 hybrid.
- Remove two small memory leaks.
[PERFORMANCE]
- Less effort to prove primality in is_prime().
0.37 2016-06-06
[ADDED]
- is_nplus1_prime(n) BLS75 N+1 deterministic primality test
- is_bls75_prime(n) BLS75 N-1, N+1, combined primality tests
[FIXES]
- Fixed primorial on systems with not-new GMP, 8-byte UV, and 4-byte long.
- sieve_range should work with >32-bit depths on 64-bit Perl + 32-bit GMP.
0.36 2016-05-21
[ADDED]
- addmod (a + b) % n
- mulmod (a * b) % n
- divmod (a / b) % n
- powmod (a ^ b) % n
- invmod (1 / b) % n
- sqrtmod square root modulo a prime
- is_primitive_root(a,n) return 1 if 'a' is a primitive root mod n
- sieve_range(n,width,depth) sieve from n, returning candidate offsets
[FIXES]
- Allow a leading '+' in inputs.
[PERFORMANCE]
- znprimroot is much faster with large inputs.
- Speedup partial sieve with large input.
- next_prime and prev_prime sieve deeper. ~5% faster with large inputs.
- AKS using Bernstein (2003) theorem 4.1. 10-20x faster.
- Speedup for large pn_primorial and primorial. Much faster for very
large values, though it will all get swamped by the overhead in
returning the large value. This is a great reason to return mpz objects.
[OTHER]
- Split out factor, primality, and AKS code into separate source files.
0.35 2015-12-13
[FIXES]
- gcdext done manually for old GMP.
- fix memory leak in chinese
0.34 2015-10-14
[ADDED]
- sieve_prime_cluster(low,high,...) sieve clusters/constellations
[PERFORMANCE]
- Speedup partial sieve with large range.
[OTHER]
- Remove _GMP_trial_primes(), which was never exported.
- Internal restructuring of sieve_primes and sieve_twin_primes.
- is_frobenius_pseudoprime with arguments doesn't check for perfect
square, and works for small primes plus large params.
0.33 2015-09-04
[ADDED]
- sieve_twin_primes(low,high) sieve for twin primes
- is_miller_prime(n[,assumeGRH]) deterministic Miller test
[PERFORMANCE]
- New results from Sorenson and Webster let us give faster deterministic
results for 65-82 bits. is_prime always returns {0,2} for this range.
0.32 2015-08-16
[ADDED]
- chinese chinese remainder theorem
- sigma divisor sums
- ramanujan_tau Ramanujan's Tau function
0.31 2015-06-21
[PERFORMANCE]
- Minor speedup to partial sieve.
[OTHER]
- Allow working on old GMP versions.
0.30 2015-06-15
[ADDED]
- harmfrac returns (num,den) of Harmonic number
- harmreal returns float value of Harmonic number
- is_proth_prime(p) For k*2^n+1, returns -1, 0, or 2
- is_frobenius_khashin_pseudoprime returns 1 if Frob-Khashin prob prime
[FIXES]
- lucas sequence with even n fixed.
[PERFORMANCE]
- A Proth test was added to quickly prove numbers of the form k*2^n+1.
- LLR testing was improved using a method by RÃ¶dseth. This allows proofs
of k*2^n-1. The old method is still used, but was unable to quickly
test cases where k was divisible by 3. The new method handles these.
- BLS75-5 proof: use an expanding stack, allowing it to work on inputs
like: 'k * n# + 1'.
- BLS75-5 proof: remove some redundant computations.
0.29 2014-11-26
[ADDED]
- is_llr_prime(p) For k*2^n-1, returns -1, 0, or 2
- lucasu(P, Q, k) U_k for Lucas(P,Q)
- lucasv(P, Q, k) V_k for Lucas(P,Q)
[PERFORMANCE]
- is_prime will prove many Proth-form (h*2^n+1) numbers.
- is_provable_prime tries less hard to make a BLS75-T5 proof. Certs may
be longer, but performance is better.
- is_power is more efficient (recursion removed, only prime powers checked).
0.28 2014-11-17
[ADDED]
- is_mersenne_prime(p) returns 1 iff 2^p-1 is prime
[PERFORMANCE]
- is_prime will do a LLR test, as will is_provable_prime if not returning
a certificate. This means many primes of the form k*2^n-1 will run
faster and return 2 rather than 1.
- Update UV SQUFOF factoring code, faster factoring once reduced in size.
- Slightly better P-1 stage 2 performance.
- Slightly deeper trial division in general factoring.
- Big reduction in average depth of unfactored stack. We work on smaller
composite factors first, and add repeated factors all at once. This
fixes some pathological inputs such as:
vecprod( map { $_*($_+2)**17 } @{twin_primes(100000,115000)} )
which has 2574 factors and would overflow the 256-element stack. With
the new code it has a maximum stack depth of 3.
