GpuOwl

GpuOwl is a Mersenne primality tester for AMD, Nvidia and Intel GPUs supporting OpenCL.

If you are making source code changes to GpuOwl, please read the code style

Mersenne primes

Mersenne numbers are numbers of the form 2^p -1. Some of these are prime numbers, called Mersenne primes.

The largest known Mersenne primes are huge numbers. They are extremely difficult to find, and discovering a new Mersenne prime is a noteworthy achievement. A long-standing distributed computing project named the Great Internet Mersenne Prime Search (GIMPS) has been searching for Mersenne primes for the last 30 years.

While traditionally the algorithms involved were implemented targeting CPUs, the GPUs have seen increased usage in computing recently because of their impressive power and wide memory bandwidth, which are advantages relative to CPUs.

GpuOwl is an implementation of some of the algorithms involved in searching for Mersenne primes in the OpenCL language for execution on modern AMD, Nvidia and Intel GPUs. GpuOwl runs best on top of the ROCm OpenCL stack.

Mersenne primality tests

These are the main test involved in Mersenne prime search:

• TF, Trial Factoring
• P-1, Pollard's p-1 factoring
• LL, Lucas-Lehmer primality test
• PRP, probable prime test

Trial Factoring (TF)

In this test, prime factors of increasingly larger magnitude are tried, checking if they divide the Mersenne candidate M(p). TF is good as a first line of attack, representing a cheap filter that removes some Mersenne candidates by finding a factor (thus deciding that the M(p) is not prime). The limitation of TF is that the checking effort grows exponentially with the size of the factors that are trialed, thus TF remains just a “first line of attack” approach.

Pollard's P-1 factoring (P-1)

This is a very ingenious, beautiful algorithm for finding factors of Mersenne candidates. It detects a special class of factors F where F-1 is highly composite (has many factors). P-1 is used as a preliminary filter (much like TF), that removes some Mersenne candidates, proving them composite by finding a factor.

Lucas-Lehmer (LL)

This is a test that proves whether a Mersenne number is prime or not, but without providing a factor in the case where it is not prime. The Lucas-Lehmer test is very simple to describe: iterate the function f(x)=(x^2 - 2) modulo M(p) starting with the number 4. If after p-2 iterations the result is 0, then M(p) is certainly prime, otherwise M(p) is certainly not prime.

Lucas-Lehmer, while a very efficient primality test, still takes a rather long time for large Mersenne numbers (on the order of weeks of intense compute), thus it is only applied to the Mersenne candidates that survived the cheaper preliminary filters TF and P-1.

PRP (“the new LL”)

The probable prime test can prove that a candidate is composite (without providing a factor), but does not prove that a candidate is prime (only stating that it probably is prime) -- although in practice the difference between probable prime and proved prime is extremely small for large Mersenne candidates.

The PRP test is very similar computationally to LL: PRP iterates f(x) = x^2 modulo M(p) starting from 3. If after p iterations the result is 9 modulo M(p), then M(p) is probably prime, otherwise M(p) is certainly not prime. The cost of PRP is exactly the same as LL.

In practice, PRP is preferred over LL because PRP does have a very strong and useful error-checking technique, which protects effectively against computation errors (which are sometimes common on GPUs).

GpuOwl: OpenCL GPU Mersenne primality testing

GpuOwl implements the PRP and P-1 tests. It also implemented, at various points in the past, LL and TF but these are not active now in GpuOwl. For double check (DC) LL tests, see the v6 branch (version 6.11-382) and for first time LL tests, see the LL branch (version 0.6).

Let us consider the PRP test, to get an idea of what GpuOwl does under the hood.

PRP uses what is called a modular squaring, computing f(x) = x^2 modulo M(p), starting from 3 (where x is an integer).

The problem is in the size of the integer x that is to be squared, which is about 100 million bits in size.

How do we compute efficiently the square of a 100 million bits integer? It turns out that one of the fastest multiplication algorithms for huge numbers consists in doing a convolution, which involves a direct and an inverse FFT transform, with a simple element-wise multiplication in the FFT domain.

And this is exactly what GpuOwl does: it implements, as building blocks, efficient huge FFT transforms. Many algorithmic tricks are also used to speed up computation, e.g. the “Irrational Base Discrete Weighted Transform” (IBDWT) described by Richard Crandall.

Files used by gpuOwl

• worktodo.txt : contains exponents to test, one entry per line
• results.txt : contains the results
• N.owl : the most recent checkpoint for exponent ; will resume from here
• N-prev.owl : the previous checkpoint, to be used if N.ll is lost or corrupted
• N.iteration.owl : a persistent checkpoint at the given iteration

worktodo.txt

The lines in worktodo.txt must be of one of these forms:

• 70100200
• PRP=1,2,77936867,-1,75,0
• PRP=N/A,1,2,77936867,-1,75,0
• PRP=FCECE568118E4626AB85ED36A9CC8D4F,1,2,77936867,-1,75,0

The first form indicates just the exponent to test, while the form starting with PRP= indicates the exponent and optionally the assignment ID (AID) from PrimeNet. The PRPDC= prefix can be used instead for PRP DC assignments.

