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Python implementation of Schoof's algorithm for counting the points on elliptic curves over finite fields

Project description

Python3 Implementation of Schoof's Algorithm
============================================

This is an implementation of Schoof's algorithm for counting the
points on elliptic curves over finite fields (Schoof, René. Elliptic
curves over finite fields and the computation of square roots mod
p. *Mathematics of Computation*, 44(170):483–494, 1985).

Elliptic curve cryptographic algorithms base their security guarantees
on the number of points on the used elliptic curve. Because naive
point counting is infeasible, having a fast counting algorithm is
important to swiftly decide whether a curve is safe to use in
cryptography or not. René Schoof's algorithm for counting the points
on an elliptic curve over a finite field is the foundation for the
(asymptotically) fastest [Schoof–Elkies–Atkin][sea-algorithm] counting
algorithm.


**Implementation.**
The implementation is written in [Python 3][python] and is kept as
simple as possible. Its goal is to give insight into the mathematics
of the algorithm without the use of (too) high-level concepts. For a
(pretty) gentle introduction into why and how Schoof's algorithm
works, please read my diploma thesis titled
[*An Elementary Derivation and Implementation of Schoof's Algorithm for Counting Points on Elliptic Curves*][thesis].
In the thesis, I try to explain *how one might get the idea for the
algorithm*. This understanding of *why* things look the way they do
is often neglected in mathematics.

Schoof's algorithm uses arithmetic on elliptic curves, finite fields,
rings of polynomials, and quotient rings. This repository therefore
contains Python modules for all these concepts that can be used on
their own.


System Requirements
-------------------

The algorithm is implemented in version 3.1 of [Python][python], an
open licensed dynamic programming language available on all common
platforms. To find out whether a compatible version of Python is
already installed on your system, execute `python --version` in a
terminal. The command will return the version number if Python is
available. Note that the version 3 interpreter could also be named
`python3`. Please see the *Using Python* part of Python's
[documentation][python-using] for system installation instructions;
follow the steps below to set up Python locally in your account.


### Local Installation from Source Code

Installing Python from source code requires a C compiler; on Linux and
Unix systems, one is almost always available. The following steps
install Python on a Linux system:

* **Download.** Download the source tar ball of version 3.1 or later
from the Python website at
<http://www.python.org/download/releases/>.
* **Compile.** Open a terminal and create a temporary directory, say
`${HOME}/tmp/`, by executing `mkdir ${HOME}/tmp/`. Change into the
temporary directory and extract the source tar ball: `cd
${HOME}/tmp/` and then `tar xzvf Python-3.1.2.tgz`; adjust the path
and file name accordingly. If you downloaded the bzipped source tar
ball, use `tar xjvf Python-3.1.2.tar.bz2` instead.

Next, change into the directory that contains the extracted source
code, for instance `${HOME}/tmp/Python-3.1.2/`. Configure the build
system by executing `./configure --prefix=${HOME}/python3`. The
prefix is the path that will be the root of the Python installation,
so adjust it to taste. In case required components are missing, the
command will exit with an error message. In this case, please
install the components and re-execute `configure`.

If everything worked, then the last line of output by `configure`
will be `creating Makefile`. To start the compilation, execute
`make`.
* **Install.** Use `make install` to install Python after the
compilation finished.
* **Set Up Environment.** To enable the local Python installation, add
its interpreter and modules to the respective search paths: execute
`export PATH=${HOME}/python3/bin:${PATH}` to tell the shell where to
find the `python3` interpreter; adjust the path to your prefix for
`configure`. Likewise, execute `export
PYTHONPATH=${HOME}/python3/lib/python3.1` to tell Python where to
find its modules.

Note that the scope of `export` is the current shell. Thus you have
to issue both commands in every freshly opened terminal you wish to
use for Python 3.1 programs.


Program Execution
-----------------

The implementations work without any installation; they may be
executed directly from the checked out repository. However, they
expect a set up Python 3.1 run-time environment as explained above.

