Steganography and Broadcasting

Fabrice Tachago

Abstract

Informally, steganography is the process of exchanging a secret message between two communicating entities so that an eavesdropper may not know that a message has been sent. After a review of some steganographic systems, we found that these systems have some defects. First, there are situations in which some concealment algorithms do not properly hide a secret message. Second, to conceal one bit of a secret message, some ask at least five documents and make at least two sampling operations, thus increasing their run-times. Considering the different ways to communicate with the receiver, we propose two steganographic systems adapted to the email communication whose algorithms are deterministic. To hide one bit of a secret message, our steganographic systems need only one document and performs one sampling operation and therefore significantly reduces the run-time.

Figures (13)

N. Hopper and al. [3] proposed a protocol based on a function called rejection sampling. The aim of function is to find in a distribution of covertexts, a covertext which its image by a given pseudo-random function is equal to the secret message’s bit that we want to hide. We present now the algorithms proposed in [3].

‘The concealment algorithm apples the #S procedure for each bit. The extracting algorithm simp applies pseudo-random function to each document received. This system possesses two defects: firstly, it needs more sampling operations (expensive operation y 7 ) which increases its run-time. The algorithm runs in O(|K|n). Secondly this system is not safe for a covertexts distribution which owns a small min-entropy [6]. To correct these flaws. Hopper and al. [4] have proposed another protocol. That protocol is still based To correct these flaws, Hopper and al. have proposed another protocol. That protocol is still based n a rejection sampling function. But this time, it makes at most two sampling operations and uses a rror correcting code to increase the reliability of the steganographic system.

The use of a error correcting code is not only time expensive but reduces the transmission rate of the steganographic system (number of bits of secret message transmitted by covertext sent). Indeed, without use of error correcting code the failure probability of concealment of this system varies between 1/4 and 1/3. It depends on the covertexts distribution.

N. Hopper et al. [5] proposed another steganographic system that no longer uses error correcting code. To hide a single bit, this system makes t copies of the secret message. And for each copy, it seeks a covertext in the channel whose image via a pseudo-random function is equal to the secret message by taking more than twice from the channel. the concealment and extracting algorithms run in O(¢) for one bit.

Alice and Bob share a channel C. F is a pseudo-random function where Fx : {0,1}4 x )> > {0,1}. Alice and Bob possess a secret key K € {0,1}* and are synchronized by a counter N € {0,1}". Let A = {< address1 >, < address2 >} be a set of array of addresses and A an uniform distribution of A. The following algorithms allow Alice and Bob to hide and extract a bit of secret message.

The idea behind this algorithm is simple. Suppose Alice and Bob can communicate using two envelopes of different colors (white and brown). Alice, to send a message to Bob, draws a document d from the channel, evaluates it by the function Fx (N,-). If the result Fx (NV, d) is equal to the bit of message she wants to hide, Alice sends the document d in the white envelope otherwise it’s sent in the brown envelope. When Bob receives the document d in a brown envelope, he takes the complement of the secret message provided by the document otherwise he takes the bit as provided.

Run-time: The execution time of this protocol is practically constant for concealment and extraction algorithms and depends on the size of the secret message that is obviously 1: T = O(1). The idea of this protocol is to do a broadcast with destination addresses. The order of the addresses in the broadcast is very important. This order determines whether the recipient must take the extracted secret message bit of the document or its complement. From the two recipient addresses, one can have only two types of diffusion address with respect of to the order. So A = {< address1, address2 >,< address2,address1 >}. Let A be a uniform distribution on A. We only need to consider mails sent one address recipient’s to extract the secret message.

Insecurity: The security of this protocol is exactly the same as the one expressed in the theorem 1.