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 *
>>> schemes.Kyber512
LWEParameters(n=512, q=3329, Xs=D(σ=1.22), Xe=D(σ=1.22), m=512, tag='Kyber 512')

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

>>> r = LWE.estimate.rough(schemes.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(schemes.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
Cathie Yun
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.

Third Party Tools Using this Estimator

Zama's TFHE Compiler: Concrete.

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.

An00001/lattice-estimator