Fast, deterministic* Miller-Rabin primality test.

# miller-rabin

I implement this fast (see Performance), deterministic (up to 64 bits unsigned), permissively licensed (MIT) Miller-Rabin primality test as a C extension to Python so you don't have to.

Only CPython 3.6 or later is supported.

Project canonical URL: https://github.com/zmwangx/miller-rabin

## Algorithm

This library implements Bradley Berg's deterministic variant[1] of the Miller-Rabin primality test for 64-bit unsigned integers as recommended by [2], and the usual probablistic test for integers beyond 64-bit. Preliminary tests with small prime divisions and in some cases one pass of Fermat test are inspired by boost::multiprecision::miller_rabin_test[3]. Integers within 16-bit are directly checked against a lookup table.

GMP[4] is used for modular exponentiation, hence the library links to libgmp (LGPLv3).

Credit:

## Installation

pip install miller-rabin


Wheels with dependencies included are available for macOS and Linux (manylinux1, manylinux2010 and manylinux2014) for Python 3.6, 3.7 and 3.8. A reasonably recent pip should be able to pick a wheel automatically (manylinux1 support was added in v8.1.2), but it is advised to update pip to latest.

To install from a source distribution, CPython development headers and libgmp along with development headers are required.

## API

The API is extremely simple so there's no need for a separate Sphinx doc site.

NAME
miller_rabin - Fast, deterministic* Miller-Rabin primality test.

FUNCTIONS
miller_rabin(n, k=10, /)
Perform Miller-Rabin primality test on the arbitrary precision int.

A deterministic variant is auto-selected if n fits into 64-bit unsigned;
otherwise, the probablistic variant is used, and k determines the number of
test rounds to perform.

miller_rabin_deterministic32(n, /)
Perform deterministic Miller-Rabin primality test on the 32-bit unsigned int.

miller_rabin_deterministic64(n, /)
Perform deterministic Miller-Rabin primality test on the 64-bit unsigned int.


In practice you should simply use the miller_rabin function for all numbers regardless of bit count, unless you want to enforce the bit count without checking beforehand.

## Performance

TL;DR: This library can deterministically test ~2.5 million odd 64-bit unsigned integers per second on a 3.7GHz Intel Core i5 CPU (single thread).

Below are some benchmarks of this library's primality test vs that of gmpy2 (Python binding to GMP). The first column is the bit count of each random sample (random odd numbers in the given range), and results are in million tests per second, estimated from the total run time on a random sample of size one million. Results labeled MR are for miller_rabin.miller_rabin from this library; results labeled G(25) are for gmpy2.is_prime on default setting (25 rounds); results labeled G(10) are for gmpy2.is_prime with 10 rounds (comparable to this library's default for numbers above 64-bit). Note that gmpy2.is_prime uses mpz_probab_prime_p under the hood. See bench/benchmark.py for details.

#bits	MR	G(25)	G(10)
1-32	4.538	0.901	1.581
32	4.553	0.916	1.601
1-64	2.597	0.845	1.377
64	2.500	0.755	1.258
65	1.120	0.694	1.153
96	1.044	0.642	0.977
128	0.832	0.495	0.745
256	0.327	0.204	0.286
(unit: million tests per second)
(CPU: Intel(R) Core(TM) i5-9600K CPU @ 3.70GHz)

#bits	MR	G(25)	G(10)
1-32	3.275	0.960	1.530
32	3.288	0.982	1.561
1-64	2.026	0.865	1.315
64	1.933	0.743	1.176
65	0.915	0.727	1.129
96	0.878	0.680	0.983
128	0.663	0.507	0.735
256	0.258	0.180	0.254
(unit: million tests per second)
(CPU: Intel(R) Core(TM) i7-8700B CPU @ 3.20GHz)


(All benchmarks are single-thread.)

As we can see, this library is 50-200% faster than gmpy2 in addition to being deterministic for unsigned 64-bit integers, depending on CPU. For integers just above 64 bits, depending on CPU this library may be up to 20% slower than gmpy2.is_prime at 10 rounds, but the gap is closed as numbers get larger, and eventually this library is faster again.

Note that for 64-bit unsigned integers, there is a pure Python implementation in alt/miller_rabin.py as a demonstration (actually, it still uses gmpy2's mpz type for modular exponentiation, so it's not pure Python strictly speaking; the reason is that CPython's long_pow can be >20x slower than GMP's mpz_powm even just for unsigned 64-bit integers). It is way slower than this library, so a C extension is indeed necessary.

## Development

Argument handling code is automatically generated by Argument Clinic from the latest v3.6.x release tree (for compatibility).

$cd /path/to/cpython/dev/tree$ git checkout v3.6.x
\$ python3 Tools/clinic/clinic.py -f /path/to/miller-rabin/src/miller-rabin.c


## Contributing

Contributions are welcome. Algorithmic changes should demonstrate measurable performance improvements (using bench/benchmark.py).

Ideas:

• Maybe a Montgomery multiplication implementation could be faster than mpz_powm? Perl's Math::Prime::Util implements Montgomery multiplication in montmath.h and uses it for Miller-Rabin, but the implementation is in x64 asm which I'm not comfortable with (could be necessary though), and the code is unfortunately GPL.

• I'm not too keen on figuring out static wheel building on Windows, so contribution from experienced Windows developer is welcome here. See .github/workflows/build-and-publish-distributions.yml.

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