QKD parameter estimation by two-universal hashing

Dimiter Ostrev

Institute of Communications and Navigation, German Aerospace Center, Oberpfaffenhofen, 82234 Weßling, Germany

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This paper proposes and proves security of a QKD protocol which uses two-universal hashing instead of random sampling to estimate the number of bit flip and phase flip errors. This protocol dramatically outperforms previous QKD protocols for small block sizes. More generally, for the two-universal hashing QKD protocol, the difference between asymptotic and finite key rate decreases with the number $n$ of qubits as $cn^{-1}$, where $c$ depends on the security parameter. For comparison, the same difference decreases no faster than $c'n^{-1/3}$ for an optimized protocol that uses random sampling and has the same asymptotic rate, where $c'$ depends on the security parameter and the error rate.

A quantum key distribution (QKD) protocol allows two users to establish a secret key by communicating over an authenticated classical channel and a completely insecure quantum channel. Important parameters for a QKD protocol are the number of qubits sent over the quantum channel, the resistance to noise on the quantum channel, the size of the output secret key, and the security level.

Existing QKD protocols and security proofs exhibit trade-offs between the parameters: for a given number of qubits, improving noise resistance or security makes the output size smaller. These trade-offs are especially severe when the number of qubits is small, i.e. around 1000-10000. Such a small number of qubits arises in practice when the quantum channel is particularly difficult to implement, for example when a satellite is transmitting entangled photon pairs to two ground stations.

The present work asks: are there QKD protocols and security proofs that exhibit better parameter trade-offs, especially in the case when the number of qubits is small? It presents one such QKD protocol and security proof. This protocol uses two-universal hashing instead of random sampling to estimate the number of bit flip and phase flip errors, leading to a dramatic improvement in parameter trade-offs for small numbers of qubits, but also making the protocol harder to implement.

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Cited by

[1] Dimiter Ostrev, Davide Orsucci, Francisco Lázaro, and Balazs Matuz, "Classical product code constructions for quantum Calderbank-Shor-Steane codes", arXiv:2209.13474, (2022).

[2] Manuel B. Santos, Paulo Mateus, and Chrysoula Vlachou, "Quantum Universally Composable Oblivious Linear Evaluation", arXiv:2204.14171, (2022).

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