Fast Library for Number Theory

FLINTOriginal author(s)| William Hart, David Harvey, and others

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Developer(s)| Fredrik Johansson, Albin Ahlbäck (maintainers), and others
Stable release|

3.4.0 / 25 November 2025; 5 months ago (2025-11-25)

Written in| C
Operating system| POSIX systems
Type| Mathematical software
License| GNU Lesser General Public License
Website| flintlib.org

The Fast Library for Number Theory (FLINT) is a C library for number theory applications. It implements efficient versions of various ring arithmetics as well as derived functionality such as integer factorization using a quadratic sieve. The library is designed to be compiled with the GNU Multi-Precision Library (GMP) and is released under the GNU General Public License. It is developed by William Hart of the University of Kaiserslautern (formerly University of Warwick) and David Harvey of University of New South Wales (formerly Harvard University) to address the speed limitations of the PARI and NTL libraries.[1]

FLINT along with a Cython wrapper for it is distributed with SageMath. The development of FLINT has led to significant contributions in the areas of integer factorization and polynomial arithmetic. For example as of May, 2007 on certain platforms FLINT factors integers in the quadratic sieve range faster than any other general implementation[2], and as of February, 2008 it does arithmetic in ℤ[x] faster than any other package.[3][4]

## Contents

* 1 Functionality
* 2 Use in research
* 3 References
* 4 Further reading

## Functionality

Core:

* Ring arithmetic (integers, rationals, algebraic, real, complex and p-adic numbers, finite fields, and number fields)
* Exact real numbers (formerly the separate Calcium library)
* Arbitrary-precision approximate real numbers with ball arithmetic (formerly the separate Arb library)[5]
* Polynomials (univariate and multivariate), power series, and matrices building on top of ring arithmetics.

Based on core:

* Primality testing (APRCL, Miller-Rabin, BPSW, etc.)
* Integer factorization (trial factor, quadratic sieve, Pollard's rho, Lenstra ECM)
* Multivariate polynomial GCD and factorisation
* FFTs
* Multimodular reconstruction
* Special functions
* Exact and approximate linear algebra
* LLL
* Finite field embeddings
* ... and more.

## Use in research

FLINT maintains a webpage listing peer-reviewed papers, preprints, and other academic products making use of its functionality.[5]

## References

1. ↑ Page 10 of the Project Description for UW 2006 COMPMATH Proposal by William Stein
2. ↑ Bottom of page 11 of Abstract of Talks, Illinois Number Theory Fest 2007
3. ↑ Slide 25 of the talk Numbers in Sage (February 29, 2008) by William Stein
4. ↑ Efficiently computing Bernoulli numbers using FLINT by David Howden
5. ↑ 5.0 5.1 "FLINT: Fast Library for Number Theory - Applications & benchmarks". https://flintlib.org/applications.html.

## Further reading

* FLINT 1.0.9: Fast Library for Number Theory by William Hart and David Harvey

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