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Journal of Mathematical Physics

Hiding messages in quantum data

Julio R. Gea-Banacloche

Published Sep 1, 2002• 68 citations
Publisher

Abstract

A method is proposed to hide messages in arbitrary quantum data files. The messages may act as “watermarks,” to secure the authenticity and/or integrity of the data. With the help of classical secret keys, they can be made unreadable by other parties and to reveal whether thay have been tampered with. The basic idea is to encode the data using a quantum error-correcting code and hide the message as (correctible) errors, deliberately inserted, which can be read out from the error syndrome. Also discussed briefly is a “reverse encoding,” which would involve putting the actual data in the error syndrome, and letting the encoded qubit itself carry the message.

DOI
10.1063/1.1495073

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•Jan 1, 2009•10 citations
A method is proposed to hide messages in arbitrary quantum data files. The messages may act as “watermarks,” to secure the authenticity and/or integrity of the data. With the help of classical secret keys, they can be made unreadable by other parties and to reveal whether thay have been tampered with. The basic idea is to encode the data using a quantum error-correcting code and hide the message as (correctible) errors, deliberately inserted, which can be read out from the error syndrome. Also discussed briefly is a “reverse encoding,” which would involve putting the actual data in the error syndrome, and letting the encoded qubit itself carry the message.
Secret communication is everywhere around us today. Cryptography is being used to encrypt messages so that they can be read only by someone who has the key.Combining the art of steganography with the powers of computers has brought a method of hiding information to a whole new level. Steganography hides messages so that their very existence is undetectable. The idea of embedding some information within a digital media, in such a way that the inserted data are intrinsically part of the media itself, has aroused a considerable interest in different fields. This work has as a purpose to expand the field of applicability of the embedded information within a digital media from the classical informatics to the quantum one.
Steganography is the technique of hiding secret information by embedding it in a seemingly ``innocent'' message. We present protocols for hiding quantum information by disguising it as noise in a codeword of a quantum error-correcting code. The sender (Alice) swaps quantum information into the codeword and applies a random choice of unitary operation, drawing on a secret random key she shares with the receiver (Bob). Using the key, Bob can retrieve the information, but an eavesdropper (Eve) with the power to monitor the channel, but without the secret key, cannot distinguish the message from channel noise. We consider two types of protocols: one in which the hidden quantum information is stored locally in the codeword, and another in which it is embedded in the space of error syndromes. We analyze how difficult it is for Eve to detect the presence of secret messages, and estimate rates of steganographic communication and secret key consumption for specific protocols and examples of error channels. We consider both the case where there is no actual noise in the channel (so that all errors in the codeword result from the deliberate actions of Alice), and the case where the channel is noisy and not controlled by Alice and Bob.
We present and analyze a protocol for quantum steganography where the sender (Alice) encodes her steganographic information into the error syndromes of the perfect (five-qubit) quantum error-correcting code, and sends it to the receiver (Bob) over a depolarizing channel. Alice and Bob share a classical secret key, and hide quantum information in such a way that to an eavesdropper (Eve) without access to the secret key, the quantum message looks like an innocent codeword with a typical sequence of quantum errors. We calculate the average rate of key consumption, and show how the protocol improves in performance as information is spread over multiple codeword blocks. Alice and Bob utilize different encodings to optimize the average number of steganographic bits that they can send to each other while matching the error statistics of the depolarizing channel.
In the current thesis we first talk about the six-qubit quantum error-correcting code and show its connections to entanglement-assisted error-correcting coding theory and then to subsystem codes. This code bridges the gap between the five-qubit (perfect) and Steane codes. We discuss two methods to encode one qubit into six physical qubits. Each of the two examples corrects an arbitrary single-qubit error. The first example is a degenerate six-qubit quantum error-correcting code. We prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender (Alice) and the receiver (Bob) and corrects an arbitrary single-qubit error. In the second half of this thesis we explore the yet uncharted and relatively undiscovered area of quantum steganography. Steganography is the process of hiding secret information by embedding it in an innocent message. We present protocols for hiding quantum information in a codeword of a quantum error-correcting code passing through a channel. Using either a shared classical secret key or shared entanglement Alice disguises her information as errors in the channel. Bob can retrieve the hidden information, but an eavesdropper (Eve) with the power to monitor the channel, but without the secret key, cannot distinguish the message from channel noise. We analyze how difficult it is for Eve to detect the presence of secret messages, and estimate rates of steganographic communication and secret key consumption for certain protocols.
We present protocols for multiparty data hiding of quantum information that implement all possible threshold access structures. Closely related to secret sharing, data hiding has a more demanding security requirement: that the data remain secure against unrestricted attacks via local operation and classical communication. In the limit of hiding a large amount of data, our protocols achieve an asymptotic rate of one hidden qubit per local physical qubit. That is, each party holds a share that is the same size as the hidden state to leading order, with accuracy and security parameters incurring an overhead that is asymptotically negligible. The data-hiding states have very unusual entanglement properties, which we briefly discuss.
Abstract—The idea of embedding some information within a digital media, in such a way that the inserted data are intrinsically part of the media itself, has aroused a considerable interest in different fields. One of the more examined issues is the possibility of hiding the highest possible amount of information without affecting the visual quality of the host data. As opposed to the traditional hiding methods, which are using different mathematical methods in order to hide the information, steganography in quantum informatics implies an approach based on the laws of the quantum physics. This work has as a purpose to expand the field of applicability of the embedded information within a digital media from the classical informatics to the quantum one.