Usage

• Get "PRP smallest available first time tests" assignments from GIMPS Manual Testing ( https://www.mersenne.org/manual_assignment/ ).
• Copy the assignment lines from GIMPS to a file named 'worktodo.txt'
• Run gpuowl. It prints progress report on stdout and in gpuowl.log, and writes result lines to results.txt
• Submit the result lines from results.txt to https://www.mersenne.org/manual_result/ manual testing.

Build

Prerequisites (please install these):

• the GNU Multiple Precision (GMP) 6.1 library libgmp-dev
• a C++20 compiler (e.g. GCC, Clang)
• an OpenCL implementation (which provides the libOpenCL library). Recommended: an AMD GPU with ROCm 1.7.

Meson build

Example build steps on linux:

cd gpuowl
mkdir build
cd build
meson ..
ninja

What the previous commands do:

• go to gpuowl source directory
• create a subdirectory named "build"
• go to the build directory
• invoke meson, passing as argument the gpuowl source directory (.. in this situation)
• run ninja to build

Make build

To build simply invoke "make" (or look inside the Makefile for a manual build).

See "gpuowl -h" for the command line options.

Self-test

Simply start GpuOwl with any valid exponent, and the built-in error checking kicks in, validating the computation. If you start seeing output lines with "OK", than it is working correctly. "EE" lines indicate computation errors.

Command-line Arguments

-dir <folder>      : specify local work directory (containing worktodo.txt, results.txt, config.txt, gpuowl.log)
-pool <dir>        : specify a directory with the shared (pooled) worktodo.txt and results.txt
                     Multiple GpuOwl instances, each in its own directory, can share a pool of assignments and report
                     the results back to the common pool.
-uid <unique_id>   : specifies to use the GPU with the given unique_id (only on ROCm/Linux)
-user <name>       : specify the user name.
-cpu  <name>       : specify the hardware name.
-time              : display kernel profiling information.
-fft <spec>        : specify FFT e.g.: 1152K, 5M, 5.5M, 256:10:1K
-block <value>     : PRP error-check block size. Must divide 10'000.
-log <step>        : log every <step> iterations. Multiple of 10'000.
-carry long|short  : force carry type. Short carry may be faster, but requires high bits/word.
-B1                : P-1 B1 bound
-B2                : P-1 B2 bound
-rB2               : ratio of B2 to B1. Default 20, used only if B2 is not explicitly set
-prp <exponent>    : run a single PRP test and exit, ignoring worktodo.txt
-verify <file>     : verify PRP-proof contained in <file>
-proof <power>     : By default a proof of power 8 is generated, using 3GB of temporary disk space for a 100M exponent.
                     A lower power reduces disk space requirements but increases the verification cost.
                     A proof of power 9 uses 6GB of disk space for a 100M exponent and enables faster verification.
-autoverify <power> : Self-verify proofs generated with at least this power. Default 9.
-tmpDir <dir>      : specify a folder with plenty of disk space where temporary proof checkpoints will be stored.
-results <file>    : name of results file, default 'results.txt'
-iters <N>         : run next PRP test for <N> iterations and exit. Multiple of 10000.
-maxAlloc <size>   : limit GPU memory usage to size, which is a value with suffix M for MB and G for GB.
                     e.g. -maxAlloc 2048M or -maxAlloc 3.5G
-save <N>          : specify the number of savefiles to keep (default 12).
-noclean           : do not delete data after the test is complete.
-from <iteration>  : start at the given iteration instead of the most recent saved iteration
-yield             : enable work-around for Nvidia GPUs busy wait. Do not use on AMD GPUs!
-nospin            : disable progress spinner
-use NEW_FFT8,OLD_FFT5,NEW_FFT10: comma separated list of defines, see the #if tests in gpuowl.cl (used for perf tuning)
-unsafeMath        : use OpenCL -cl-unsafe-math-optimizations (use at your own risk)
-binary <file>     : specify a file containing the compiled kernels binary
-device <N>        : select a specific device:

Device numbers start at zero.

Primenet.py Arguments

-h, --help            show this help message and exit
-u USERNAME           Primenet user name
-p PASSWORD           Primenet password
-t TIMEOUT            Seconds to sleep between updates
--dirs DIR [DIR ...]  GpuOwl directories to scan
--tasks NTASKS        Number of tasks to fetch ahead
-w {PRP,PM1,LL_DC,PRP_DC,PRP_WORLD_RECORD,PRP_100M}   GIMPS work type