Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0610193

Conjugate Codes and Applications to Cryptography

Mitsuru Hamada

Published Oct 23, 2006• 11 citations
arXivHTMLPDF

Abstract

A conjugate code pair is defined as a pair of linear codes such that one contains the dual of the other. The conjugate code pair represents the essential structure of the corresponding Calderbank-Shor-Steane (CSS) quantum code. It is argued that conjugate code pairs are applicable to quantum cryptography in order to motivate studies on conjugate code pairs.

DOI
10.48550/arxiv.quant-ph/0610193

Related papers

Concatenated Conjugate Codes

Mitsuru Hamada

A conjugate code pair is defined as a pair of linear codes either of which contains the dual of the other. A conjugate code pair represents the essential structure of the corresponding Calderbank-Shor-Steane (CSS) quantum code. It is known that conjugate code pairs are applicable to (quantum) cryptography. We give a construction method for efficiently decodable conjugate code pairs.

arXiv:quant-ph/0610194•Oct 23, 2006•5 citations

Constructive Conjugate Codes for Quantum Error Correction and Cryptography

Mitsuru Hamada

A conjugate code pair is defined as a pair of linear codes either of which contains the dual of the other. A conjugate code pair represents the essential structure of the corresponding Calderbank-Shor-Steane (CSS) quantum error-correcting code. It is known that conjugate code pairs are applicable to quantum cryptography. In this work, a polynomial construction of conjugate code pairs is presented. The constructed pairs achieve the highest known achievable rate on additive channels, and are decodable with algorithms of polynomial complexity.

arXiv:cs/0703141•Mar 28, 2007•4 citations

Conjugate Codes for Secure and Reliable Information Transmission

Mitsuru Hamada

A conjugate code pair is defined as a pair of linear codes either of which contains the dual of the other. A conjugate code pair represents the essential structure of the corresponding Calderbank-Shor-Steane (CSS) quantum code. It is known that conjugate code pairs are applicable to (quantum) cryptography. In this work, polynomially constructible and efficiently decodable conjugate code pairs are presented. The constructed pairs achieve the highest known achievable rate on additive channels.

IEEE Information Theory Workshop (ITW)•Oct 1, 2006•14 citations

Algebraic and quantum theoretical approach to coding on wiretap channels

Mitsuru Hamada

Algebraic codes designed on a criterion that has arisen from quantum cryptography are analyzed. The primary purpose of this analysis is to demonstrate the constructibility of codes for wiretap channels. Specifically, it is shown that reliable conjugate code pairs, or Calderbank-Shor-Steane (CSS) quantum error-correcting codes, can immediately be converted into reliable and secure codes for wiretap channels (Wyner, 1975). Here, a pair of linear codes is called a conjugate code pair if either of the pair contains the dual of the other; A code is called reliable if it has exponentially small decoding error probability, and it is called secure if the mutual information between the transmitted confidential message and the leaked data is exponentially small. It is argued that conjugate code pairs are applicable to wiretap channels whether they are classical or quantum theoretical. In particular, this implies that recently obtained conjugate code pairs that are constructible with polynomial complexity achieve positive rates for wiretap channels.

International Symposium on Communications, Control and Signal Processing•Mar 1, 2008•4 citations

$\mathbb {F}_p$-Linear and $\mathbb {F}_{p^m}$-Linear Qudit Codes From Dual-Containing Classical Codes

Priya J. Nadkarni, Shayan Srinivasa Garani

Quantum code construction from two classical codes D1[n,k1,d1] and D2[n,k2,d2] over the field Fpm(p is prime and m is an integer) satisfying the dual containing criteria D1⊥⊂ D2using the Calderbank-Shor-Steane (CSS) framework is well-studied. We show that the generalization of the CSS framework for qubits to qudits yields two different classes of codes, namely, the Fp-linear CSS codes and the well-known Fpm-linear CSS codes based on the check matrix-based definition and the coset-based definition of CSS codes over qubits. Our contribution to this article are three-folds. 1) We study the properties of the \mathbb Fp-linear and Fpm-linear CSS codes and demonstrate the tradeoff for designing codes with higher rates or better error detection and correction capability, useful for quantum systems. 2) For Fpm-linear CSS codes, we provide the explicit form of the check matrix and show that the minimum distances dxand dzare equal to d2and d1, respectively, if and only if the code is nondegenerate. 3) We propose two classes of quantum codes obtained from the codes D1and D2, where one code is an Fpl-linear code (l divides m) and the other code is obtained from a particular subgroup of the stabilizer group of the Fpm-linear CSS code. Within each class of codes, we demonstrate the tradeoff between higher rates and better error detection and correction capability.

