Efficient Description of some Classes of Codes using Group Algebras

@article{ChimalDzul2022EfficientDO,
  title={Efficient Description of some Classes of Codes using Group Algebras},
  author={Henry Chimal-Dzul and Niklas Gassner and Joachim Rosenthal and Reto Schnyder},
  journal={ArXiv},
  year={2022},
  volume={abs/2208.04061}
}
Circulant matrices are an important tool widely used in coding theory and cryptography. A circulant matrix is a square matrix whose rows are the cyclic shifts of the first row. Such a matrix can be efficiently stored in memory because it is fully specified by its first row. The ring of n × n circulant matrices can be identified with the quotient ring F [ x ] / ( x n − 1). In consequence, the strong algebraic structure of the ring F [ x ] / ( x n − 1) can be used to study properties of the collection… 

Left ideal LRPC codes and a ROLLO-type cryptosystem based on group algebras

Left ideal low-rank parity-check codes are introduced by using group algebras and they are used to extend KEM.

References

SHOWING 1-10 OF 13 REFERENCES

Binary codes which are ideals in the group algebra of an abelian group

A cyclic code is an ideal in the group algebra of a special kind of Abelian group, namely a cyclic group. Many properties of cyclic codes are special cases of properties of ideals in an Abelian group

LDPC block and convolutional codes based on circulant matrices

A class of algebraically structured quasi-cyclic low-density parity-check (LDPC) codes and their convolutional counterparts is presented and bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described.

Reducing Key Length of the McEliece Cryptosystem

The result suggests that decoding attack against the variant has little chance to be better than the general one against the classical McEliece cryptosystem, and a new NP-complete decision problem called quasi-cyclic syndrome decoding is introduced.

Quasi-Cyclic Low-Density Parity-Check Codes in the McEliece Cryptosystem

The authors conclude that some families of QC-LDPC codes, based on circulant permutation matrices, are inapplicable in this context, due to security issues, whilst other codes,based on the "difference families" approach, can be able to ensure a good level of security against intrusions, even if very large lengths are needed.

Advanced linear algebra

Most of what is done in this course will be done over an arbitrary field of scalars. While the cases of real scalars and complex scalars have always been regarded as fundamental for “application”, it

Low-density parity-check codes

A simple but nonoptimum decoding scheme operating directly from the channel a posteriori probabilities is described and the probability of error using this decoder on a binary symmetric channel is shown to decrease at least exponentially with a root of the block length.

An introduction to group rings

Preface. 1. Groups. 2. Rings, Modules and Algebras. 3. Group Rings. 4. A Glance at Group Representations. 5. Group Characters. 6. Ideals in Group Rings. 7. Algebraic Elements. 8. Units of Group

A Survey on Code-Based Cryptography

This chapter covers the main frameworks introduced in code-based cryptography and analyzes their security assumptions, and provides the mathematical background in a lecture notes style, with the intention of reaching a wider audience.

BIKE: Bit Flipping Key Encapsulation

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not, for teaching and research institutions in France or abroad, or from public or private research centers.

Reproducible families of codes and cryptographic applications

This article considers some cryptographic applications of codes of this type and shows that their use can be advantageous for hindering some current attacks against cryptosystems relying on structured codes, suggesting that the general framework introduced may enable future developments of code-based cryptography.