CodeRed

Basis Reduction Algorithms for Codes (LLL and more)

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This small library is a research Artefact, paired with the article

An Algorithmic Reduction Theory for Binary Codes: LLL and more
Thomas Debris--Alazard, Léo Ducas and Wessel P.J. van Woerden

The article is available on this repository and on the IACR eprint.

It provides a core in C++ for reduction algorithms for codes (LLL, Size-Reduction, Lee-Brickell, Lee-Brickell-Babai), and a Python interface.

Acknowledgments

This work was supported by the grant EPSRC EP/S02087X/1, by the European Union Horizon 2020 Research and Innovation Program Grant 780701 (PROMETHEUS) and by the ERC Advanced Grant 740972 (ALGSTRONGCRYPTO).

License:

MIT License see LICENSE.txt.

How to cite:

@misc{DDW20,
    author = {Thomas {Debris-Alazard} and Léo Ducas and Wessel P.J. van Woerden},
    title = {An Algorithmic Reduction Theory for Binary Codes: LLL and more},
    howpublished = {Cryptology ePrint Archive, Report 2020/869},
    year = {2020},
    note = {\url{https://eprint.iacr.org/2020/869}},
}

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Installation

Requirement

gcc
python
numpy

Compilation

To compile the C++ core before reproducing the experiments simply run bash compile_cpp_core.sh.

A maximum dimension (n, multiple of 64) is hardcoded at compile time, and the above creates binaries for n=256,384,512,768... If you use this library for other purpose than reproducing experiment, please adjust compile_cpp_core.sh to your needs.

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Reproducing Experiments from the paired article

Figure 4: python experiment_LLL_time.py
Figure 5: python experiment_LLL_profile.py
Figure 6: python experiment_LBB_distrib.py
Figure 7: python experiment_LBB_perf.py

To reach very large dimensions, the last experiment may be run using several cores by editing the python script.

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Usage from Python

All reduction algorithms are available through the class CodeRedLib and are applied on the internally stored basis. All input/outputs are numpy arrays.

Example (Code and execution of example.py):

>>> from numpy import array, random
>>> from middleware import CodeRedLib
>>>  
>>> B = random.randint(0,2, size=(5, 16), dtype="bool")  # Create a random Basis for a [5,16]-code
>>> red = CodeRedLib(B)                                  # Load it into a fresh CodeRedLib object  
>>> # The above fails in the unlucky case of the code C(B) not of full rank or not of full length.  
>>>  
>>> def niceprint(B):
...     for v in B:
...         print("".join(["1" if x else "." for x in v]))
...     print
... 
>>>   
>>> niceprint(red.B)    # Print current basis
1..1.1.1...1.1..
11..1..1111..1.1
11.11.1.1...111.
.111.11.1....1.1
11....1.1.1.1...
>>> niceprint(red.E)    # Print current Epipodal matrix
1..1.1.1...1.1..
.1..1...111....1
......1.....1.1.
..1.............
................
>>> print(red.l)        # Print current Profile
[6 6 3 1 0]
>>> 
>>> red.LLL()           # Apply LLL
>>> 
>>> niceprint(red.B)    # Print current basis
1..1.1.1...1.1..
..1....1..111..1
.111.11.1....1.1
1.111111.11.....
11.11.1.1...111.
>>> niceprint(red.E)    # Print current Epipodal matrix
1..1.1.1...1.1..
..1.......1.1..1
.1....1.1.......
....1....1......
..............1.
>>> print(red.l)        # Print current Profile
[6 4 3 2 1]

API:

Functions names match with the one of the paper.

Basis Processing functions:

Randomize()
Systematize()
LLL()
EpiSort()
SizeRedBasis()
SemiSystematize()
KillTwos()

Functions for finding short words in the code C (or in the coset t+C):

SizeRed(t)
LB(w2, goal_w=None, t=None, stats=False)
LBB(k1, w2, goal_w=None, t=None, stats=False)

Parameters:

• The default value for the target t=None is interpreted as the zero vector of length n (i.e. LB/LBB searches for a short codeword rather than a close codeword).
• Parameters k1 and w2 are integers, whose role is described in the paper.
• Leave default value goal_w=None to get the shortest visited codeword as output. Set goal_w to an integer to return as soon as a coderword of at most that length is found (and returns None if goal not met).
• Set Stats=True to instead get as return value the counts on visited codeword of each length.

Fundamental Domain and Probabilities

Function for computing distributions of words in the fundamental domain W(l) for a given profile l are provided in weights.py.

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Extending the C++ Core

The "plumbing" from C++ to python is done via the ctypes library for python. This require boilerplates both in the middleware.py file defining the CodeRedLib class, and within the

extern "C" {
...
}

statement at the end of the coderedlib.cpp file.