# 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}
}
• Published in PQCrypto 4 June 2013
• Computer Science, Mathematics
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…

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## 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 Science
IEEE 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 Science
ArXiv
• 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

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, Mathematics
Lecture 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, Mathematics
Selected 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 Science
ASIACRYPT
• 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

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

• J. Stern
• Computer Science, Mathematics
CRYPTO
• 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

• J. Stern
• Computer Science, Mathematics
EUROCRYPT
• 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.