Improved Secret Key Rate in Quantum Key Distribution using highly irregular Low-Density Parity-Check codes

Quantum key distribution (QKD) has proven to be the most mature subfield of quantum information theory, elegantly spanning the gap between theory and practice. It is a simple, and yet sophisticated idea, in which the fundamental principles of modern quantum mechanics are employed to distribute a provably secure cryptographic key between two parties. With many variations of this scheme realized, it has become a question of enhancing each of the underlying structures in order to achieve optimal performance. One of these structures is the key reconciliation protocol, which has the objective of making sure the two communicating parties indeed have identical keys at the end of the protocol, by publicly exchanging an amount of extra information on the key. Since an antagonist can exploit this redundant information, the final key needs to be shortened with a corresponding amount. The conventional approach to this problem has been for the two parties to engage in a dialogue over a secure channel, to locate and correct the faulty bits in their key. While such protocols are easy to implement and deliver stable performance, it is a time-consuming task with suboptimal capabilities in regards to the superfluous information made publicly available. When the lost low-density parity-check (LDPC) codes of Gallager were recently "rediscovered", it caused a boom in related research, generating reports of redundancy extremely close to the theoretical lower limit. Even though key reconciliation protocols based on these codes have been tried and tested for QKD, they have not been able to reproduce the performance of classical implementations and have thus enjoyed limited success. The goal of this text is to demonstrate how, when suitably implemented, the true potential of LDPC codes can also be harnessed for QKD. After an extensive presentation of the theoretical background, an explicit key reconciliation protocol is presented. Simulations on a practical implementation are made, showing very good performance and outperforming some of the alternative protocols. The best code presented in this text operates at only 9.0% above the Shannon limit, while the interactive CASCADE protocol is reported to perform at ≈ 21% above, for the same set of parameters. Additionally, the appreciable characteristics of the codes are utilized in a novel detection scheme, which promises tighter security at a lower cost on expended bits used for detecting an eavesdropper. This makes LDPC codes a highly competitive alternative to the conventional protocols.

[1]  C. G. Peterson,et al.  Fast, efficient error reconciliation for quantum cryptography , 2002, quant-ph/0203096.

[2]  T. Sugimoto,et al.  A Study on Secret key Reconciliation Protocol "Cascade" , 2000 .

[3]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[4]  D. Bruß,et al.  Optimal universal and state-dependent quantum cloning , 1997, quant-ph/9705038.

[5]  Renato Renner,et al.  Quantum cryptography with finite resources: unconditional security bound for discrete-variable protocols with one-way postprocessing. , 2007, Physical review letters.

[6]  Schumacher,et al.  Quantum coding. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[7]  John Preskill,et al.  Security of quantum key distribution with imperfect devices , 2002, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[8]  Chip Elliott,et al.  Current status of the DARPA Quantum Network , 2005 .

[9]  Maassen,et al.  Generalized entropic uncertainty relations. , 1988, Physical review letters.

[10]  Renato Renner,et al.  Security of quantum key distribution , 2005, Ausgezeichnete Informatikdissertationen.

[11]  Ueli Maurer,et al.  Generalized privacy amplification , 1994, Proceedings of 1994 IEEE International Symposium on Information Theory.

[12]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[13]  Jack K. Wolf,et al.  Noiseless coding of correlated information sources , 1973, IEEE Trans. Inf. Theory.

[14]  Fred Daneshgaran,et al.  Information Theory An algorithm for the computation of the minimum distance of LDPC codes , 2006, Eur. Trans. Telecommun..

[15]  Lo,et al.  Unconditional security of quantum key distribution over arbitrarily long distances , 1999, Science.

[16]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[17]  Rüdiger L. Urbanke,et al.  Exact thresholds and optimal codes for the binary-symmetric channel and Gallager's decoding algorithm A , 2000, IEEE Transactions on Information Theory.

[18]  David J. C. MacKay,et al.  Comparison of constructions of irregular Gallager codes , 1999, IEEE Trans. Commun..

[19]  Robert König,et al.  Universally Composable Privacy Amplification Against Quantum Adversaries , 2004, TCC.

[20]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[21]  David A. Pearson,et al.  High‐speed QKD Reconciliation using Forward Error Correction , 2004 .

[22]  Sae-Young Chung,et al.  On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit , 2001, IEEE Communications Letters.

[23]  Daniel A. Spielman,et al.  Analysis of low density codes and improved designs using irregular graphs , 1998, STOC '98.

[24]  Rüdiger L. Urbanke,et al.  The capacity of low-density parity-check codes under message-passing decoding , 2001, IEEE Trans. Inf. Theory.

[25]  H. Bechmann-Pasquinucci,et al.  Incoherent and coherent eavesdropping in the six-state protocol of quantum cryptography , 1998, quant-ph/9807041.

[26]  Won-Young Hwang.  Quantum key distribution with high loss: toward global secure communication. , 2003, Physical review letters.

[27]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[28]  W. Tittel,et al.  Proof-of-concept of real-world quantum key distribution with quantum frames , 2009, 0901.0612.

[29]  Gilles Brassard,et al.  Privacy Amplification by Public Discussion , 1988, SIAM J. Comput..

