paper

Reliability of Calderbank-Shor-Steane Codes and Security of Quantum Key Distribution

arXiv:quant-ph/0308029 · doi:10.1088/0305-4470/37/34/009

Abstract

After Mayers (1996, 2001) gave a proof of the security of the Bennett-Brassard 1984 (BB84) quantum key distribution protocol, Shor and Preskill (2000) made a remarkable observation that a Calderbank-Shor-Steane (CSS) code had been implicitly used in the BB84 protocol, and suggested its security could be proven by bounding the fidelity, say F(n), of the incorporated CSS code of length n in the form 1-F(n) <= exp[-n E+o(n)] for some positive number E. This work presents such a number E=E(R) as a function of the rate of a code R, and a threshold R' such that E(R)>0 whenever R < R', which is larger than the achievable rate based on the Gilbert-Varshamov bound that is essentially due to Shor and Preskill (2000). The codes in the present work are robust against fluctuations of channel parameters, which fact is needed to establish the security rigorously and was not proved for rates above the Gilbert-Varshamov rate before in the literature. As a byproduct, the security of a modified BB84 protocol against any joint (coherent) attacks is proved quantitatively.

No figure. Ver.2: Security criterion slightly refined. 25 pages. Ver.3: Completely revised to fix a serious error. The existence of needed robust codes beyond the Gilbert-Varshamov rate was not proved in the previous versions (or in the literature). Ver.4: Minor errors corrected. Ver.5: Security proof for general joint attacks added. Ver.5: Accepted version; the results and reasonings unchanged; a remark on random sampling added in Appendix C with three references

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