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.
We currently provide evaluators for the security of the LWE, NTRU, and SIS problems. Our estimator integrates simulators for the best known attacks against these problems, and provides bit-security estimates relying on heuristics to predict the cost and shape of lattice reduction algorithms. The default models are configured in conf.py.
It is possible to evaluate attacks cost individually, or using the helper functions:
*.estimate.rough: fast routine that evaluates the security of the problem only against the usually most efficient attacks. Note that it uses a non-default cost model for lattice reduction, most often used in the literature for ease of comparison, and will thus return different numbers than the rest of the API. Refer to its documentation for details.*.estimate: extended routine that evaluates the security of the problem against all supported attacks. This uses the default cost and shape model for lattice reduction.
Usage examples:
>>> 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: ≈2^115.5, red: ≈2^115.3, guess: ≈2^112.3, β: 395, p: 5, ζ: 0, t: 40, β': 395, N: ≈2^81.4, m: 512 >>> 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 dual :: rop: ≈2^149.9, mem: ≈2^97.1, m: 512, β: 424, d: 1024, ↻: 1, tag: dual dual_hybrid :: rop: ≈2^139.7, red: ≈2^139.6, guess: ≈2^135.9, β: 387, p: 5, ζ: 0, t: 50, β': 391, N: ≈2^81.1, m: 512>>> from estimator import * >>> schemes.Dilithium2_MSIS_WkUnf SISParameters(n=1024, q=8380417, length_bound=350209, m=2304, norm=+Infinity, tag='Dilithium2_MSIS_WkUnf') >>> r = SIS.estimate.rough(schemes.Dilithium2_MSIS_WkUnf) lattice :: rop: ≈2^123.5, red: ≈2^123.5, sieve: ≈2^-332.2, β: 423, η: 423, ζ: 1, d: 2303, prob: 1, ↻: 1, tag: infinity >>> r = SIS.estimate(schemes.Dilithium2_MSIS_WkUnf) lattice :: rop: ≈2^152.2, red: ≈2^151.3, sieve: ≈2^151.1, β: 427, η: 433, ζ: 0, d: 2304, prob: 1, ↻: 1, tag: infinity>>> from estimator import * >>> schemes.Falcon512_SKR NTRUParameters(n=512, q=12289, Xs=D(σ=4.05), Xe=D(σ=4.05), m=512, tag='Falcon512_SKR', ntru_type='circulant') >>> r = NTRU.estimate.rough(schemes.Falcon512_SKR) usvp :: rop: ≈2^140.5, red: ≈2^140.5, δ: 1.003499, β: 481, d: 544, tag: usvp >>> r = NTRU.estimate(schemes.Falcon512_SKR) usvp :: rop: ≈2^165.1, red: ≈2^165.1, δ: 1.003489, β: 483, d: 1020, tag: usvp bdd :: rop: ≈2^160.6, red: ≈2^159.6, svp: ≈2^159.6, β: 463, η: 496, d: 1022, tag: bdd bdd_hybrid :: rop: ≈2^160.6, red: ≈2^159.6, svp: ≈2^159.6, β: 463, η: 496, ζ: 0, |S|: 1, d: 1024, prob: 1, ↻: 1, tag: hybrid bdd_mitm_hybrid :: rop: ≈2^349.3, red: ≈2^349.3, svp: ≈2^204.8, β: 481, η: 2, ζ: 0, |S|: 1, d: 1024, prob: ≈2^-182.6, ↻: ≈2^184.8, tag: hybrid >>> schemes.Falcon512_Unf SISParameters(n=512, q=12289, length_bound=5833.9072, m=1024, norm=2, tag='Falcon512_Unf') >>> r = SIS.estimate.rough(schemes.Falcon512_Unf) lattice :: rop: ≈2^121.2, red: ≈2^121.2, δ: 1.003882, β: 415, d: 1024, tag: euclidean >>> r = SIS.estimate(schemes.Falcon512_Unf) lattice :: rop: ≈2^146.4, red: ≈2^146.4, δ: 1.003882, β: 415, d: 1024, tag: euclidean
We cover:
[x]:doc:`primal attacks on LWE <../algorithms/lwe-primal>`
[X][x][X][x]
We are planning:
[ ]attack on SIS instances
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.
Please report bugs through the GitHub issue tracker.
At present, this estimator is maintained by Martin Albrecht. Contributors are:
- Benjamin Curtis
- Cathie Yun
- Cedric Lefebvre
- Fernando Virdia
- Florian Göpfert
- Hamish Hunt
- Hunter Kippen
- James Owen
- Léo Ducas
- Ludo Pulles
- Markus Schmidt
- Martin Albrecht
- Michael Walter
- Rachel Player
- Sam Scott
See Contributing for details on how to contribute.
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.
The estimator is licensed under the LGPLv3+ license.
- Zama's TFHE Compiler: Concrete.
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, by Zama and by SandboxAQ.