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Error Estimation, Error Correction and Verification In Quantum Key Distribution

Øystein Marøy, Magne Gudmundsen, Lars Lydersen, Johannes Skaar

Published Jan 1, 2014• 1 citation
OpenAlex

Abstract

We consider error correction in quantum key distribu-tion. To avoid that Alice and Bob unwittingly end up with different keys precautions must be taken. Before running the error correction protocol, Bob and Al-ice normally sacrifice some bits to estimate the error rate. To reduce the probability that they end up with different keys to an acceptable level, we show that a large number of bits must be sacrificed. Instead, if Alice and Bob can make a good guess about the er-ror rate before the error correction, they can verify that their keys are similar after the error correction protocol. This verification can be done by utilizing properties of Low Density Parity Check codes used in the error correction. We compare the methods and show that by verification it is often possible to sac-rifice less bits without compromising security. The improvement is heavily dependent on the error rate and the block length, but for a key produced by the IdQuantique system Clavis2, the increase in the key rate is approximately 5 percent. We also show that for systems with large fluctuations in the error rate a combination of the two methods is optimal.

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We consider error correction in quantum key distribu-tion. To avoid that Alice and Bob unwittingly end up with different keys precautions must be taken. Before running the error correction protocol, Bob and Al-ice normally sacrifice some bits to estimate the error rate. To reduce the probability that they end up with different keys to an acceptable level, we show that a large number of bits must be sacrificed. Instead, if Alice and Bob can make a good guess about the er-ror rate before the error correction, they can verify that their keys are similar after the error correction protocol. This verification can be done by utilizing properties of Low Density Parity Check codes used in the error correction. We compare the methods and show that by verification it is often possible to sac-rifice less bits without compromising security. The improvement is heavily dependent on the error rate and the block length, but for a key produced by the IdQuantique system Clavis2, the increase in the key rate is approximately 5 percent. We also show that for systems with large fluctuations in the error rate a combination of the two methods is optimal.
The authors consider error correction in quantum key distribution. To avoid that Alice and Bob unwittingly end up with different keys, precautions must be taken. Before running the error correction protocol, Bob and Alice normally sacrifice some bits to estimate the error rate. To reduce the probability that they end up with different keys to an acceptable level, the authors show that a large number of bits must be sacrificed. Instead, if Alice and Bob can make a good guess about the error rate before the error correction, they can verify that their keys are similar after the error correction protocol. In this study, the authors show that this verification can be done by utilising properties of low‐density parity check codes used in the error correction. The authors compare the different methods and show that by verification it is often possible to sacrifice less bits without compromising security. The improvement is heavily dependent on the error rate and the block length, but for a key produced by the IdQuantique system Clavis 2 , the increase in the key rate is ∼5%. The authors also show that for systems with large fluctuations in the error rate, a combination of the two methods is optimal.
Quantum cryptography provides methods for secret key distribution between the two distant users. An important step in quantum key distribution protocols is the correction of errors that occur during the transmission and implementation of the protocol. Here we consider error estimation and error correction with linear codes such as repetition code, Hamming code, low density parity check (LDPC) code. The paper investigates the increase in the efficiency of error correction using a combination of a repetition code and a LDPC code for a high error probability.
Quantum key distribution protocols (QKD) identify eavesdroppers on quantum channel based on laws of quantum mechanics. In the original BB84 protocol, unauthorized measurements on qubits are discovered after Alice and Bob sacrifice some random resulting key bits as a test and find out a mismatch rate. Decoy state protocol alerts photon number splitting attack when the decoy multiphoton pulses deliberately positioned by Alice are registered abnormally higher at Bob’s detectors than the single photon pulses are. E91 protocol is the first QKD protocol that ever brought about entangled atoms. Based on Bell’s theorem, a probability coefficient is constrained within a fixed range under the circumstances of interception. While in practice, bit error rate (BER) is not necessarily associated with vicious eavesdropping. Detection inefficiency, transmission loss, imperfect entanglement sources and many other factors contribute as false alarms. Error correction is an essential post-processing procedure to tolerate implementation limits, yet secure the accuracy of the transmitted key. Quantifying third party contaminations also helps setting an optimal BER threshold to decide if one should abandon all previous communications. In this work, we summarize major causes of bit error in various practical QKD protocols mentioned above and study whether error correction models, such as low-density parity-check code (LDPC), perform accordingly to different kinds of errors.
Quantum cryptography is essentially the quantum key distribution (QKD). In the context of QKD, one from two partners (Alice) generates and sends a sequence of qubits through a private quantum channel to another partner (Bob) and Bob receives the sequence and measures the state of each qubit. After the quantum transmission stage, Alice and Bob have almost identical qubit sequences. The erros are due to physical imperfections in the channel and presence of an eavesdropper. The next stage in QKD is key reconciliation (i.e. finding and correcting discrepancies between Alice's string and that of Bob). This reconciliation can be done by public discussion. Let us suppose there is a secret quantum channel between Alice and Bob through which Alice transmits a n-bit string A=(A1, A2,...,An)ε{0,1}n. Then Bob receives a n-bit string B=(B1, B2,...,Bn)ε{0,1)n. The string B differs from A due to the presence of noise and eavesdropper in the channel. One can estimate the bit error probability in the channel. For example, Bob can choose a random subset from his string and send it to Alice in public. Then Alice compares the received string with her corresponding subset and calculates the total number of protocol steps. The cascade scheme uses the interaction over the public channel to correct the secret strings by dividing them into the blocks of a fixed length. The length is determined from the bit error probability. A simple interactive routine is applied in each of these blocks. An error found in some block results in some action with other blocks. It is important to optimize the error-finding routines in standalone blocks as well as to organize the effective constrution of blocks with the object of protocol benchmark, information leakage and number of interactions between partners.
Post-processing is a significant step in quantum key distribution (QKD), which is used for correcting the quantum-channel noise errors and distilling identical corrected keys between two distant legitimate parties. Efficient error reconciliation protocol, which can lead to an increase in the secure key generation rate, is one of the main performance indicators of QKD setups. In this paper, we propose a multi-low-density parity-check codes based reconciliation scheme, which can provide remarkable perspectives for highly efficient information reconciliation. By testing our approach through data simulation, we show that the proposed scheme combining multi-syndrome-based error rate estimation allows for a more accurate estimation of the error rate as compared with random sampling and single-syndrome estimation techniques before the error correction, as well as a significant increase in the efficiency of the procedure without compromising security and sacrificing reconciliation efficiency.
Recall that in the Bennett–Brassard key distribution scheme, after Alice and Bob have obtained bit strings that ideally are supposed to be identical, they have to do some further processing to make sure they end up with strings that really are identical. Our first goal in this chapter is to show how they can do this. As you might expect, we will use error-correcting codes of the sort we have been discussing in the preceding chapter. However, the way we use error-correcting codes in quantum key distribution is not quite the same as in classical communication. Normally, one corrects errors by encoding one's message into special codewords that are sufficiently different from each other that they will still be distinguishable after passing through a noisy channel. But in quantum key distribution the “noise” of the channel might actually be the effect of an eavesdropper who is free to manipulate the signals sent by Alice. Ordinary error correction is not designed for such a setting. So instead of using codewords to encode the original transmission, we wait until all the bits have been sent and then use an error-correcting code after the fact . In this respect error correction in quantum key distribution is similar to the use of an error-correcting code in the “hat problem” discussed at the end of Chapter 4. There also, the error-correcting code is applied only after all the data – in that case the vector of hat colors – has been generated and conveyed to the participants.