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Equivalence of three quantum algorithms: Privacy amplification, error correction, and data compression.

Toyohiro Tsurumaru

Published Sep 18, 2020• 1 citation
OpenAlex

Abstract

Privacy amplification (PA) is an indispensable component in classical and quantum cryptography. Error correction (EC) and data compression (DC) algorithms are also indispensable in classical and quantum information theory. We here study quantum algorithms of these three types (PA, EC, and DC) in the one-shot scenario, and show that they all become equivalent if modified properly. As an application of this equivalence, we take previously known security bounds of PA, and translate them into coding theorems for EC and DC which have not been obtained previously. Further, we apply these results to simplify and improve our previous result that the two prevalent approaches to the security proof of quantum key distribution (QKD) are equivalent. We also propose a new method to simplify the security proof of QKD.

Related papers

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Privacy amplification (PA) is an indispensable component in classical and quantum cryptography. Error correction (EC) and data compression (DC) algorithms are also indispensable in classical and quantum information theory. We here study these three algorithms (PA, EC, and DC) in the presence of quantum side information, and show that they all become equivalent in the one-shot scenario. As an application of this equivalence, we take previously known security bounds of PA, and translate them into coding theorems for EC and DC which have not been obtained previously. Further, we apply these results to simplify and improve our previous result that the two prevalent approaches to the security proof of quantum key distribution (QKD) are equivalent. We also propose a new method to simplify the security proof of QKD.

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Information processing protocols are typically built out of simpler parts, called primitives, and two of the most important such primitives are privacy amplification (PA) and data compression. The former extracts the truly secret part of some classical data, while the latter squeezes it into the smallest possible form. We show these tasks are dual in the setting of quantum information processing. Specifically, the tasks of PA of classical information against quantum adversaries and classical data compression with quantum side information are dual in the sense that the ability to perform one implies the ability to perform the other. The duality arises because the two protocols are connected by complementarity and the uncertainty principle in the quantum setting. Applications include a new uncertainty principle formulated in terms of smooth min- and max-entropies, which are useful in the study of one-shot protocols, as well as new conditions for approximate quantum error correction.

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We show that the Mayers-Shor-Preskill approach and Renner's approach to proving the security of quantum key distribution (QKD) are essentially the same. We begin our analysis by considering a special case of QKD called privacy amplification (PA). PA itself is an important building block of cryptography, both classical and quantum. The standard theoretical tool used for its security proof is called the leftover hashing lemma (LHL). We present a direct connection between the LHL and the coding theorem of a certain quantum error correction code. Then we apply this result to proving the equivalence between the two approaches to proving the security of QKD.

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In a standard quantum key distribution (QKD) scheme such as BB84, two procedures, error correction and privacy amplification, are applied to extract a final secure key from a raw key generated from quantum transmission. To simplify the study of protocols, it is commonly assumed that the two procedures can be decoupled from each other. While such a decoupling assumption may be valid for individual attacks, it is actually unproven in the context of ultimate or unconditional security, which is the Holy Grail of quantum cryptography. In particular, this means that the application of standard efficient two-way error-correction protocols like Cascade is not proven to be unconditionally secure. Here, I provide the first proof of such a decoupling principle in the context of unconditional security. The method requires Alice and Bob to share some initial secret string and use it to encrypt their communications in the error correction stage using one-time-pad encryption. Consequently, I prove the unconditional security of the interactive Cascade protocol proposed by Brassard and Salvail for error correction and modified by one-time-pad encryption of the error syndrome, followed by the random matrix protocol for privacy amplification. This is an efficient protocol in terms of both computational power and key generation rate. My proof uses the entanglement purification approach to security proofs of QKD. The proof applies to all adaptive symmetric methods for error correction, which cover all existing methods proposed for BB84. In terms of the net key generation rate, the new method is as efficient as the standard Shor-Preskill proof.

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Abstract Privacy amplification is the key step to guarantee the security of quantum communication. The existing security proofs require accumulating a large number of raw key bits for privacy amplification. This is similar to block ciphers in classical cryptography that would delay the final key generation since an entire block must be accumulated before privacy amplification. Moreover, any leftover errors after information reconciliation would corrupt the entire block. By modifying the security proof based on quantum error correction, we develop a stream privacy amplification scheme, which resembles the classical stream cipher, to solve the problems of final key generation delay and error spread. The stream scheme can also help to enhance the security of trusted-relay quantum networks. Inspired by the connection between stream ciphers and quantum error correction in our security analysis, we further develop a generic information-theoretic tool to study the security of classical encryption algorithms.

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Devashish Tupkary, Zhiyao Wang, Ernest Y. -Z. Tan

Quantum Key Distribution (QKD) protocols typically involve two separate steps of error-verification and privacy amplification. Both steps involve the application of a hash function sampled from a suitable hash family to classical data. A potentially simpler alternative is to implement a single-step hashing procedure, announce and compare a fixed number of bits of its output for error-verification, and retain the remaining bits as the final output key. This reduces the complexity of classical postprocessing, the amount of classical communication, and the amount of randomness needed in the protocol. While this construction has appeared in the literature, a suitable mathematical justification is lacking. In this work, we show that the security of such protocols, under certain simple conditions, follows from a slightly modified notion of secrecy of the protocol without the final error-verification step. In particular, we show that for most entropy-based proofs, the combined protocol can be implemented with no change to the key rate, and a minor change to the security parameter.

quant-pharXiv:2610.10702•Oct 9, 2026•0 citations
Privacy amplification (PA) is an indispensable component in classical and quantum cryptography. Error correction (EC) and data compression (DC) algorithms are also indispensable in classical and quantum information theory. We here study quantum algorithms of these three types (PA, EC, and DC) in the one-shot scenario, and show that they all become equivalent if modified properly. As an application of this equivalence, we take previously known security bounds of PA, and translate them into coding theorems for EC and DC which have not been obtained previously. Further, we apply these results to simplify and improve our previous result that the two prevalent approaches to the security proof of quantum key distribution (QKD) are equivalent. We also propose a new method to simplify the security proof of QKD.
Privacy amplification (PA) is an indispensable component in classical and quantum cryptography. Error correction (EC) and data compression (DC) algorithms are also indispensable in classical and quantum information theory. We here study these three algorithms (PA, EC, and DC) in the presence of quantum side information, and show that they all become equivalent in the one-shot scenario. As an application of this equivalence, we take previously known security bounds of PA, and translate them into coding theorems for EC and DC which have not been obtained previously. Further, we apply these results to simplify and improve our previous result that the two prevalent approaches to the security proof of quantum key distribution (QKD) are equivalent. We also propose a new method to simplify the security proof of QKD.
Information processing protocols are typically built out of simpler parts, called primitives, and two of the most important such primitives are privacy amplification (PA) and data compression. The former extracts the truly secret part of some classical data, while the latter squeezes it into the smallest possible form. We show these tasks are dual in the setting of quantum information processing. Specifically, the tasks of PA of classical information against quantum adversaries and classical data compression with quantum side information are dual in the sense that the ability to perform one implies the ability to perform the other. The duality arises because the two protocols are connected by complementarity and the uncertainty principle in the quantum setting. Applications include a new uncertainty principle formulated in terms of smooth min- and max-entropies, which are useful in the study of one-shot protocols, as well as new conditions for approximate quantum error correction.
We show that the Mayers-Shor-Preskill approach and Renner's approach to proving the security of quantum key distribution (QKD) are essentially the same. We begin our analysis by considering a special case of QKD called privacy amplification (PA). PA itself is an important building block of cryptography, both classical and quantum. The standard theoretical tool used for its security proof is called the leftover hashing lemma (LHL). We present a direct connection between the LHL and the coding theorem of a certain quantum error correction code. Then we apply this result to proving the equivalence between the two approaches to proving the security of QKD.
In a standard quantum key distribution (QKD) scheme such as BB84, two procedures, error correction and privacy amplification, are applied to extract a final secure key from a raw key generated from quantum transmission. To simplify the study of protocols, it is commonly assumed that the two procedures can be decoupled from each other. While such a decoupling assumption may be valid for individual attacks, it is actually unproven in the context of ultimate or unconditional security, which is the Holy Grail of quantum cryptography. In particular, this means that the application of standard efficient two-way error-correction protocols like Cascade is not proven to be unconditionally secure. Here, I provide the first proof of such a decoupling principle in the context of unconditional security. The method requires Alice and Bob to share some initial secret string and use it to encrypt their communications in the error correction stage using one-time-pad encryption. Consequently, I prove the unconditional security of the interactive Cascade protocol proposed by Brassard and Salvail for error correction and modified by one-time-pad encryption of the error syndrome, followed by the random matrix protocol for privacy amplification. This is an efficient protocol in terms of both computational power and key generation rate. My proof uses the entanglement purification approach to security proofs of QKD. The proof applies to all adaptive symmetric methods for error correction, which cover all existing methods proposed for BB84. In terms of the net key generation rate, the new method is as efficient as the standard Shor-Preskill proof.
Abstract Privacy amplification is the key step to guarantee the security of quantum communication. The existing security proofs require accumulating a large number of raw key bits for privacy amplification. This is similar to block ciphers in classical cryptography that would delay the final key generation since an entire block must be accumulated before privacy amplification. Moreover, any leftover errors after information reconciliation would corrupt the entire block. By modifying the security proof based on quantum error correction, we develop a stream privacy amplification scheme, which resembles the classical stream cipher, to solve the problems of final key generation delay and error spread. The stream scheme can also help to enhance the security of trusted-relay quantum networks. Inspired by the connection between stream ciphers and quantum error correction in our security analysis, we further develop a generic information-theoretic tool to study the security of classical encryption algorithms.
Quantum Key Distribution (QKD) protocols typically involve two separate steps of error-verification and privacy amplification. Both steps involve the application of a hash function sampled from a suitable hash family to classical data. A potentially simpler alternative is to implement a single-step hashing procedure, announce and compare a fixed number of bits of its output for error-verification, and retain the remaining bits as the final output key. This reduces the complexity of classical postprocessing, the amount of classical communication, and the amount of randomness needed in the protocol. While this construction has appeared in the literature, a suitable mathematical justification is lacking. In this work, we show that the security of such protocols, under certain simple conditions, follows from a slightly modified notion of secrecy of the protocol without the final error-verification step. In particular, we show that for most entropy-based proofs, the combined protocol can be implemented with no change to the key rate, and a minor change to the security parameter.