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Generalized Quantum Shannon Impossibility for Quantum Encryption

Ching–Yi Lai, Kai-Min Chung

Published Jan 11, 2018• 1 citation
OpenAlex

Abstract

The famous Shannon impossibility result says that any encryption scheme with perfect secrecy requires a secret key at least as long as the message. In this paper we prove its quantum analogue with imperfect secrecy and imperfect correctness. We also provide a systematic study of information-theoretically secure quantum encryption with two secrecy definitions.

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The famous Shannon impossibility result says that any encryption scheme with perfect secrecy requires a secret key at least as long as the message. In this paper we prove its quantum analogue with imperfect secrecy and imperfect correctness. We also provide a systematic study of information-theoretically secure quantum encryption with two secrecy definitions.
Shannon's perfect-secrecy theorem states that a perfect encryption system that yields zero information to the adversary must be a one-time pad (OTP) with the keys randomly generated and never reused. In this work we design the first encryption method (classical or quantum) that overcomes Shannon's theorem to achieve perfect secrecy with reusable keys. Because the mechanisms used are fundamentally quantum, Shannon's theorem remains true in the classical regime. Consequently, the quantum encryption design demonstrates decisive quantum advantage by achieving a goal impossible for classical systems. Finally, the design has major practical advantages by not requiring authentication and having silent tampering detection.
Abstract In this paper, we consider the relationship between C Shannon’s unicity distance, which is determined by the number of symbols of the encrypted message required to uniquely find the encryption key of the cryptosystem in information-theoretical sense, ε -secrecy criterion in quantum cryptography and Holevo information. A lower bound for the unicity distance for ε -secret keys in quantum cryptography is obtained. The relationship between unicity distance, ε -secrecy criterion in quantum cryptography and Holevo information is demonstrated.
We consider the scenario where Alice wants to send a secret (classical) nnn-bit message to Bob using a classical key, and where only one-way transmission from Alice to Bob is possible. In this case, quantum communication cannot help to obtain perfect secrecy with key length smaller then nnn. We study the question of whether there might still be fundamental differences between the case where quantum as opposed to classical communication is used. In this direction, we show that there exist ciphers with perfect security producing quantum ciphertext where, even if an adversary knows the plaintext and applies an optimal measurement on the ciphertext, his Shannon uncertainty about the key used is almost maximal. This is in contrast to the classical case where the adversary always learns nnn bits of information on the key in a known plaintext attack. We also show that there is a limit to how different the classical and quantum cases can be: the most probable key, given matching plain- and ciphertexts, has the same probability in both the quantum and the classical cases. We suggest an application of our results in the case where only a short secret key is available and the message is much longer.
We investigate the definition of security for encryption scheme in quantum context. We systematically define the indistinguishability and semantic security for quantum public-key and private-key encryption schemes, and for computational security, physical security and information-theoretic security. Based on our definition, we present a necessary and sufficient condition that leads to information-theoretic indistinguishability for quantum encryption scheme. The equivalence between the indistinguishability and semantic security of quantum encryption scheme is also proved.
This paper is dedicated to the late Professor H.P. Yuen in commemoration to our 50-year friendship. He invented the concept of quantum stream cipher. It is designed based on a completely different concept from conventional ciphers. The purpose of this cipher is to provide information-theoretic security of long data and secret key with short key length. It is based on hiding the ciphertext of mathematical cipher with quantum noise, achieving unprecedented information-theoretic security in any cipher. The protocol corresponds to randomizing the ciphertext by means of differentiating the receiving performance of Bob with key and Eve without key according to the principle of quantum communication theory. In this paper, we introduce some progress on the specific method to develop from the standard type to generalized quantum stream cipher (Quantum Enigma Cipher). There are two methods for generalization. One is the additional randomization method by the product cipher form and the other is the M-th order extended quantum coding method. Here we discuss the former. The results proved that it has sufficient information-theoretic security against known plaintext attack on key in comparison with quantum data locking. The latter method will be reported in Part II.
In this Thesis, several results in quantum information theory are collected, most of which use entropy as the main mathematical tool. *While a direct generalization of the Shannon entropy to density matrices, the von Neumann entropy behaves differently. A long-standing open question is, whether there are quantum analogues of unconstrained non-Shannon type inequalities. Here, a new constrained non-von-Neumann type inequality is proven, a step towards a conjectured unconstrained inequality by Linden and Winter. *IID quantum state merging can be optimally achieved using the decoupling technique. The one-shot results by Berta et al. and Anshu at al., however, had to bring in additional mathematical machinery. We introduce a natural generalized decoupling paradigm, catalytic decoupling, that can reproduce the aforementioned results when used analogously to the application of standard decoupling in the asymptotic case. *Port based teleportation, a variant of standard quantum teleportation protocol, cannot be implemented perfectly. We prove several lower bounds on the necessary number of output ports N to achieve port based teleportation for given error and input dimension, showing that N diverges uniformly in the dimension of the teleported quantum system, for vanishing error. As a byproduct, a new lower bound for the size of the program register for an approximate universal programmable quantum processor is derived. *In the last part, we give a new definition for information-theoretic quantum non-malleability, strengthening the previous definition by Ambainis et al. We show that quantum non-malleability implies secrecy, analogous to quantum authentication. Furthermore, non-malleable encryption schemes can be used as a primitive to build authenticating encryption schemes. We also show that the strong notion of authentication recently proposed by Garg et al. can be fulfilled using 2-designs.