New Bounds on 2-Frameproof Codes of Length 4

Abstract

Frameproof codes were first introduced by Boneh and Shaw in 1998 in the context of digital fingerprinting to protect copyrighted materials. These digital fingerprints are generally denoted as codewords in , where Q is an alphabet of size q and n is a positive integer. A 2-frameproof code is a code C such that any 2 codewords in C cannot form a new codeword under a particular rule. Thus, no pair of users can frame a user who is not a member of the coalition. This paper concentrates on the upper bound for the size of a q-ary 2-frameproof code of length 4. Our new upper bound shows that when q is odd and .

1. Introduction

In order to protect a digital content, a distributor marks each copy with a codeword. This marking discourages users from releasing an unauthorized copy, since a mark allows the distributor to detect any unauthorized copy and trace it back to the user. However, a coalition of users may detect some of the marks, namely, the ones where their copies differ. Thus, they can forge a new copy by changing these marks arbitrarily. To prevent a coalition of users from “framing” a user outside the coalition, Boneh and Shaw [1] defined the concept of frameproof codes. A -frameproof code has the property that no coalition of at most users can frame a user not in the coalition. Frameproof codes are defined as follows.

Let q and n be positive integers. Let Q be a set of size q, and let be a set of words of length n over the alphabet Q. Each codeword can be represented as , where for all . The set of descendants of , , is defined as

Definition 1. Let be an integer such that . A -frameproof code is a subset such that for all with , we have that .

Example 1. Let . Then, C is a 3-ary 2-frameproof code of length 4.
As the number of codewords in the code corresponds to the number of authorized users in the digital copyright protection system, one of the classical question regarding frameproof codes is what is the largest cardinality of a -frameproof code of length n over an alphabet of size q? This paper concentrates on the special case when , , and q is large.
There has been extensive study on the upper bound and lower bound on the size of frameproof codes [2–11]. Some of those are in separating hash families’ language [3, 6, 10, 11]. Previous bounds are restated as follows.

Theorem 1 (see [2]). Let q, n, and be positive integers such that and . Then,

Theorem 2 (see [2]). Let n be a positive even integer such that . Let m be a prime power such that . Let . There exists a q-ary 2-frameproof code of length n of size

Theorem 3 (see [4]). There exists a q-ary -frameproof code of length of sizefor all odd q, when , and for all , when .

For 2-frameproof codes of length 4, Theorem 3 only gives lower bound for odd q. For even q, we first use only symbols to construct a -ary 2-frameproof code of size . Then, add a codeword to the code, where α is the unused symbol. Thus, the following result on the lower bound of is obtained.

Corollary 1 (see [4]). For any positive integer q,

In 2003, Blackburn proved the following upper bound of -frameproof codes.

Theorem 4 (see [2]). For any positive integers q, n, and , if C is a q-ary -frameproof code of length n, thenwhere r is a unique integer in such that .

When and , the following result on the upper bound of is obtained.

Corollary 2. For any positive integer q,

Much later on, in 2019, Cheng et al. proved the following theorem, which is the best previously known result on the upper bound of 2-frameproof code of length 4.

Theorem 5 (see [5]). For any positive integer , if C is a q-ary 2-frameproof code of length 4, then

Our result is the improved version of Theorem 5. We aim to prove the following theorem.

Theorem 6. For any odd positive integer , if C is a q-ary 2-frameproof code of length 4, then

We analyze the combinatorial structure of a code, setting up an optimization problem, deriving some constraints, and solving this optimization problem to obtain Theorem 6. The gap between the lower bound of odd and even q in Corollary 1 is the key motivation for proving the main result. The rest of this paper is ordered as follows. In Section 2, the essential·notations are defined. Necessary conditions of a q-ary 2-frameproof code of length 4 are also stated. In Section 3, the proof of Theorem 6 is provided. We conclude the result in the last section.

2. Preliminaries

In this section, we define some notations and state relating lemmas that are useful for proving the main theorem.

Let . For , for , and for any non-empty set ,(i)let (ii)let , for any (iii)let (iv)let , for any

We say x is unique under I if , and we say x is nonunique under I when .

For any , let and .

Remark. It is easy to see that for any nonempty subsets , the following conditions hold:(i)(ii)(iii), for any

Lemma 1. Let C be a q-ary 2-frameproof code of length 4. For any and any nonempty subset , if , then , where .