IEEE Transactions on Quantum Engineering•Jan 1, 2021•4 citations

Dual Universality of Hash Functions and Its Applications to Quantum Cryptography

Toyohiro Tsurumaru, M. Hayashi

In this paper, we introduce the concept of dual universality of hash functions and present its applications to quantum cryptography. We begin by establishing the one-to-one correspondence between a linear function familyFand a code familyC, and thereby defining ε-almost dual universal2hash functions, as a generalization of the conventional universal2hash functions. Then, we show that this generalized (and thus broader) class of hash functions is in fact sufficient for the security of quantum cryptography. This result can be explained in two different formalisms. First, by noting its relation to the δ-biased family introduced by Dodis and Smith, we demonstrate that Renner's two-universal hashing lemma is generalized to our class of hash functions. Next, we prove that the proof technique by Shor and Preskill can be applied to quantum key distribution (QKD) systems that use our generalized class of hash functions for privacy amplification. While Shor-Preskill formalism requires an implementer of a QKD system to explicitly construct a linear code of the Calderbank-Shor-Steane (CSS) type, this result removes the existing difficulty of the construction of a linear code of CSS code by replacing it by the combination of an ordinary classical error correcting code and our proposed hash function. We also show that a similar result applies to the quantum wire-tap channel. Finally, we compare our results in the two formalisms and show that, in typical QKD scenarios, the Shor-Preskill-type argument gives better security bounds in terms of the trace distance and Holevo information than the method based on the δ-biased family.

arXiv:1101.0064•IEEE Transactions on Information Theory•Apr 2, 2013•55 citations
A conjugate code pair is defined as a pair of linear codes such that one contains the dual of the other. The conjugate code pair represents the essential structure of the corresponding Calderbank-Shor-Steane (CSS) quantum code. It is argued that conjugate code pairs are applicable to quantum cryptography in order to motivate studies on conjugate code pairs.
A conjugate code pair is defined as a pair of linear codes either of which contains the dual of the other. A conjugate code pair represents the essential structure of the corresponding Calderbank-Shor-Steane (CSS) quantum error-correcting code. It is known that conjugate code pairs are applicable to quantum cryptography. In this work, a polynomial construction of conjugate code pairs is presented. The constructed pairs achieve the highest known achievable rate on additive channels, and are decodable with algorithms of polynomial complexity.
Algebraic codes designed on a criterion that has arisen from quantum cryptography are analyzed. The primary purpose of this analysis is to demonstrate the constructibility of codes for wiretap channels. Specifically, it is shown that reliable conjugate code pairs, or Calderbank-Shor-Steane (CSS) quantum error-correcting codes, can immediately be converted into reliable and secure codes for wiretap channels (Wyner, 1975). Here, a pair of linear codes is called a conjugate code pair if either of the pair contains the dual of the other; A code is called reliable if it has exponentially small decoding error probability, and it is called secure if the mutual information between the transmitted confidential message and the leaked data is exponentially small. It is argued that conjugate code pairs are applicable to wiretap channels whether they are classical or quantum theoretical. In particular, this implies that recently obtained conjugate code pairs that are constructible with polynomial complexity achieve positive rates for wiretap channels.
Fp\mathbb {F}_pFp​-Linear and Fpm\mathbb {F}_{p^m}Fpm​-Linear Qudit Codes From Dual-Containing Classical Codes
Quantum code construction from two classical codes D1[n,k1,d1] and D2[n,k2,d2] over the field Fpm(p is prime and m is an integer) satisfying the dual containing criteria D1⊥⊂ D2using the Calderbank-Shor-Steane (CSS) framework is well-studied. We show that the generalization of the CSS framework for qubits to qudits yields two different classes of codes, namely, the Fp-linear CSS codes and the well-known Fpm-linear CSS codes based on the check matrix-based definition and the coset-based definition of CSS codes over qubits. Our contribution to this article are three-folds. 1) We study the properties of the \mathbb Fp-linear and Fpm-linear CSS codes and demonstrate the tradeoff for designing codes with higher rates or better error detection and correction capability, useful for quantum systems. 2) For Fpm-linear CSS codes, we provide the explicit form of the check matrix and show that the minimum distances dxand dzare equal to d2and d1, respectively, if and only if the code is nondegenerate. 3) We propose two classes of quantum codes obtained from the codes D1and D2, where one code is an Fpl-linear code (l divides m) and the other code is obtained from a particular subgroup of the stabilizer group of the Fpm-linear CSS code. Within each class of codes, we demonstrate the tradeoff between higher rates and better error detection and correction capability.
In this paper, we introduce the concept of dual universality of hash functions and present its applications to quantum cryptography. We begin by establishing the one-to-one correspondence between a linear function familyFand a code familyC, and thereby defining ε-almost dual universal2hash functions, as a generalization of the conventional universal2hash functions. Then, we show that this generalized (and thus broader) class of hash functions is in fact sufficient for the security of quantum cryptography. This result can be explained in two different formalisms. First, by noting its relation to the δ-biased family introduced by Dodis and Smith, we demonstrate that Renner's two-universal hashing lemma is generalized to our class of hash functions. Next, we prove that the proof technique by Shor and Preskill can be applied to quantum key distribution (QKD) systems that use our generalized class of hash functions for privacy amplification. While Shor-Preskill formalism requires an implementer of a QKD system to explicitly construct a linear code of the Calderbank-Shor-Steane (CSS) type, this result removes the existing difficulty of the construction of a linear code of CSS code by replacing it by the combination of an ordinary classical error correcting code and our proposed hash function. We also show that a similar result applies to the quantum wire-tap channel. Finally, we compare our results in the two formalisms and show that, in typical QKD scenarios, the Shor-Preskill-type argument gives better security bounds in terms of the trace distance and Holevo information than the method based on the δ-biased family.