The root directory contains the point counting programs: the file
`naive_schoof.py` is the implementation discussed in
\autoref{sec:implementation}; the file `reduced_computation_schoof.py`
is the version with better constants using a smarter computation order
and Hasse's theorem. Curves for counting are specified as
space-separated triples *p*, *A*, and *B*: *p* is the prime size of
the galois field *GF[p]*, and *A* and *B* are the curve parameters.

### Example

Suppose you want to count the number of points on the elliptic curve
over *GF[23]* with parameters *A=4* and *B=2*. If the current
directory in the terminal is the repository root, then
executing `python3 naive_schoof.py 23 4 2` yields the output

~~~~~
Counting points on y^2 = x^3 + 4x + 2 over GF<23>: 21
~~~~~


### Command Line Parameters

The program supports the command line options described below, which
for instance allow to read the curves from a file, or to create
execution profiles for performance analysis.

* `--version`: Show program's version number and exit.
* `-h`, `--help`: Show a help message and exit.
* `-i` *f*, `--input-file=`*f*: Read the curve parameters from file
*f*. Lines in *f* must contain either a comment, or a
space-separated triple of curve parameters *p*, *A*, and *B*.
Comment lines start with a hash (`#`); these and empty lines are
skipped.
* `-o` *f*, `--output-file=`*f*: Write the results to file *f* instead
of showing them on the terminal. Each line of input generates one
corresponding line of output.
* `-t` *s*, `--timelimit=`*s*: Terminate the program if processing an
input triple takes longer than *s* seconds. The program ends if the
time limit expires; no subsequent lines will be read from an input
file. Thus, sort the parameters in input files ascending in length
of the prime *p* to avoid triggering the time limit too early.
* `-p`, `--create-profile`: Create an execution profile for each
processed input triple. The profile consists of a call profile
generated with the `cProfile` Python module, and a file with data
about elapsed time and used memory.
* `-d` *d*, `--profile-directory=`*d*: If profiling is enabled with
the `-p` flag, then the collected data is written to the directory
*d*. The default value is the current directory.


Profiling Tools
---------------

The `tools/` directory contains several programs to post-process
profiles resulting from a set `-p` flag. For example, it includes a
call profile converter that outputs [Callgrind][callgrind] files. Use
[KCacheGrind][kcachegrind] to interactively inspect Callgrind files.


Further Documentation
---------------------

The implementation comes with extensive [API documentation][api] that
explains the purpose and usage of all classes.

Furthermore, the directory `_test` contains unit tests to verify
correct arithmetic. The unit tests are designed to be easily extended
to further implementations of the same objects. Thus, mistakes in new
designs should be simpler to locate. To run the unit tests, execute
`python3 -m _test` in the repository root directory.


License
-------

Copyright (c) 2010--2012 Peter Dinges <pdinges@acm.org>.

The software in this repository is free software: you can redistribute
it and/or modify it under the terms of the GNU General Public License
as published by the Free Software Foundation, either version 3 of the
License, or (at your option) any later version.

The software is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
General Public License for more details.

You should have received a copy of the [GNU General Public License][gpl3]
along with this program. If not, see <http://www.gnu.org/licenses/>.


[api]: http://pdinges.github.com/python-schoof
[callgrind]: http://valgrind.org/info/tools.html "Callgrind is part of the Valgrind tool suite"
[gpl3]: http://opensource.org/licenses/GPL-3.0 "GNU General Public License, version 3"
[kcachegrind]: http://kcachegrind.sourceforge.net/html/Home.html "Interactive viewer for Callgrind files."
[python]: http://python.org "Python Programming Language"
[python-using]: http://docs.python.org/py3k/using/index.html "Python Setup and Usage"
[sea-algorithm]: http://en.wikipedia.org/wiki/Schoof%E2%80%93Elkies%E2%80%93Atkin_algorithm "Schoof-Elkies-Atkin algorithm for counting points on elliptic curves over finite fields."
[thesis]: http://www.elwedgo.de/fileadmin/elwedgo.de/portfolio/diploma_thesis_math/dinges-elementary_schoof_derivation-thesis.pdf

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