1. Introduction
Network coding is a novel technique that allows the intermediate node to make a combination of its received messages before sending out to the network, instead of the store-forward method [
1], and it has been shown to offer large advantages in throughput, power consumption, and security in wireline and wireless networks. Field size and adaption to varying topologies are two of the key issues in network coding, since field size affects the complexity of encoding and decoding processes, and the code construction is related to the knowledge of network topology. Li et al. [
2] proved that linear network coding was able to achieve the multicast capacity as the field size was sufficiently large. Later, random linear network coding (RLNC) [
3] was proposed for the unknown or changing topology to achieve the multicast capacity asymptotically in field size and network size, in which nodes independently and randomly select coding kernels. Since then, network coding has attracted a substantial amount of research attention.
In practical communication networks, transmission is often under wiretapping attacks. A general communication model of wiretap network is specified by a quintuple
, where
is a directed graph of the network topology, where and are sets of nodes and edges, respectively;
α is the unique source node in the graph;
is the set of user nodes. Each user node is fully accessed by a legal user who is required to recover the source message without error or with a vanishing decoding error probability;
is a collection of subsets of . Each member in may be fully accessed by an eavesdropper;
specifies the capacities of edges in .
Especially, a wiretap network is called a
-wiretap network if the size of each edge set in
is
. Secure network coding was firstly introduced by Cai and Yeung to prevent information leaking to eavesdroppers with zero error probability of decoding at legitimate users [
4,
5]. They imposed an information theoretic security requirement that the mutual information between the source symbols and the messages available to the adversary must be zero. Given a network code with message length
K, and a wire-tap adversary that was capable to wiretap on at most
edges, Cai and Yeung [
5] suggested using a linear “secret-sharing” method to provide security in the network. Instead of sending
K message symbols, the source node sent
random symbols and
message symbols. Additionally, the code itself underwent a certain linear transformation. Cai and Yeung gave sufficient conditions for this transformation to guarantee security. They also showed that as long as the field size
, a secure transformation existed. In addition, their construction of the linear transformation took at least
time steps. This complexity, as well as the required lower bound on the field size
q, was quite restrictive when the scale of network was large.
Feldman et al [
6] proved that the problem of making a linear network code secure was equivalent to the problem of finding a linear code with certain generalized distance properties, and they also showed that the required field size for secure network coding could be much smaller if they gave up a small amount of overall capacity. Namely, sending
random symbols and
message symbols, then a random linear transformation would be secure with high probability as long as
, which allowed a trade-off between capacity and field size.
Furthermore, a new level of information theoretic security was defined as weakly secure network coding [
7], in which adversaries were unable to obtain any “meaningful” information about the source messages. The weak security requirements could also be satisfied when the number of independent messages available to the adversary was less than the multicast capacity. Ho et al. [
8] considered the related problem of network coding in the presence of a Byzantine attacker that could modify data sent from a node the the network.
The idea of wiretap network came from a wiretap channel of type II, which was firstly studied by Ozarow and Wyner [
9]. The transmitter sent a message to the legitimate receiver via a binary noiseless channel. An eavesdropper could observe a subset of received data from the receiver with a certain size. It was assumed that the eavesdropper could always choose the best observing subset of received digital bits to minimize the equivocation over sent data. Wiretap channel II can be regarded as a special case of wiretap network with
,
,
and
,
and
(see
Figure 1).
Note that since both the coding schemes developed in [
5] and [
6] rely on a Galois field with sufficiently large size, neither of them work on the classic wiretap channel of type II, where the symbol of each transmission is binary.
In this paper, we study a special class of wiretap network with a single source node and
K sink nodes (considered as distributed servers or disk blocks), which is depicted in
Figure 2. In this network model, the legitimate users are able to connect to any
sink nodes. On the other hand, there exist eavesdroppers who are able observe digital sequences from arbitrary
sink nodes. We propose a randomized secure network coding scheme, ensuring that every legitimate user is able to recover the source message with an arbitrarily small average decoding error probability while every eavesdropper has vanishing information about the source message. The coding scheme in this paper works over the binary field (alphabet), which indicates the complexity of the scheme does not increase accordingly with the scale of the network. Moreover, the coding scheme in this paper can work on the classic wiretap channel II readily, indicating that communication model in this paper includes wiretap channel II as a special case. Differences among the coding schemes in [
5,
6] and this paper are summarized in
Table 1.
The coding scheme in this paper comes from that of arbitrarily varying channels (AVCs). In fact, the network defined in this paper can be readily regarded as a special class arbitrarily varying wiretap channels (AVWCs) with constrained state sequences. The difference is that the receivers know the channel state sequences in our case, and hence just one single codebook is enough to assure the reliable transmission (see Remark 8 for details). The partitioning scheme is based on Csiszár’s almost independent coloring scheme [
10], which has been recently used to solve the security problem of wiretap channel II with the noisy main channel [
11]. Some results on the secrecy capacity of AVWCs can be found in [
12,
13,
14].
Designing a coding scheme with a small field size is critical in some practical engineering problems. As an example, consider the situation where the transmitter needs to send a big file to the receiver through the Internet. To achieve this, the big file is divided into many data frames. Since the size of each data frame is less than 1500 bytes according to the TCP (Transmission Control Protocol) or UDP (User Datagram Protocol) protocols, the number of data frames will be quite large when the size of the file is huge. Packet loss is quite a common problem in network communication. When some data frames get lost, a common method is to require the transmitter to send the lost data frames again. Now, supposing that we plan to deal with the problem of frame loss via the coding scheme, each sink node in
Figure 2 can be regarded as a data frame divided from the big file with
being the total number of the data frames. Since the size of each data frame is constrained, it cannot represent arbitrarily a large Galois field. Consequently, the number of
K cannot be arbitrarily large when the coding schemes in [
5,
6] are applied, indicating that the size of the big file is constrained. However, if our coding scheme is used, the size of the big file can be arbitrary.