[OTHER]
- is_power works with negative powers, although doesn't return root.
0.27 2014-10-07
[PERFORMANCE]
- Minor changes to factor recipe, should give a little speedup.
- Cache ~32k worth of small primes to give a little speedup in many places.
- Switch to my original AGM code, slightly faster for large values.
- Add Goetgheluck binomial code, and switch to mpz_bin_uiui for builtin.
For large inputs this can be thousands of times faster than mpz_bin_ui.
[OTHER]
- Don't use mp_bitcnt_t -- old GMPs don't have this type.
0.26 2014-09-26
[ADDED]
- stirling(n,m,[type]) Stirling numbers of first,second,third kind
- vecprod(list) product of a list of integers
[OTHER]
- Cleanup invmod, etc. XS parser. Smaller code.
- Fixed some leaked mpz_t / mpz_f objects.
0.25 2014-09-23
- Fixed compiler warning (error for some compilers).
- prev_prime uses a sieve for 200+ bits. 20% speedup for large inputs.
0.24 2014-09-22
[ADDED]
- sieve_primes(low,high[,k]) sieve for primes, partial or BPSW
- is_frobenius_pseudoprime(n,[a,b]) Frobenius quadratic primality test
- is_perrin_pseudoprime(n) Perrin primality test
- factorial(n) n!
- bernfrac returns (num,den) of Bernoulli number
- Pi([digits]) Pi with requested number of digits
[OTHER]
- next_prime will use a partial sieve for 120+ bit inputs. For large
inputs this is a 15-30% speedup. For 2469*2617#/93030-12182 I get:
= 392.2s OpenPFGW 3.7.7
= 220.6s Pari/GP 2.6.2
= 128.4s GMP 5.0.2 mpz_nextprime
= 57.6s old MPU
= 45.5s new MPU
- New version of Frobenius-Underwood test to match the 2014 draft paper.
This is just a code refresh and has no other effect.
- BLS75 with effort 1 toned down. This makes is_prime with 65- to 200-bit
inputs faster, though a bit less likely to return with the value 2
rather than 1. It's a couple percent fewer, but 10-60% faster.
0.23 2014-08-16
- Fat comma isn't fat for numbers, garbled test hashes on 32-bit.
0.22 2014-08-16
[ADDED]
- moebius(n[,nhi]) Moebius function (single or ranged)
- liouville(n) Liouville function
- totient(n) Euler's Totient function (single)
- jordan_totient(k, n) Jordan totient
- carmichael_lambda(n) Carmichael Lambda (reduced totient)
- znorder(a, n) multiplicative order of a mod n
- znprimroot(n) least primitive root of n
[OTHER]
- Moved factoring loop out of XS file.
- factor does much better power splitting, similar to MPU 0.38's code:
time mpu 'use bigint; my $n = next_prime(10**20)**200; say join(" ", map {"[@$_]"} factor_exp($n));'
time mpu 'use bigint; my $n = next_prime(10**21)**200 * next_prime(10**20)**200; say join(" ", map {"[@$_]"} factor_exp($n));'
- Fix spelling of Paul Zimmermann's name (thanks to Mathew @ mersenneforum)
- Standalone ECPP now does expression parsing using the GMP 6.0.0a demo
code. Version bumped to 1.04.
0.21 2014-06-19
- Used a bare 64-bit in a test. Wrap in quotes.
0.20 2014-06-18
[ADDED]
- valuation(a,b) how many times does b divide a?
- invmod(a,n) inverse of a modulo n
- is_pseudoprime(n,base) Simple Fermat test
- binomial(n,k) binomial coefficient
- gcdext(a,b) extended Euclidian algorithm
- vecsum(...) sum list of integers
[OTHER]
- 10%-ish speedup for next/prev prime with 38-950 digit inputs.
0.19 2014-04-21
[ADDED]
- is_power
- exp_mangoldt
[FIXES]
- Fixed string shortcut for simple divisibility. is_prime and related
functions are a bit faster when given inputs divisible by 2 or 5.
[OTHER]
- Add improved AKS parameter selection. About 200x faster, though still
thousands of times slower than APR-CL or ECPP. Updated times for the
example in the v0.10 entry: Timing for 10**100+267:
AKS: ~5 days.
BLS75 n-1: ~3 minutes.
APR-CL: 0.09 seconds
ECPP: 0.05 seconds.
- ECPP performance adjustments, version 1.03 of standalone ECPP.