Proof. Let . Then, there exists such that .
Assume . Then, . Thus, there exists such that . Hence, . This contradicts the 2-frameproof property of C.
For convenience, we set up some parameters. Let , , , and . Without loss of generality, we assume .
ConsiderFrom Lemma 1, we can see that . Therefore, . Since , and are all pairwise disjoints, we haveHence, we can deduce upper bound of C from the upper bounds of , , and . The following sections are dedicated to the upper bounds of , , and .

2.1. Upper Bound of

This section gives the upper bound of the first component of equation (11).

Lemma 2. Suppose C is a q-ary 2-frameproof code of length 4. Then,

Proof. Assume . Then, we have that and thus . Clearly, there are at most choices of . Thus,From the definition of , all x in has different values of . This makes . So,Since , then . Therefore,as required.
We use this lemma to find constrains on the upper bound of . Then, after Section 2.3, we eliminate the term before proving the main theorem.

2.2. Upper Bound of

This section gives the upper bound of the second component of equation (11). It gives the same results as [5]. We put it here for completeness. We find an upper bound of by counting elements in .

Lemma 3. Suppose C is a q-ary 2-frameproof code of length 4. Then,

Proof. Assume . Then, x is nonunique under . Therefore, x must be unique under . Thus, when , the triple is a forbidden value in . We say x contributes at least forbidden values in . And hence we can eliminatevalues from .
Here, we reduce the size of further. Let and Assume . Then, x must be an element of . Therefore, x must be unique under . Thus, when , the triple is a forbidden values in . We say x contributes at least forbidden values in . To sum up, contributes at leastforbidden values in . Similarly, contributes at leastforbidden values in . However, the triple in the form of , where will be counted twice. Thus, together, and eliminatevalues from .
Furthermore, and are subsets of . This makes forbidden values counted twice in equations (17) and (20). Thus, we obtain at leastdifferent forbidden values in from this step.
Hence,Since and form a partition of , we haveFrom equations (22) and (23), we haveThus,We use this lemma to find constrains on the upper bound of in the next section.

2.3. Upper Bound of

This section aim to eliminate the third component of equation (11). Recall that we define , , , and .

Lemma 4. Suppose C is a q-ary 2-frameproof code of length 4 such that .
If , then and .

Proof. From equation (11), Lemma 2 and Lemma 3, we haveFrom Corollary 1, we have that there always exists a q-ary 2-FP code of size for any positive odd integer q. So, we havewhich can be rewritten asNote that if or , there are at most q codewords in C. We then only consider the case . We haveHence,Substitute in equation (28), we obtainThus,when .
Since it is impossible to have a single codeword in C, that is, nonunique under , then for all . Therefore, if , then . Thus, with the maximal of , we have . Furthermore, there are only 3 possible cases for , which are , , and .

Remark. Note that the condition can be removed by substituting corresponding values into equations (28) and (32) repeatedly until a contradiction is reached.

3. Main Results

In this section, we aim to prove Theorem 6, which is the main theorem.

By applying the condition from Lemma 4 to equations (11) and (26), we obtain new equations:

We also obtain the following corollary from Lemma 2.

Corollary 3. Suppose C is a q-ary 2-frameproof code of length 4. Then,

Using equations (33) and (34) and Corollary 3, we now prove Theorem 6.

Proof of the Theorem 6. Here, is the number of different order pairs , where x is nonunique in the last two positions. If we remove all codewords in from C, then symbols from are also eliminated from the first position of the remaining codewords. The condition implies . Thus, each order pair in is used at most times in the last two positions of remaining codewords. Thus,

So, by equations (33) and (36) and Corollary 3, we have

Thus, by Corollary 1,

Hence,

However, when , we have . Thus, we only have yet to consider the case that .

Substitute in (34), we obtain

Since must be an integer, as required.

We show that for odd . Thus, .

Example 1 gives a 3-ary 2-frameproof of size for , which is very close to the obtained upper bound. However, things could be different for larger q.

Consider from the proof of Theorem 6. Notice that for , must be an integer. We can conclude that for . Hence, for , we have . This implies which is equal to the lower bound. Thus, this could also be investigated further whether it is possible to push the upper bound down to when .

4. Conclusion

In this paper, we investigate the bounds of 2-frameproof codes with length 4 by observing the structure of a code. The improvement of the upper bound for the case of odd q is derived from the difference between the known lower bound of odd and even q. The paper shows that in the case when q is odd and . Then, if q is large, when q is odd and when q is even. Example 1, for , suggests that the upper bound might be tighter than the lower bound; however, the case of larger q is yet to be determined.

Data Availability

No data were used to support this study since all proofs are included in the manuscript.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This research was supported by Chiang Mai University.