Security Estimates for Lattice Problems

.. image:: https://mybinder.org/badge_logo.svg :target: https://mybinder.org/v2/gh/malb/lattice-estimator/jupyter-notebooks?labpath=..%2F..%2Ftree%2Fprompt.ipynb .. image:: https://readthedocs.org/projects/lattice-estimator/badge/?version=latest :target: https://lattice-estimator.readthedocs.io/en/latest/?badge=latest :alt: Documentation Status

This Sage <http://sagemath.org>__ module provides functions for estimating the concrete security of Learning with Errors <https://en.wikipedia.org/wiki/Learning_with_errors>__ instances.

The main purpose of this estimator is to give designers an easy way to choose parameters resisting known attacks and to enable cryptanalysts to compare their results and ideas with other techniques known in the literature.

Quick Start

• Usage

  .. code-block:: python

  from estimator import * Kyber512 LWEParameters(n=512, q=3329, Xs=D(σ=1.22), Xe=D(σ=1.22), m=512, tag='Kyber 512')

  LWE.primal_usvp(Kyber512) rop: ≈2^143.8, red: ≈2^143.8, δ: 1.003941, β: 406, d: 998, tag: usvp

  r = LWE.estimate.rough(Kyber512) usvp :: rop: ≈2^118.6, red: ≈2^118.6, δ: 1.003941, β: 406, d: 998, tag: usvp dual_hybrid :: rop: ≈2^121.9, mem: ≈2^116.8, m: 512, β: 417, d: 1013, ↻: 1, ζ: 11, tag: dual_hybrid

  r = LWE.estimate(Kyber512) bkw :: rop: ≈2^178.8, m: ≈2^166.8, mem: ≈2^167.8, b: 14, t1: 0, t2: 16, ℓ: 13, #cod: 448, #top: 0, #test: 64, tag: coded-bkw usvp :: rop: ≈2^143.8, red: ≈2^143.8, δ: 1.003941, β: 406, d: 998, tag: usvp bdd :: rop: ≈2^140.3, red: ≈2^139.7, svp: ≈2^138.8, β: 391, η: 421, d: 1013, tag: bdd bdd_hybrid :: rop: ≈2^140.3, red: ≈2^139.7, svp: ≈2^138.8, β: 391, η: 421, ζ: 0, |S|: 1, d: 1016, prob: 1, ↻: 1, tag: hybrid bdd_mitm_hybrid :: rop: ≈2^260.3, red: ≈2^259.4, svp: ≈2^259.3, β: 405, η: 2, ζ: 102, |S|: ≈2^247.2, d: 923, prob: ≈2^-113.8, ↻: ≈2^116.0, tag: hybrid dual :: rop: ≈2^149.9, mem: ≈2^88.0, m: 512, β: 424, d: 1024, ↻: 1, tag: dual dual_hybrid :: rop: ≈2^145.6, mem: ≈2^140.5, m: 512, β: 408, d: 1004, ↻: 1, ζ: 20, tag: dual_hybrid

• Try it in your browser <https://mybinder.org/v2/gh/malb/lattice-estimator/jupyter-notebooks?labpath=..%2F..%2Ftree%2Fprompt.ipynb>__.

• Read the documentation <https://lattice-estimator.readthedocs.io/en/latest/>__.

Status

We have feature parity with the old estimator <https://bitbucket.org/malb/lwe-estimator/src/master/>__:

• [x] |lwe-primal-binder| :doc:primal attacks on LWE <../algorithms/lwe-primal>
• [X] |lwe-dual-binder| :doc:dual attacks on LWE <../algorithms/lwe-dual>
• [x] |lwe-bkw-binder| :doc:Coded-BKW attack on LWE <../algorithms/lwe-bkw>
• [X] |gb-binder| :doc:Arora-GB attack on LWE <../algorithms/gb>

.. |lwe-primal-binder| image:: https://mybinder.org/badge_logo.svg :target: https://mybinder.org/v2/gh/malb/lattice-estimator/jupyter-notebooks?labpath=..%2F..%2Ftree%2Flwe-primal.ipynb

.. |lwe-dual-binder| image:: https://mybinder.org/badge_logo.svg :target: https://mybinder.org/v2/gh/malb/lattice-estimator/jupyter-notebooks?labpath=..%2F..%2Ftree%2Flwe-dual.ipynb

.. |lwe-bkw-binder| image:: https://mybinder.org/badge_logo.svg :target: https://mybinder.org/v2/gh/malb/lattice-estimator/jupyter-notebooks?labpath=..%2F..%2Ftree%2Flwe-bkw.ipynb

.. |gb-binder| image:: https://mybinder.org/badge_logo.svg :target: https://mybinder.org/v2/gh/malb/lattice-estimator/jupyter-notebooks?labpath=..%2F..%2Ftree%2Fgb.ipynb

but we are also planning:

• [ ] attacks on NTRU <https://en.wikipedia.org/wiki/NTRU>__ public keys (using overstretched parameters)
• [ ] attack on SIS <https://en.wikipedia.org/wiki/Short_integer_solution_problem>__ instances

Evolution

This code is evolving, new results are added and bugs are fixed. Hence, estimations from earlier versions might not match current estimations. This is annoying but unavoidable. We recommend to also state the commit that was used when referencing this project.

.. warning :: We give no API/interface stability guarantees. We try to be mindful but we may reorganize the code without advance warning.

Bugs

Please report bugs through the GitHub issue tracker <https://github.com/malb/lattice-estimator/issues>__.

Contributions

At present, this estimator is maintained by Martin Albrecht. Contributors are:

• Benjamin Curtis
• Cedric Lefebvre
• Fernando Virdia
• Florian Göpfert
• James Owen
• Léo Ducas
• Markus Schmidt
• Martin Albrecht
• Michael Walter
• Rachel Player
• Sam Scott

See :doc:Contributing <../contributing> for details on how to contribute.

Citing

If you use this estimator in your work, please cite

| Martin R. Albrecht, Rachel Player and Sam Scott. *On the concrete hardness of Learning with Errors*.
| Journal of Mathematical Cryptology. Volume 9, Issue 3, Pages 169–203, ISSN (Online) 1862-2984,
| ISSN (Print) 1862-2976 DOI: 10.1515/jmc-2015-0016, October 2015

A pre-print is available as

| Cryptology ePrint Archive, Report 2015/046, 2015. https://eprint.iacr.org/2015/046

An updated version of the material covered in the above survey is available in Rachel Player's PhD thesis <https://pure.royalholloway.ac.uk/portal/files/29983580/2018playerrphd.pdf>__.

License

The estimator is licensed under the LGPLv3+ <https://www.gnu.org/licenses/lgpl-3.0.en.html>__ license.

Acknowledgements

This project was supported through the European Union PROMETHEUS project (Horizon 2020 Research and Innovation Program, grant 780701), EPSRC grant EP/P009417/1 and EPSRC grant EP/S020330/1, and by Zama <https://zama.ai/>__.