Conjugate codes and applications to cryptography

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.

[1]  H. Weyl.  Gruppentheorie und Quantenmechanik , 1928 .

[2]  A. Calderbank,et al.  Quantum Error Correction and Orthogonal Geometry , 1996, quant-ph/9605005.

[3]  Mitsuru Hamada.  Reliability of Calderbank-Shor-Steane codes and security of quantum key distribution , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[4]  The Binary Weight Distribution of the Extended ( 2 m , 2 m-4 ) Code of the Reed-Solomon Code , 2001 .

[5]  Alexei E. Ashikhmin,et al.  Nonbinary quantum stabilizer codes , 2001, IEEE Trans. Inf. Theory.

[6]  Robert J. McEliece,et al.  The Theory of Information and Coding , 1979 .

[7]  Gottesman.  Class of quantum error-correcting codes saturating the quantum Hamming bound. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[8]  Mitsuru Hamada,et al.  Quotient Codes and Their Reliability , 2005 .

[9]  N. J. A. Sloane,et al.  Quantum Error Correction Via Codes Over GF(4) , 1998, IEEE Trans. Inf. Theory.

[10]  Stephen Wiesner,et al.  Conjugate coding , 1983, SIGA.

[11]  M. Hamada.  Concatenated Conjugate Codes , 2006, quant-ph/0610194.

[12]  John K. Tomfohr,et al.  Lecture Notes on Physics , 1879, Nature.

[13]  E. Knill,et al.  Theory of quantum error-correcting codes , 1997 .

[14]  Mitsuru Hamada,et al.  Minimum Distance of Concatenated Conjugate Codes for Cryptography and Quantum Error Correction , 2006 .

[15]  Raymond Laflamme,et al.  A Theory of Quantum Error-Correcting Codes , 1996 .

[16]  F. Lemmermeyer.  Error-correcting Codes , 2005 .

[17]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[18]  Andrew M. Steane.  Efficient fault-tolerant quantum computing , 1999, Nature.

[19]  Schumacher,et al.  Sending entanglement through noisy quantum channels. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[20]  K. Kraus.  General state changes in quantum theory , 1971 .

[21]  Mitsuru Hamada.  Teleportation and entanglement distillation in the presence of correlation among bipartite mixed states , 2003 .

[22]  Rudolf Lide,et al.  Finite fields , 1983 .

[23]  Mitsuru Hamada,et al.  Conjugate Codes for Secure and Reliable Information Transmission , 2006, 2006 IEEE Information Theory Workshop - ITW '06 Chengdu.

[24]  Jacobus H. van Lint,et al.  Introduction to Coding Theory , 1982 .

[25]  Shor,et al.  Scheme for reducing decoherence in quantum computer memory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[26]  Mitsuru Hamada.  NOTES ON THE FIDELITY OF SYMPLECTIC QUANTUM ERROR-CORRECTING CODES , 2003 .

[27]  Elwyn R. Berlekamp,et al.  Key Papers in the Development of Coding Theory , 1974 .

[28]  Dominic Mayers,et al.  Unconditional security in quantum cryptography , 1998, JACM.

[29]  A. D. Wyner,et al.  The wire-tap channel , 1975, The Bell System Technical Journal.

[30]  Ning Cai,et al.  Quantum privacy and quantum wiretap channels , 2004, Probl. Inf. Transm..

[31]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[32]  K. E. Hellwig.  General scheme of measurement processes , 1995 .

[33]  A. Holevo.  Statistical structure of quantum theory , 2001 .

[34]  I. Devetak,et al.  The private classical information capacity and quantum information capacity of a quantum channel , 2003 .

[35]  Shor,et al.  Simple proof of security of the BB84 quantum key distribution protocol , 2000, Physical review letters.