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goprime

goprime is a software that can perform a Lucas-Lehmer-Riesel primality test for numbers of the form h*2n-1.

Motivation

The main motivations for why goprime has been written are:

  • Implement, evaluate, compare and document all the algorithms involved in the LLR test.
  • Estimate the time each substep of the LLR test and a full LLR test would take for any given number.

goprime is open source to serve as a generic learning base for all those interested in understanding how the LLR test works. It can be used also for finding new prime numbers, but finding new prime numbers is not the purpose for why it was written (there currently is much better software for that purpose).

Results

Generating V(1)

Generating V(1) is the fastest substep, but at the same time the most difficult to understand part of the LLR test. We implemented three different algorithms to generate V(1) (Riesel, Rödseth and Penné).

In general, we found the Rödseth algorithm to be the most straightforward to implement and we recommend to use it, given that it performs well in comparison with the other methods.

Further details may be found in our code comments.

Generating U(2)

Generating U(2) = V(h) requires to compute approximately log2(h) terms of the {Vi} sequence. Each iteration of this substep works by computing V(2x+1) and V(2x) until we reach V(h).

We found out that, during every iteration, the operations of computing V(2x+1) and V(2x) can be easily parallelized and do not need to be done sequentially. Implementing this optimization reduced the total computation time of this substep of about 50%.

Generating U(n)

Generating U(n) is the most time consuming substep of the LLR test, as it requires to compute n terms of the {Ui} sequence where each term depends on the previous term (which makes it hard to parallelize).

We evaluated the speed of this substep, with a special focus on comparing the time required to square large numbers with three different libraries (Go math/big, FLINT and GMP). The squaring routine is the most crucial part of this substep, because it is where most of the computation time is spent.

It appears that, in squaring large numbers, GoLang math/big is very slow, and that FLINT is slightly faster than GMP.

You may wish to explore other squaring solutions. We expect that approaches based on Crandall's transform, George Woltman's Gwnums library, Colin Percival paper or hardware-specific hand tuned code (such as using C with inline assembly to access special hardware instructions) can achieve results at least one order of magnitude faster than what we observed so far.

Usage

Goprime

# Download and install goprime
$ go get github.com/arcetri/goprime

# Run goprime with any h and n
$ goprime 391581 216193

If you have errors with these commands, check that you have GoLang (at least v6) installed and configured with:

# Set the $GOPATH and add the $GOPATH/bin to the PATH environment variable if not already done.
$ export GOPATH=$HOME/go
$ export PATH=$PATH:$GOPATH/bin

NOTE: goprime, by default, uses the Go math/big library, which is slow. For information on how to make it use a faster library, read the "Advanced" section below.

Goprime-c

goprime-c is a C translation of the goprime software. In our tests, it appeared that goprime-c is about 1.5% faster than the corresponding GoLang code of goprime (when both are set to use the FLINT lib - see below for details on that).

NOTE: goprime-c requires the FLINT library to be installed in your system.

# Download and install goprime
$ go get github.com/arcetri/goprime
$ cd $GOPATH/src/github.com/arcetri/goprime/c
$ make install

# Run goprime with any h and n
$ goprime-c 391581 216193

Future work

  • Evaluate other methods to perform the squaring in the "Generating U(n)" substep.
  • Add correctness checks to be regularly performed during the "Generating U(n)" substep.
  • Add checkpoints to be regularly saved during the "Generating U(n)" substep.
  • Improve the goprime-c code using the goprime code and comments as an example.

Advanced

NOTE: read this section only if you are willing to work on this project. Should you still have questions after reading it, please feel free to contact us.

To change the multiplication algorithm that goprime uses, one can use a provided script:

$ cd rieseltest
$ ./change_multiplication_algorithm.sh <library name>

Currently goprime supports the libraries big, gmp and flint. However, using the flint or the gmp libraries for long tests might cause the system to start swapping.

We previously "fixed" this issue in the timings branch of this repository, which also contains some experimental code we used for our experiments before switching to goprime-c. However, our "fix" broke the compatibility with the default math/big library.

Due to time-constraints, and in order not to make the simple code of goprime harder to understand, we decided to write goprime-c, which currently uses flint (the fastest library in our experiments) and do the timings from there.

Thus, if you want to work on the go version of this project, beware that there are some "bindings bug" that still need to be fixed.

NOTE: if you wish to work on this project, we also recommend that you install the go gvt tool and use it to manage the dependencies that are currently vendored in the "vendor" folder.

Contribute

Please feel invited to contribute by creating a pull request to submit the code or bug fixes you would like to be included in goprime.

You can also contact us using the following email address: goprime-contributors (at) external (dot) cisco (dot) com. If you send us an email, please include the phrase "goprime author question" somewhere in the subject line or your email may be rejected.

License

This project is distributed under the terms of the Apache License v2.0. See file "LICENSE" for further reference.