PARI/GP

PARI/GP
Name	PARI/GP
Developer	Université Bordeaux I; CNRS; INRIA
Released	1985
Programming language	C; C++; Python
Operating system	Linux, macOS, Microsoft Windows
Genre	Computer algebra system
License	GNU General Public License

PARI/GP is a computer algebra system designed for fast computations in number theory, algebraic number fields, elliptic curves, and related areas. It originated in academic environments and is used by researchers and educators in institutions such as Université Bordeaux I, CNRS, and INRIA. The system provides a compiled library and an interactive interpreter, and it integrates with tools and languages common in mathematical research environments.

History

Development began in the mid-1980s at Université Bordeaux I under the direction of mathematicians connected to research networks that included collaborators at CNRS and later INRIA. Early work paralleled efforts by teams behind GNU Project, Free Software Foundation, and projects like SageMath and Maxima. Influences and integrations trace through collaborations with researchers associated with European Mathematical Society, American Mathematical Society, and conferences such as the International Congress of Mathematicians. Over time, contributions came from individuals with links to institutions including École Polytechnique, Université Paris-Sud, ETH Zurich, Princeton University, University of Cambridge, Harvard University, Massachusetts Institute of Technology, Stanford University, and research centers like Institut des Hautes Études Scientifiques.

Key milestones mirror developments in computational number theory achieved by figures connected to Andrew Wiles, work on Fermat's Last Theorem, advances in elliptic curve cryptography influenced by research at IBM and Bell Labs, and algorithmic improvements related to algorithms studied at CNRS nodes and workshops hosted by Fields Institute and Mathematical Sciences Research Institute. The project evolved alongside contemporaries such as Magma, Maple, and Mathematica.

Features and Capabilities

The system implements routines for algebraic number theory used in studies by authors affiliated with École Normale Supérieure, computations of modular forms explored at Institut Henri Poincaré, and explicit class field theory work connected to researchers at Heidelberg University. It provides fast arithmetic for integers and polynomials, support for RSA-related prime testing algorithms that echo research at Stanford University and Bell Labs, and utilities for exploring Modular curve computations of interest to teams from Princeton University and Harvard University. Capabilities include computations with elliptic curves relevant to projects at CERN and Microsoft Research, manipulation of algebraic numbers used in investigations at University of California, Berkeley and University of Tokyo, and special functions studied by groups at Max Planck Institute for Mathematics.

Advanced features support L-series calculations pertinent to research by scholars linked with Clay Mathematics Institute, class group and unit computations used in collaborations with Imperial College London, and cryptographic parameter exploration relevant to standards bodies such as NIST. Numeric and symbolic tools parallel functionality found in systems developed at University of Warwick and University of Illinois Urbana-Champaign.

Architecture and Implementation

The core is written in C with performance-oriented components leveraging techniques discussed in publications from ACM and IEEE. Memory management and multi-precision arithmetic follow paradigms also adopted in libraries like GNU Multiple Precision Arithmetic Library and algorithms explored at Bell Labs. The architecture exposes a library interface used by projects from SageMath contributors and bindings that allow integration with Python ecosystems developed by teams at PSL Research University and University of Washington.

Implementation choices reflect algorithmic developments presented at conferences such as Symposium on Theory of Computing and International Symposium on Symbolic and Algebraic Computation, where contributors from INRIA and CNRS disseminated optimizations. Cross-platform build systems accommodate environments used at Los Alamos National Laboratory and Sandia National Laboratories.

Usage and Interfaces

Users interact via the gp interpreter, command-line interfaces similar to tools used at Los Alamos National Laboratory, and graphical front ends inspired by interfaces from Wolfram Research and Maplesoft. Bindings and integration enable use within workflows at CERN and computational pipelines developed at NASA. Interfaces include options to embed the library in C/C++ programs used in academic projects at University of Michigan and to call routines from R and Python notebooks favored at Princeton University and Stanford University.

Educational use occurs in courses at École Polytechnique and Universidad de Buenos Aires, and researchers incorporate it into reproducible research environments promoted by Open Science Framework contributors and initiatives from CrossRef.

Development and Community

Development is coordinated by contributors affiliated with Université Bordeaux I, INRIA, and international collaborators from institutions like École Normale Supérieure, Heidelberg University, ETH Zurich, University of Cambridge, Princeton University, and Harvard University. The community exchanges patches and discussions similar to practices in projects hosted on platforms favored by GitHub and GitLab, with releases and issue tracking mirroring workflows used by Debian and Ubuntu maintainers. Workshops and coding sprints have been organized in venues such as Institute for Advanced Study, Fields Institute, and Mathematical Sciences Research Institute.

Documentation and tutorial contributions come from educators at University of Paris, University of Bordeaux, and contributors who have presented at meetings sponsored by European Mathematical Society and American Mathematical Society.

Reception and Applications

The software is cited in research across algebraic number theory, computational arithmetic, and cryptography by authors connected to Clay Mathematics Institute, Institute for Advanced Study, and university groups at Princeton University and Harvard University. Applied uses include cryptographic parameter testing relevant to standards discussions at NIST and algorithmic experimentation in projects collaborating with CERN and Microsoft Research. It appears in computational toolchains for published work from researchers at ETH Zurich, Imperial College London, University of Cambridge, and Massachusetts Institute of Technology. Reviews and comparisons with systems from Wolfram Research, Maplesoft, and Magma are common at conferences such as ISSAC and in journals associated with the American Mathematical Society.

Category:Computer algebra systems