- Updated ECPP class polynomial data. Default "tiny" table had very minor
changes. The "big" table (in the github xt/ directory, default for
standalone ECPP) removed some large coefficient 17-24 degree polys to
make room for many more higher-degree polys. For some ranges this may
mean more backtracking, but should expand the input size that is able to
find good discriminants without high factoring effort. "prob" below is
summing the estimate 1/2H: 9x more polys and 66x larger size gives on
average about 3x more candidates.
Default "tiny" table:
OLD: 30373 bytes 604 polys 24 maxdeg 42.0 prob 1450 prob/MB
NEW: 30422 bytes 611 polys 25 maxdeg 42.8 prob 1475 prob/MB
"big" table at www.probableprime.org/ecpp/cpd/big/class_poly_data.h.gz
OLD: 2032376 bytes 3197 polys 117 maxdeg 104.5 prob 54 prob/MB
NEW: 2005072 bytes 5271 polys 85 maxdeg 125.2 prob 65 prob/MB
"huge" table at www.probableprime.org/ecpp/cpd/huge/class_poly_data.h.gz
15724395 bytes 14571 polys 128 maxdeg 207.9 prob 14 prob/MB
0.18 2014-01-27
[FIXES]
- Fix for 5.6.2 (undefined symbol).
- Fix for unsigned long != UV, reported by CHORNY.
0.17 2014-01-24
[ADDED]
- is_bpsw_prime specific BPSW-only test
- gcd 20-50x faster than Math::BigInt
- lcm 3-800x faster than Math::BigInt
- kronecker
[FIXES]
- Factoring with a number or intermediate near the word boundary would
hang or run very slow. Thanks to Hugo van der Sanden for the report.
- Next version of vcert.c, which handles some new Primo changes.
0.16 2013-10-28
[ADDED]
- partitions partition function p(n), OEIS A000041
[FIXES]
- Fixed memory leak in Lucas sequence (is_prime, next_prime, etc.).
- is_aks_prime wasn't properly checking divisibility for composites.
[Scripts and Programs Added]
- verify_primegap.pl parallel prime gap verification
0.15 2013-09-30
[Functions Added]
- miller_rabin_random
- A tree sieve is done in trial factor for large (900+ digits) inputs.
This improves performance greatly for very large inputs.
- is_prob_prime uses more trial division for large inputs. For very
large inputs (e.g. 50,000+ digits) this can greatly speed up probable
prime testing, for instance in next_prime or similar sieving.
Time for next_prime(99992 * 10**10101 - 100):
1m 4s MPUGMP 0.15
3m 34s Pari/GP (needs 450MB of stack!)
4m 1s mpz_nextprime
9m 33s Math::Primality
- Use shallow product tree for primorials. Large primorials are 2 to 12
times faster. Break consecutive_integer_lcm into four sub-products so
it runs 2-4x faster for large inputs.
- Trim ECPP and adjust its heuristics.
- Standalone ECPP now has consistent return codes, making it easier to
use in applications without having to parse return text. The return
codes are consistent with the certificate verifier.
- factor() in scalar context is now consistent.
0.14 2013-08-07
- Fix small certificates leaving out the "N " for small numbers.
0.13 2013-08-06
[API Changes]
- Primality proofs now use a text certificate. This is nicer for
external interaction, but is a change from previous behavior. You
will need to use Math::Prime::Util 0.30 or newer.
[Functions Added]
- lucas_sequence
- is_almost_extra_strong_lucas_pseudoprime
- is_frobenius_underwood_pseudoprime
- pplus1_factor
[Enhancements]
- is_prob_prime now uses the extra-strong Lucas test instead of the
strong Lucas test. This gives better performance. is_prime and
is_provable_prime also incorporate the change.
- Added more trial division to is_prob_prime for big (100+ digit)
numbers. This is a significant speedup for next_prime in many cases.
Pari/gp 2.6.0 nextprime(10^4000) 19 minutes
MPU:GMP 0.12 next_prime(10**4000) 15 minutes
MPU:GMP 0.13 next_prime(10**4000) 8 minutes
- ECPP now tries partial n-1 and n+1 proofs (BLS theorem 3 / 15) at each
step, and adds a couple additional quick factoring tests. This mainly
helps lower the time variability with large inputs.
- Updated ECPP polynomials. Should give better performance with larger
inputs.
[Scripts and Programs Added]
- convert-primo-cert.pl convert a Primo certificate to MPU format.
- verify-cert.pl Verify a Primo or MPU certificate.
- vcert.c Verify a Primo or MPU certificate.
0.12 2013-06-12
- add standard and extra strong Lucas probable prime tests.
- Rearrange C code to allow standalone build of ECPP.
- Speedups for ECPP.
0.11 2013-05-20
- is_prob_prime is faster at finding composites.
- rewrote Lucas inner loop for ~20% speedup.
