Four techniques will be reviewed in the following. The first technique is to share secret images by the exclusive-OR operation. A discussion on the feasibility of applying the technique to texts will also be made. The second and third ones are about the so-called hierarchical sharing. Brief discussions on their applications to text
sharing will be presented. The last one is about data hiding in text documents.
2.1.1 Secret Sharing Technique by Exclusive-OR Operations
In this section, a technique of secret sharing using the exclusive-OR operation proposed by Lin and Tsai [19] will be reviewed. Originally, this technique is applied to share images. A secret image is exclusive-ORed pixel by pixel with some randomly selected images, all of the same size as that of the secret image. The selected images and the resulting image which is meaningless are regarded as shares and distributed to secret sharing participants.
Due to the constraint that the sizes of the selected images must be the same as that of the secret image, the technique can be utilized for texts only under the condition that the lengths of secret texts and those of selected texts are the same. This requirement can be relaxed for pure texts, which will be described in Chapter 3.
2.1.2 Secret Sharing Technique by Cooperative Sharing Operations
The cooperative sharing operation is one of the three main operations of hierarchical secret sharing proposed by Lin [20]. A property of the operation is that only when all shares are collected can the secret be recovered. In addition, secret sharing by cooperative sharing operations itself is an (n, n)-threshold method.
The function of cooperative sharing for two participants is shown as follows:
s b x a x x
f( )=( − )+( − )+ ,
where s is the secret, and a and b are randomly selected integers. And (a, f(a)) and (b, f(b)) are distributed to the two participants as their own share data of s, respectively.
The secret s can be recovered only when the two participants cooperate in the following way. From the viewpoint of the participant who keeps (a, f(a)), s equals to f(a) – a + b. Instead, from the other participant’s point of view, s can be obtained by computing f(b) – b + a.
For more than two participants, the function can be revised as follows:
∑=
where n is the number of participants, s is the secret, and all ai are randomly selected integers, where i = 1, 2, …, n. Hence, the ith participant keeps share data (ai, f (ai)), and for each participant, s can be revealed by the cooperative recovery formula in the following:
An illustrative example is described as follows. Let the participant number n be 3, the secret s be 24, and the randomly selected integer ai be in order 129, 3, and 10, for i = 1, 2, 3. The three participants’ share data are (129, 269), (3, –109), (10, –88), respectively. For the participant who keeps the share data (3, –109), according to the cooperative sharing recovery function, the computed value is −109 − [(3 − 1) × 3] + [129 + 10] = 24, which equals to s.
While trying to apply the technique to texts, a critical issue is encountered. As seen in the example illustrated above, the minimal space size for storing a piece of share data is not consistent. For share data (129, 269), three bytes will be used; as for either of the remaining ones, two bytes will be used. How to limit the size of each
piece of share data by controlling their magnitudes will be mentioned in 2.3.1.
2.1.3 Hierarchical Secret Sharing Technique
Hierarchical secret sharing is a new concept of secret sharing. From a behavior point of view, a senior person initiates the secret sharing activity among participants;
the new concept is first to share the secret among several groups formed by the participants and then, regarding each piece of share data as the group secret of the corresponding group, to continuously share the group secret of each group among the smaller groups formed by the participants of the group, until each participant of the secret gets his/her own share data. However, by applying three different sharing operations, namely, the cooperative sharing operation, the independent sharing operation, and the dominant sharing operation, the concept of “hierarchical” secret sharing can be truly realized. To understand the technique of hierarchical secret sharing, the three main sharing operations are first described and a description of how the hierarchical secret sharing technique works follows.
The description of the cooperative sharing operation will be given in 2.2.2 and is skipped here. The concept of independent sharing operation is that each participant knows the secret. Its corresponding function is described in the following:
∏=
where s is the secret, n denotes the number of participants, and ai denotes a randomly selected integer for the ith participant. The ith participant keeps share data (ai, f(ai)), where f(ai) equals to s.
The concept of dominant sharing operation is that only one “dominant”
participant can know the secret and the others can get the secret only after the permission of the dominant participant. The function of dominant sharing is as follows:
where s is the secret, n denotes the number of participants, and ai denotes a randomly selected integer, for i = 1 through n. The first participant keeping share data (a1, f(a1)) knows the secret because f(a1) equals to s, while the other participants must get the permission of the first participant to know the secret. Note that the ith participant, except the first one, can compute the secret by the formula s = f(ai) − ai + a1. That is, if these participants are trustworthy, the first participant can transmit just a1 to the ith participant via the Internet and the ith participant can know the secret without caring about the secure problem.
According to the requirement of a group of participants, the group secret is processed by one of the three sharing operations. An illustrative example is presented in Figure 2.1.
Suppose that Participants 1 and 2 are two managers of a company, Participant 3 is the president of the company, and Participant 4 is the secretary of the president.
Now, assume that a secret of the company is 5. The secret can be known only under the condition of acquiring the president’s and one of the two managers’ agreements. In order to avoid failure of secret recovery coming from the absence of the president and the hard time to reveal the secret, the secretary is standby for such the condition.
Usually, the president, Participant 3, and one of the two managers, for example, Participant 1, can cooperate to get the secret after three recovery operations.
Figure 2.1 A hierarchical secret sharing example.
Firstly, Participant 1 can recover his/her and Participant 2’s composite secret independently by simply extracting the second part of pairs. Secondly, Participant 3 can recover his/her and Participant 4’s composite secret by himself/herself by directly taking the second part of his/her own share data out. Finally, Participant 1 and Participant 3 can use their secrets, which are just recovered, as shares to recover the secret by the cooperative recovery formula. Under the condition just mentioned, the absent president can send the first part of his/her share data to his/her secretary and then his/her secretary can get their composite secret as well by computation.
As discussed in Section 2.2.2, the data magnitude control problem also exists in the hierarchical secret sharing technique for real world applications. The method in Section 2.3.1 is proposed for solving this problem.
2.1.4 Data Hiding for Text Documents
Three techniques of data hiding that will be reviewed here are proposed by Bender et al. [24]. These techniques hide data into text documents through manipulations of three kinds of spaces, including inter-sentence space, end-of-line space, and inter-word space, in text documents. For instance, one space between two
Cooperative
successive words, sentences, or lines may be regarded to represent a “0”, while two spaces a “1”. Therefore, the article in Figure 2.2, for example, contains 8 bit data
“01011110.”
W e a r e T h e W o r l d . Y o u a r e m y
s u n s h i n e .
Figure 2.2 An example of hiding data in a text document.