[30]  Evangelos Eleftheriou,et al.  On the computation of the minimum distance of low-density parity-check codes , 2004, 2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577).

[31]  Evangelos Eleftheriou,et al.  Progressive edge-growth Tanner graphs , 2001, GLOBECOM'01. IEEE Global Telecommunications Conference (Cat. No.01CH37270).

[32]  J. Skaar,et al.  Effects of detector efficiency mismatch on security of quantum cryptosystems , 2005, quant-ph/0511032.

[33]  Masato Koashi,et al.  Simple security proof of quantum key distribution based on complementarity , 2009 .

[34]  Xiongfeng Ma,et al.  Universally composable and customizable post-processing for practical quantum key distribution , 2009, Comput. Secur..

[35]  Sarah J. Johnson.  Introducing Low-Density Parity-Check Codes , 2008 .

[36]  Zixiang Xiong,et al.  Compression of binary sources with side information at the decoder using LDPC codes , 2002, IEEE Communications Letters.

[37]  Daniel A. Spielman,et al.  Practical loss-resilient codes , 1997, STOC '97.

[38]  Xiongfeng Ma,et al.  ar X iv : q ua ntp h / 05 12 08 0 v 2 1 1 A pr 2 00 6 TIMESHIFT ATTACK IN PRACTICAL QUANTUM , 2005 .

[39]  Charles H. Bennett,et al.  Quantum cryptography using any two nonorthogonal states. , 1992, Physical review letters.

[40]  H. Lo.  Method for decoupling error correction from privacy amplification , 2002, quant-ph/0201030.

[41]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[42]  V. Rich.  Personal communication , 1989, Nature.

[43]  Gilles Brassard,et al.  Quantum Cryptography , 2005, Encyclopedia of Cryptography and Security.

[44]  David Elkouss,et al.  Efficient reconciliation protocol for discrete-variable quantum key distribution , 2009, 2009 IEEE International Symposium on Information Theory.

[45]  John Preskill,et al.  Secure quantum key distribution with an uncharacterized source. , 2003, Physical review letters.

[46]  Vernam.  Cipher printing telegraph systems , 1926, Journal of the A.I.E.E..

[47]  Peter W. Shor,et al.  Algorithms for Quantum Computation: Discrete Log and Factoring (Extended Abstract) , 1994, FOCS 1994.

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

[49]  Norbert Lutkenhaus,et al.  Estimates for practical quantum cryptography , 1998 .

[50]  Sellami Ali,et al.  DECOY STATE QUANTUM KEY DISTRIBUTION , 2010 .

[51]  Ajay Dholakia,et al.  Reduced-complexity decoding of LDPC codes , 2005, IEEE Transactions on Communications.

[52]  Gilles Brassard,et al.  Secret-Key Reconciliation by Public Discussion , 1994, EUROCRYPT.

[53]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[54]  Ayoub Otmani,et al.  On the Minimum Distance of Generalized LDPC Codes , 2007, 2007 IEEE International Symposium on Information Theory.

[55]  Johannes Skaar,et al.  Security of quantum key distribution with bit and basis dependent detector flaws , 2008, Quantum Inf. Comput..

[56]  Rüdiger L. Urbanke,et al.  Efficient encoding of low-density parity-check codes , 2001, IEEE Trans. Inf. Theory.

[57]  S. McLaughlin,et al.  Quantum key distribution over 25 km with an all-fiber continuous-variable system , 2007, 0706.4255.

[58]  Ryutaroh Matsumoto.  Problems in application of LDPC codes to information reconciliation in quantum key distribution protocols , 2009, ArXiv.

[59]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

[60]  Dominic Mayers,et al.  Quantum Key Distribution and String Oblivious Transfer in Noisy Channels , 1996, CRYPTO.

[61]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[62]  Daniel A. Spielman,et al.  Improved low-density parity-check codes using irregular graphs and belief propagation , 1998, Proceedings. 1998 IEEE International Symposium on Information Theory (Cat. No.98CH36252).

[63]  David J. C. MacKay,et al.  Good Error-Correcting Codes Based on Very Sparse Matrices , 1997, IEEE Trans. Inf. Theory.

[64]  Robert Michael Tanner,et al.  A recursive approach to low complexity codes , 1981, IEEE Trans. Inf. Theory.

[65]  Claude E. Shannon,et al.  Communication theory of secrecy systems , 1949, Bell Syst. Tech. J..

[66]  V. Scarani,et al.  The security of practical quantum key distribution , 2008, 0802.4155.

[67]  Larry Carter,et al.  New Hash Functions and Their Use in Authentication and Set Equality , 1981, J. Comput. Syst. Sci..

[68]  Marten van Dijk,et al.  A Practical Protocol for Advantage Distillation and Information Reconciliation , 2003, Des. Codes Cryptogr..

[69]  A. Glavieux,et al.  Near Shannon limit error-correcting coding and decoding: Turbo-codes. 1 , 1993, Proceedings of ICC '93 - IEEE International Conference on Communications.

[70]  Tong Zhang,et al.  On finite precision implementation of low density parity check codes decoder , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).

[71]  W. Munro,et al.  Low Cost and Compact Quantum Cryptography , 2006, quant-ph/0608213.

[72]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .