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arXiv:0707.0691 · IEEE Transactions on Information Theory

Quantum Entropic Security and Approximate Quantum Encryption

Simon Pierre Desrosiers, Frédéric Dupuis

Published Jun 23, 2010• 21 citations
arXivHTMLPDF

Abstract

An encryption scheme is said to be entropically secure if an adversary whose min-entropy on the message is upper bounded cannot guess any function of the message. Similarly, an encryption scheme is entropically indistinguishable if the encrypted version of a message whose min-entropy is high enough is statistically indistinguishable from a fixed distribution. We present full generalizations of these two concepts to the encryption of quantum states in which the quantum conditional min-entropy, as introduced by Renner, is used to bound the adversary's prior information on the message. A proof of the equivalence between quantum entropic security and quantum entropic indistinguishability is presented. We also provide proofs of security for two different ciphers in this model and a proof for a lower bound on the key length required by any such cipher. These ciphers generalize existing schemes for approximate quantum encryption to the entropic security model.

DOI
10.1109/tit.2010.2048488

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An encryption scheme is said to be entropically secure if an adversary whose min-entropy on the message is upper bounded cannot guess any function of the message. Similarly, an encryption scheme is entropically indistinguishable if the encrypted version of a message whose min-entropy is high enough is statistically indistinguishable from a fixed distribution. We present full generalizations of these two concepts to the encryption of quantum states in which the quantum conditional min-entropy, as introduced by Renner, is used to bound the adversary's prior information on the message. A proof of the equivalence between quantum entropic security and quantum entropic indistinguishability is presented. We also provide proofs of security for two different ciphers in this model and a proof for a lower bound on the key length required by any such cipher. These ciphers generalize existing schemes for approximate quantum encryption to the entropic security model.
Abstract Entropically secure encryption is a way to encrypt a large plaintext with a small key and still have information-theoretic security, thus in a certain sense circumventing Shannon’s result that perfect encryption requires the key to be at least as long as the entropy of the plaintext. Entropically secure encryption is possible when a lower bound is known on the entropy of the plaintext from the adversary’s point of view. The typical implementation is to expand the short key to the size of the plaintext, e.g. by multiplication with a public random string, and then use one-time pad encryption. This works in the classical as well as the quantum setting. In this paper, we introduce a new key expansion method that is faster than existing ones. We prove that it achieves the same security. The speed gain is most notable when the key length is a sizeable fraction of the message length. In particular, a factor of 2 is gained in the case of approximate randomization of quantum states. In the classical case, we obtain a reduction of the ciphertext size.
We present two new definitions of security for quantum ciphers which are inspired by the definition of entropic security and entropic indistinguishability defined by Dodis and Smith. We prove the equivalence of these two new definitions. We also propose a generalization of a cipher described by Dodis and Smith and show that it can actually encrypt n qubits using less than n bits of key under reasonable conditions and yet be secure in an information theoretic setting. This cipher also totally closes the gap between the key requirement of quantum ciphers and classical ciphers.
Entropically secure encryption is a way to encrypt a large plaintext with a small key and still have information-theoretic security, thus in a certain sense circumventing Shannon's result that perfect encryption requires the key to be at least as long as the entropy of the plaintext. Entropically secure encryption is not perfect, and it works only if a lower bound is known on the entropy of the plaintext. The typical implementation is to expand the short key to the size of the plaintext, e.g. by multiplication with a public random string, and then use one-time pad encryption. This works in the classical as well as the quantum setting. In this paper, we introduce a new key expansion method that is faster than existing ones. We prove that it achieves the same security. The speed gain is most notable when the key length is a sizeable fraction of the message length. In particular, a factor of 2 is gained in the case of approximate randomization of quantum states.
We derive a new entropic quantum uncertainty relation involving min-entropy. The relation is tight and can be applied in various quantum-cryptographic settings. Protocols for quantum 1-out-of-2 Oblivious Transfer and quantum Bit Commitment are presented and the uncertainty relation is used to prove the security of these protocols in the bounded quantum-storage model according to new strong security definitions. As another application, we consider the realistic setting of Quantum Key Distribution (QKD) against quantum-memory-bounded eavesdroppers. The uncertainty relation allows to prove the security of QKD protocols in this setting while tolerating considerably higher error rates compared to the standard model with unbounded adversaries. For instance, for the six-state protocol with one-way communication, a bit-flip error rate of up to 17% can be tolerated (compared to 13% in the standard model). Our uncertainty relation also yields a lower bound on the min-entropy key uncertainty against known-plaintext attacks when quantum ciphers are composed. Previously, the key uncertainty of these ciphers was only known with respect to Shannon entropy.
The advent of quantum computing poses a significant threat to the security of currently deployed public-key cryptosystems. In response, the field of post-quantum cryptography (PQC) has emerged, proposing new algorithms based on mathematical problems believed to be hard for both classical and quantum computers. While the security of these systems is typically analyzed using computational complexity theory, this paper explores an alternative and complementary approach rooted in information theory. We investigate the security of PQC schemes by establishing entropic bounds on the information available to a quantum adversary. By leveraging concepts such as conditional min-entropy and Rényi divergence, we formulate a framework to quantify the uncertainty of secret keys and plaintexts from an adversary's perspective. This paper derives explicit entropic security bounds for several major families of post-quantum cryptosystems, including lattice-based, code-based, and hash-based schemes. We demonstrate that the security of these systems can be directly related to the entropy of the underlying hard problems' solutions. The results provide a new lens through which to assess and compare the security of PQC candidates, offering insights that are independent of specific computational models and algorithmic assumptions. This information-theoretic perspective not only strengthens the security foundations of PQC but also offers practical guidance for parameter selection to ensure long-term resilience against future cryptanalytic advances.
The advent of quantum computing poses a significant threat to the security of currently deployed public-key cryptosystems. In response, the field of post-quantum cryptography (PQC) has emerged, proposing new algorithms based on mathematical problems believed to be hard for both classical and quantum computers. While the security of these systems is typically analyzed using computational complexity theory, this paper explores an alternative and complementary approach rooted in information theory. We investigate the security of PQC schemes by establishing entropic bounds on the information available to a quantum adversary. By leveraging concepts such as conditional min-entropy and Rényi divergence, we formulate a framework to quantify the uncertainty of secret keys and plaintexts from an adversary's perspective. This paper derives explicit entropic security bounds for several major families of post-quantum cryptosystems, including lattice-based, code-based, and hash-based schemes. We demonstrate that the security of these systems can be directly related to the entropy of the underlying hard problems' solutions. The results provide a new lens through which to assess and compare the security of PQC candidates, offering insights that are independent of specific computational models and algorithmic assumptions. This information-theoretic perspective not only strengthens the security foundations of PQC but also offers practical guidance for parameter selection to ensure long-term resilience against future cryptanalytic advances.