- The previous two changes make is_prob_prime a bit faster, which means
a small speedup to almost all functions.
- Lower is_prime proving effort. Proves ~30% of 128-bit primes instead
of 50%, but runs about 4x faster.
- Change ECPP to factor all strategy with backtracking. Not much
difference below 200 digits, but a big help after that. Certificates
are identical.
0.10 2013-05-07
- ECPP -- a much faster primality prover. BLS75 n-1 works well to about
40 digits, then slows down rapidly. This ECPP implementation is good
to 300-500 digits. Timing for 10**100+267:
AKS: ~1 year. BLS75 n-1: 1.5-5 minutes. ECPP: 0.1 seconds.
- is_prime does an additional 4 random-base M-R tests.
- is_provable_prime will try a quick n-1 then do ECPP.
- is_nminus1_prime added to give access to that specific method, in
case someone has reason to insist on that proof type.
- Change polynomial multiplication to use binary segmentation. Huge
speed improvement for AKS primality proving (20-100x faster). AKS
is now faster in GMP than MPU's C code. It's still not nearly as fast
as other methods: proving 100000000003 takes 65 seconds, while this
would take a couple milliseconds at most for an n-1 proof. The one year
estimate in the first paragraph is with the _new_ code.
- Compile-time support to BLS75 theorem 7, which reduces the amount of
n-1 we need to factor. Not enabling because it just doesn't help
enough, and ECPP is a better place to spend development effort.
- Lots of new internal functions to support ECPP, which could be used
for future projects.
0.09 2013-04-21
- Add primality certificate generation.
0.08 2013-04-05
- Switch to a projective ECM with a stage 2. Much better results, but
note that it doesn't build up to B1 like the old version. This has
a big impact on factoring and primality proving.
- Add a QS based on William Hart's SIMPQS (a simple QS that is a
predecessor to what went into FLINT). Not the fastest by a long shot
(yafu and msieve take that prize), but it's quite small and works pretty
well. Eventually this will get replaced with a home-built QS. Meanwhile
some improvements from version 2.0 that remain are (1) no partial
relations, (2) uses too much memory, and (3) uses GE instead of
jasonp's block Lanczos.
- The new ECM and QS make factoring much faster, especially for 30+
digit inputs. Factoring should give reasonable times out to 70+
digits now. Time is competitive with Math::Pari now, and often faster
(noting that Math::Pari uses a fairly old version of Pari).
- Factoring mix redone given the big changes in ECM and QS.
- Primality proofs adjusted to better use p-1 and ECM. The quick proof
in is_prime has a higher success rate for all input sizes and is a
little faster for small numbers. is_provable_prime is 10-50x faster.
0.07 2013-03-19
- Tiny speedup when passing in bigints.
- Some speedups in pbrent, pbrent usage, and small prime iterator.
Factoring small (< ~30 digit) numbers is faster.
- Handle large and small M-R bases just like MPU does -- mod with n,
then return 1 if base <= 1 or base >= n-1.
0.06 2012-12-17
- Fix 1-byte memory overrun (thanks to CPAN Testers, Solaris, Valgrind).
- Add factoring of small numbers. Helps a little when the input gets
reduced enough to fit into a UV.
0.05 2012-12-15
- Add AKS primality test. Super slow, but nice to have around.
- ECM is faster.
- Add a small prime iterator, which means _much_ less memory and faster
operation for big smoothness factors in pminus1 and ecm factoring.
0.04 2012-11-11
- Add simple prime_count function. It uses next_prime so is terribly slow
for big ranges. However it's a lot faster than the PP code when given
a large base and small range e.g. (10**96, 10**96 + 2**18).
- Add primorial, pn_primorial, and consecutive_integer_lcm functions.
- Factoring:
Add a perfect power test.
Add a simple ECM factoring method.
Speed up SQUFOF a bit.
Complete p-1 rewrite. Much faster and finds more factors.
Adjust general factor() mix.
- Add Pocklington-Lehmer and BLS primality tests. is_prime() uses the
BLS test with a quick factoring attempt for numbers less than 2^200,
though the chances of success drop off as the size increases.
The point is not to cull mismarked probable primes (we use BPSW so this
is highly unlikely for these small sizes), but to quickly mark more
numbers as definitely prime. Remember to use is_prob_prime if you do
not care about this distinction and want the result slightly faster.
- add is_provable_prime function that calls BLS with much more aggressive
factoring.
0.03 2012-07-16
- XS callable: _lcm_of_consecutive_integers(B)
which is a better alternative for B! for many factoring algorithms.
- Fix some minor compile issues.
0.02 2012-07-15
- Factoring tests assumed 64-bit. Rewrite.
0.01 2012-07-15
- Initial release
```