# Overview

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# Security Estimates for the Learning with Errors Problem

This Sage module provides functions for estimating the concrete security of Learning with Errors instances.

The main intend 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.

## Usage Examples

sage: load("estimator.py") sage: n, alpha, q = Param.Regev(128) sage: costs = estimate_lwe(n, alpha, q) usvp: rop: ≈2^57.7, red: ≈2^57.7, δ_0: 1.009214, β: 102, d: 357, m: 228 dec: rop: ≈2^61.5, m: 229, red: ≈2^61.5, δ_0: 1.009595, β: 93, d: 357, babai: ≈2^46.8, babai_op: ≈2^61.9, repeat: 293, ε: 0.015625 dual: rop: ≈2^81.4, m: 376, red: ≈2^81.4, δ_0: 1.008810, β: 111, d: 376, |v|: 736.521, repeat: ≈2^19.0, ε: 0.003906

## Online

You can run the estimator online using the Sage Math Cell server.

## Coverage

At present the following algorithms are covered by this estimator.

- meet-in-the-middle exhaustive search
- Coded-BKW [C:GuoJohSta15]
- dual-lattice attack and small/sparse secret variant [EC:Albrecht17]
- lattice-reduction + enumeration [RSA:LinPei11]
- primal attack via uSVP [USENIX:ADPS16,ACISP:BaiGal14]
- Arora-Ge algorithm [ICALP:AroGe11] using Gröbner bases [EPRINT:ACFP14]

Above, we use cryptobib-style bibtex keys as references.

## Documentation

Documentation for the `estimator` is available here.

## 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 at present. We recommend to also state the commit that was used when referencing this project.

We also encourage authors to let us know if their paper uses this code. In particular, we thrive to tag commits with those cryptobib ePrint references that use it. For example, this commit corresponds to this ePrint entry.

## Contributions

Our intent is for this estimator to be maintained by the research community. For example, we encourage algorithm designers to add their own algorithms to this estimator and we are happy to help with that process.

More generally, all contributions such as bugfixes, documentation and tests are welcome. Please go ahead and submit your pull requests. Also, don’t forget to add yourself to the list of contributors below in your pull requests.

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

- Martin Albrecht
- Florian Göpfert
- Cedric Lefebvre
- James Owen
- Rachel Player
- Markus Schmidt
- Sam Scott
- Fernando Virdia

Please follow PEP8 in your submissions. You can use flake8 to check for compliance. We use the following flake8 configuration (to allow longer line numbers and more complex functions):

[flake8] max-line-length = 120 max-complexity = 16 ignore = E22,E241

## Bugs

If you run into a bug, please open an issue on bitbucket. Also, please check first if the issue has already been reported.

## 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.

## Parameters from the Literature

The following estimates for various schemes from the literature illustrate the behaviour of the
`estimator`. These estimates do not necessarily correspond to the claimed security levels of the
respective schemes: for several parameter sets below the claimed security level by the designers’ is
lower than the complexity estimated by the `estimator`. This is usually because the designers
anticipate potential future improvements to lattice-reduction algorithms and strategies. We
recommend to follow the designers’ decision. We intend to extend the estimator to cover these more
optimistic (from an attacker’s point of view) estimates in the future … pull requests welcome, as
always.

sage: load("estimator.py") sage: n = 1024; q = 12289; stddev = sqrt(16/2); alpha = alphaf(sigmaf(stddev), q) sage: _ = estimate_lwe(n, alpha, q, reduction_cost_model=BKZ.sieve) usvp: rop: ≈2^313.1, red: ≈2^313.1, δ_0: 1.002094, β: 968, d: 2101, m: 1076 dec: rop: ≈2^410.0, m: 1308, red: ≈2^410.0, δ_0: 1.001763, β: 1213, d: 2332, babai: ≈2^395.5, babai_op: ≈2^410.6, repeat: ≈2^25.2, ε: ≈2^-23.0 dual: rop: ≈2^355.5, m: 1239, red: ≈2^355.5, δ_0: 1.001884, β: 1113, repeat: ≈2^307.0, d: 2263, c: 1

sage: load("estimator.py") sage: n = 752; q = 2^15; stddev = sqrt(1.75); alpha = alphaf(sigmaf(stddev), q) sage: _ = estimate_lwe(n, alpha, q, reduction_cost_model=BKZ.sieve) usvp: rop: ≈2^173.0, red: ≈2^173.0, δ_0: 1.003453, β: 490, d: 1448, m: 695 dec: rop: ≈2^208.3, m: 829, red: ≈2^208.3, δ_0: 1.003064, β: 579, d: 1581, babai: ≈2^194.5, babai_op: ≈2^209.6, repeat: 588, ε: 0.007812 dual: rop: ≈2^196.2, m: 836, red: ≈2^196.2, δ_0: 1.003104, β: 569, repeat: ≈2^135.0, d: 1588, c: 1

sage: load("estimator.py") sage: n = 804; q = 2^31 - 19; alpha = sqrt(2*pi)*57/q; m = 4972 sage: _ = estimate_lwe(n, alpha, q, m=m, reduction_cost_model=BKZ.sieve) usvp: rop: ≈2^129.3, red: ≈2^129.3, δ_0: 1.004461, β: 339, d: 1954, m: 1149 dec: rop: ≈2^144.9, m: 1237, red: ≈2^144.9, δ_0: 1.004148, β: 378, d: 2041, babai: ≈2^130.9, babai_op: ≈2^146.0, repeat: 17, ε: 0.250000 dual: rop: ≈2^139.4, m: 1231, red: ≈2^139.4, δ_0: 1.004180, β: 373, repeat: ≈2^93.0, d: 2035, c: 1

sage: load("estimator.py") sage: n = 2048; q = 2^54 - 2^24 + 1; alpha = 8/q; m = 2*n sage: _ = estimate_lwe(n, alpha, q, secret_distribution=(-1,1), reduction_cost_model=BKZ.sieve, m=m) usvp: rop: ≈2^129.7, red: ≈2^129.7, δ_0: 1.004479, β: 337, d: 3915, m: 1866 dec: rop: ≈2^144.4, m: ≈2^11.1, red: ≈2^144.4, δ_0: 1.004154, β: 377, d: 4272, babai: ≈2^131.2, babai_op: ≈2^146.3, repeat: 7, ε: 0.500000 dual: rop: ≈2^134.2, m: ≈2^11.0, red: ≈2^134.2, δ_0: 1.004353, β: 352, repeat: ≈2^59.6, d: 4091, c: 3.909, k: 32, postprocess: 10