Security Estimates for Lattice Problems

This Sage module provides functions for estimating the concrete security of 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

  >>> 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.

• Read the documentation.

Status

We have feature parity with the old estimator:

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

but we are also planning:

• [ ] attacks on NTRU public keys (using overstretched parameters)
• [ ] attack on SIS 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.

Contributions

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

• Benjamin Curtis
• Cedric Lefebvre
• Fernando Virdia
• Florian Göpfert
• Hamish Hunt
• 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.

License

The estimator is licensed under the LGPLv3+ 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.