# The Hardness of Code Equivalence over and Its Application to Code-Based Cryptography

@inproceedings{Sendrier2013TheHO, title={The Hardness of Code Equivalence over and Its Application to Code-Based Cryptography}, author={Nicolas Sendrier and Dimitris E. Simos}, booktitle={PQCrypto}, year={2013} }

The code equivalence problem is to decide whether two linear codes over \(\mathbb{F}_{q}\) are identical up to a linear isometry of the Hamming space. In this paper, we review the hardness of code equivalence over \(\mathbb{F}_q\) due to some recent negative results and argue on the possible implications in code-based cryptography. In particular, we present an improved version of the three-pass identification scheme of Girault and discuss on a connection between code equivalence and the hidden…

## 35 Citations

### How easy is code equivalence over F_q

- Computer Science, Mathematics
- 2013

A generalization of the support splitting algorithm where the problem of deciding the equivalence of linear codes to an instance of permutation equivalence is reduced and the notion of the closure of a code is introduced and given some of its properties.

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### On the hardness of code equivalence problems in rank metric

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The hardness of the code equivalence problem in rank metric for $\mathbb{F}_{q^m}$--linear and general rank metric codes is discussed and it is proved that the equivalence problems for general matrix codes is at least as hard as the well--studied {\em Monomial Equivalence Problem} for codes endowed with the Hamming metric.

### Not enough LESS: An improved algorithm for solving Code Equivalence Problems over 픽q

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
- 2020

An improved algorithm is given for solving both the conjectured hardness of the linear code equivalence problem and the permutation code equivalences problem for weakly self-dual codes over sufficiently large finite fields.

### On the equivalence of cyclic and quasi-cyclic codes over finite fields

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- 2017

This paper generalizes the results of Huffman, Job, and Pless, and characterize the set by which two quasi-cyclic codes of length $p^rl$ can be equivalent, and proves that the affine group is one of its subsets.

### Algebraic Approach for Code Equivalence

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- 2017

A new reduction from permutation code equivalence when the hull is trivial to graph isomorphism is introduced, which shows that this subclass of permutation equivalence is not harder than graph isomorphicism.

### An algebraic approach for the resolution of algorithmic problems raised by cryptography and coding theory

- Computer Science
- 2017

This research was naturally directed to the study of the McEliece based encryption schemes, among which the latest MDCP based variant [MTSB13] and Polar codes variant [SK14] and an important weakness regarding the key pairs are revealed.

### Limitations of single coset states and quantum algorithms for code equivalence

- Computer ScienceQuantum Inf. Comput.
- 2015

The results suggest that for many codes of interest--including generalized Reed Solomon codes, alternant codes, and Reed-Muller codes--solving these instances of Code Equivalence via Fourier sampling appears to be out of reach of current families of quantum algorithms.

### A New Algorithm for Equivalence of Cyclic Codes and Its Applications

- Computer ScienceApplicable Algebra in Engineering, Communication and Computing
- 2021

A more efficient algorithm is created to partition cyclic codes by equivalence based on cyclotomic cosets to find several new linear codes that are cyclic or QC as an application of the new approach, as well as more desirable constructions for linear codes with best known parameters.

## References

SHOWING 1-10 OF 60 REFERENCES

### How easy is code equivalence over F_q

- Computer Science, Mathematics
- 2013

A generalization of the support splitting algorithm where the problem of deciding the equivalence of linear codes to an instance of permutation equivalence is reduced and the notion of the closure of a code is introduced and given some of its properties.

### Is code equivalence easy to decide?

- Computer ScienceIEEE Trans. Inf. Theory
- 1997

If one could find an efficient (i.e., polynomial-time) algorithm for code equivalence, then one could settle the long-standing problem of determining whether there is an efficient algorithm for solving graph isomorphism.

### Quantum Fourier sampling, Code Equivalence, and the quantum security of the McEliece and Sidelnikov cryptosystems

- Computer ScienceArXiv
- 2011

The claims of the previous article are revisited in the light of these classical attacks, and the particular case of the Sidelnikov cryptosystem, which is based on Reed-Muller codes, is discussed.

### Information-Set Decoding for Linear Codes over Fq

- Computer SciencePQCrypto
- 2010

This paper presents a generalization of Stern's information-set- decoding algorithm for decoding linear codes over arbitrary finite fields Fq and analyzes the complexity to make it possible to compute the security of recently proposed code-based systems over non-binary fields.

### Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

- Computer Science, MathematicsLecture Notes in Computer Science
- 2009

This work discusses the construction of Authentication/Secrecy Codes, performance analysis of M-PSK Signal Constellations in Riemannian Varieties, and fast Decomposition of Polynomials with Known Galois Group.

### Polynomial Reconstruction Based Cryptography

- Computer Science, MathematicsSelected Areas in Cryptography
- 2001

A short overview of recent works on the problem of Decoding Reed Solomon Codes (aka Polynomial Reconstruction) and the novel applications that were enabled due to this development.

### How to Achieve a McEliece-Based Digital Signature Scheme

- Computer ScienceASIACRYPT
- 2001

This paper disproves the belief that code-based cryptosystems like McEliece do not allow practical digital signatures, and shows a way to build a practical signature scheme based on coding theory.

### Finding the permutation between equivalent linear codes: The support splitting algorithm

- Computer ScienceIEEE Trans. Inf. Theory
- 2000

An algorithm able to compute a fully discriminant signature for most linear codes by determining a set of properties invariant by permutation, one for each coordinate, called a signature is presented.

### A New Identification Scheme Based on Syndrome Decoding

- Computer Science, MathematicsCRYPTO
- 1993

This paper proposes a new identification scheme, based on error-correcting codes, which is zero-knowledge and is of practical value, and describes several variants, including one which has an identity based character.

### An Alternative to the Fiat-Shamir Protocol

- Computer Science, MathematicsEUROCRYPT
- 1989

This paper is an attempt to build identification protocols that use only very basic operations such as multiplication by a fixed matrix over the two-element field, which can be viewed as the parity-check matrix of a linear binary error-correcting code.