This paper provides a formula for the sacrifice bit-length for privacy amplification with the Bennett-Brassard 1984 protocol for finite key lengths, when we employ the decoy method. Using the formula, we can guarantee the security parameter for a realizable quantum key distribution system. The key generation rates with finite key lengths are numerically evaluated. The proposed method improves the existing key generation rate even in the asymptotic setting.In this paper, when Aliceʼs basis is the same as Bobʼs basis, the basis is called matched.Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.setting, there are two obstacles for security. One is the noise of the communication quantum channel. Due to the presence of the noise, the eavesdropper can obtain part of the raw keys information behind the noise. The second one is the imperfection of the photon source. If the sender sends the two-photon state instead of the single photon state, the eavesdropper can obtain one photon so that she can obtain information perfectly. Many realized QKD systems have been realized with weak coherent pulses. In this case, the photon number of transmitted pulses obeys the Poisson distribution, whose average is given by the intensity μ of the pulse. The first problem can be resolved by the application of error correction and random privacy amplification to the raw keys [2][3][4][5]. In the privacy amplification stage, we amplify the security of the raw keys by sacrificing part of our raw keys. The security of the final keys depends on the decreasing number of keys in the privacy amplification stage, which is called the sacrifice bit-length. Shor-Preskill [2] and Mayers [3] showed that this method gives the secure keys asymptotically when the rate of the sacrifice bit-length is greater than a certain amount. In order to solve the second problem, Gottesman-Lo-Lütkenhaus-Preskill (GLLP) [6] extended their result to the case when the photon source has an imperfection. However, GLLPʼs result assumes the fractions of respective photon number pulses among received pulses. Indeed, there is a possibility that the eavesdropper can control the receiverʼs detection rate depending on the photon number because pulses with the different photon number can be distinguished by the eavesdropper. In order to solve this problem, we need to estimate the detection rate of the single photon pulses. Hwang proposed the decoy method to estimate the detection rate [7]. This method has been improved by many researchers [8][9][10][11][12][13][14][15][16]. In this method, in order to estimate the detection rates, the sender randomly chooses several kinds of pulses with different intensities. The first kind of pulses are the signal pulses, which generate raw keys. The other kind of pulses are the decoy pulses, which are used for estimating the operation by the eavesdropper and have a di...

We report the first total synthesis of the proposed structure of ardimerin, which was achieved in 14 steps starting from 2,3,4‐trimethoxybenzoic acid. The key steps include the β‐selective formation of the crucial C‐glycoside linkage and stepwise construction of the strained eight‐membered salicylide core. The synthesis revealed that the proposed structure 1 does not match the natural product. A proposal is made for reassigning the isolated natural product to the already known structure of bergenin. Interesting properties of the synthetic eight‐membered salicylides are documented, including their susceptibility toward nucleophilic ring opening and the bowl chirality.

Tin hydride mediated radical addition of organic halide to 2-(2,2,2-trifluoroethylidene)-1,3-dithiane 1-oxide has been devised. The reaction is equivalent to an unrealizable radical addition to trifluoromethylketene, providing useful α-trifluoromethyl carbonyl equivalents. The trifluoromethyl and the sulfoxide groups of the substrate play key roles for the success of the radical addition, lowering the barrier of the radical addition step and controlling the stereoselectivity of the reaction, which DFT calculations have elucidated.

No abstract