Provably Secure Steganography
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10-02-2010, 10:56 PM


Steganography is the problem of hiding secret messages in innocent-looking public communication so that the presence of the secret messages cannot be detected. This paper introduces a cryptographic formalization of steganographic security in terms of computational indistinguishability from a channel, an indexed family of probability distributions on cover messages. We use cryptographic and complexity-theoretic proof techniques to show that the existence of one-way functions and the ability to sample from the channel are necessary conditions for secure Steganography. We then construct a steganographic protocol, based on rejection sampling from the channel that is provably secure and has nearly optimal bandwidth under these conditions. This is the first known example of a general provably secure steganographic protocol. We also give the first formalization of robust Steganography, where an adversary attempts to remove any hidden messages without unduly disrupting the cover channel. We give a necessary condition on the amount of disruption the adversary is allowed in terms of a worst case measure of mutual information. We give a construction that is provably secure and computationally efficient and has nearly optimal bandwidth, assuming repeatable access to the channel distribution.
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31-07-2012, 04:06 PM

Provably Secure Steganography


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Abstract

Informally, steganography is the process of sending a secret message from Alice to Bob in such a way that an eavesdropper
(who listens to all communications) cannot even tell that a secret message is being sent. In this work, we
initiate the study of steganography from a complexity-theoretic point of view. We introduce de nitions based on computational
indistinguishability and we prove that the existence of one-way functions implies the existence of secure
steganographic protocols.

Introduction

The scienti c study of steganography began in 1983 when Simmons [17] stated the problem in terms of
communication in a prison. In his formulation, two inmates, Alice and Bob, are trying to hatch an escape
plan. The only way they can communicate with each other is through a public channel, which is carefully
monitored by the warden of the prison, Ward. If Ward detects any encrypted messages or codes, he will
throw both Alice and Bob into solitary con nement. The problem of steganography is, then: how can Alice
and Bob cook up an escape plan by communicating over the public channel in such a way that Ward doesn't
suspect anything shy is going on. (Notice how steganography is di erent from classical cryptography, which
is about hiding the content of secret messages: steganography is about hiding the very existence of the secret
messages.)
Steganographic \protocols" have a long and intriguing history that goes back to antiquity. There are
stories of secret messages written in invisible ink or hidden in love letters (the rst character of each sentence
can be used to spell a secret, for instance). More recently, steganography was used by prisoners and soldiers
during World War II because all mail in Europe was carefully inspected at the time [9]. Postal censors
crossed out anything that looked like sensitive information (e.g. long strings of digits), and they prosecuted
individuals whose mail seemed suspicious. In many cases, censors even randomly deleted innocent-looking
sentences or entire paragraphs in order to prevent secret messages from going through. Over the last few
years, steganography has been studied in the framework of computer science, and several algorithms have
been developed to hide secret messages in innocent looking data.

Related Work

There has been considerable work on digital steganography. The rst International Workshop on Information
Hiding occurred in 1996, with ve subsequent workshops, and even books have been published about the
subject [10]. Surprisingly, though, very little work has attempted to formalize steganography, and most of the
literature consists of heuristic approaches: steganography using digital images [8, 10], steganography using
video systems [10, 12, 18], etc. A few papers have given information theoretic models for steganography [3,
13, 15, 19], but these are limited in the same way that information theoretic cryptography is limited. We
believe complexity theory is the right framework in which to view steganography and, to the best of our
knowledge, this is the rst paper to treat steganography from a complexity-theoretic point of view (and to
achieve provably positive results).

Organization of the Paper

In section 2 we de ne the basic cryptographic quantities used throughout the paper, as well as the notions
of a cover channel and a stegosystem. In section 3 we de ne steganographic secrecy and state protocols
which are steganographically secret assuming the existence of one-way functions. In section 4 we de ne
robust steganographic secrecy for adversaries with bounded power to perturb stegotext messages and state
protocols which satisfy this de nition. Section 5 closes the paper with a discussion of implications.

Steganography

Steganography will be thought of as a game between the warden, Ward, and the inmate, Alice. The goal of
Alice is to pass a secret message to Bob over a communication channel (known to Ward). The goal of Ward
is to detect whether a secret message is being passed. In this and the following sections we will formalize
this game. We start by de ning a communication channel.

Robust Steganography

De nitions for Robust Steganography

Robust steganography will be modelled as a game between Alice and Ward in which Ward is allowed to
make some alterations to Alice's messages. Alice wins if she can pass a message with high probability, even
when Ward alters her message. For example, if Alice passes a single bit per channel message and Ward is
unable to change the bit with probability at least 1
2 , Alice can use error correcting codes to reliably transmit
her message. It will be important to state the limitations we impose on Ward, since otherwise he can replace
all messages with a new draw from the channel distribution, e ectively destroying any hidden information.
In this section we give a formal de nition of robust steganography with respect to a limited adversary.
We will model Ward's power as de ned by a relation R which is constrained to not corrupt the channel
too much. This general notion of constraint is sucient to include many simpler notions such as (for example)
\only alter at most 1% of the bits".

Complexity theoretic rami cations

Construction 1 gives a stegosystem which is steganographically secret for any channel distribution C which
has minimum entropy greater than 1, assuming the existence of a pseudorandom function family. Goldreich
et al [5] show how to construct a pseudorandom function from a pseudorandom generator, which in turn can
be constructed from any one-way function, as demonstrated by Hastad et al [6]. Thus in an asymptotic sense,
our constructions show that one-way functions are sucient for steganography. Conversely, it is easy to see
that a stegosystem which is steganographically secret for some C is a secure weak private key encryption
protocol in the sense of Impagliazzo and Luby [7]; and they prove that the existence of such a protocol
implies the existence of a one-way function. Thus the existence of secure steganography is equivalent to the
existence of one-way functions.
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