Another application of this model is for splitting and sharing secrete information among authorized persons. In this scenario, a group of
n persons is allowed to reconstruct the secrete information correctly, while any groups with less participants can not read the split message. Please refer to [
15] and references therein.
The remainder of this paper is organized as follows: the notations and problem statements are introduced in
Section 2 and the main result is presented in
Section 3. Furthermore, the direct and converse proofs are given in
Section 4 and
Section 5, respectively.
Section 6 explains the coding scheme in
Section 4 via two simple examples.
Section 7 gives the discussions on the field size, and
Section 8 concludes this paper.
2. Notations and Problem Statements
Throughout the paper, is the set of positive integers and for any .
Random variables, sample values and alphabets (sets) are denoted by capital letters, lower case letters and calligraphic letters, respectively. A similar convention is applied to random vectors and their sample values. For example, represents a random N-vector (), and is a specific vector of in . is the Nth Cartesian power of .
Let “?” be a “dummy” letter. For any index set
and finite alphabet
not containing the “dummy” letter “?”, denote
For any given random vector
and index set
,
is a “projection” of onto with for , and otherwise;
is a subvector of .
The random vector
takes value from
, while the random vector
takes value from
.
Example 1. Supposing that , the index set and the random vector , we have and .
Proposition 1. For any N-random vector and index set , it holds that
Proof. Let
g be a mapping from
to
such that
for every
. One can easily verify that
g is an one-to-one mapping. Furthermore,
for every
, implying that
and
share the same distribution. This completes the proof of the proposition. ☐
The communication model of wiretap network with
K sink nodes, depicted in
Figure 2, consists of four parts, namely encoder, network, receiver and eavesdropper. The formal definitions of those parts are introduced in Definitions 1–4, respectively. The definition of achievable transmission rate is given in Definition 5.
Definition 1. (Encoder) The source message W is uniformly distributed on the message set . The (stochastic) encoder is specified by a matrix of conditional probability for and , indicating the probability that the message w is encoded into the digital sequence , where is binary.
Definition 2. (
Network) Suppose that the source message
W is encoded into
. The encoder firstly divides
into
K parts, denoted by
with
for all
, where
is an integer without loss of generality. Then, those sequences are transmitted to the sink nodes
, respectively, through
K noiseless channels. Therefore, the digital sequence received by the sink node
is
for every
. Let
The sequence
can then be rewritten as
.
Definition 3. (
Receiver) Let
be a constant real number with
being an integer. The receiver is able to access digital sequences from arbitrary
sink nodes. Let
be the collection of subsets of sink nodes possibly selected by the receiver. The whole digital sequence obtained by the receiver may be any random sequence from
where
with
. It is clear that
is distributed on
. Denoting by
the decoder is a mapping
. If it is known that the receiver has access to the sink nodes, whose indices lie in
, the estimation of the source message is then denoted by
, and the average decoding error probability is
. However, since the sink nodes accessed by the receiver are actually unknown, the average decoding error is defined as
Remark 1. Denoting
, it follows that
for every
. Consequently, setting
yields
for every
.
Remark 2. The communication model can also be regarded as a wiretap network with
legitimate receivers, each of whom has access to a certain set of sink nodes in
. Equation (
2) represents the maximal value of the average decoding error probabilities of all those legitimate receivers.
Definition 4. (
Eavesdropper) Let
be a constant real number with
being an integer. The eavesdropper is able to access digital sequences of arbitrary
sink nodes. Let
be the collection of subsets of sink nodes possibly selected by the eavesdropper. The whole digital sequence obtained by the eavesdropper may be any random sequence from
where
with
. The quantity of source information exposed to the eavesdropper is then denoted by
Remark 3. Similar to Remark 1, denoting
and
it follows that
for every
.
Remark 4. The communication model can also be regarded as a wiretap network with
eavesdroppers, each of whom has access to a certain set of sink nodes in
. Equation (
3) represents the maximal quantity of exposed source information to those eavesdroppers.
Example 2. To have a clearer idea on the notations defined in this section, a special example of wiretap network is given in
Figure 3 with
This can be treated as a network with three receivers and three eavesdroppers. In this case, we have
with
and
Definition 5. (
Achievablility) A non-negative real number
R is said to be achievable, if for any
there exists an integer
such that for any
, one can construct a pair of encoder and decoder
of length
N satisfying
and
The capacity of the communication model described in
Figure 2 is denoted by
.
Remark 5. When regarding the communication model depicted in
Figure 2 as a wiretap network with multiple legitimate receivers and multiple eavesdroppers, Equations (
5) and (
6) require that every legitimate receiver is able to decode the source message with a vanishing average decoding error probability, and the quantity of information about the source message exposed to every eavesdropper is vanishing.
4. Direct Half of Theorem 1
This section gives the proof of the direct half of Theorem 1, i.e., it is achievable for every
. More precisely, we need to prove that for any
and any sufficiently large
N, there exists a pair of encoder and decoder
satisfying
and
On account of Remarks 1 and 3, it follows that
and
Therefore, instead of constructing encoder and decoder pair
satisfying Equations (
7)–(
9), this section would prove the existence of the encoder and decoder pair satisfying Equation (
7),
and
The main idea of the proof goes as follows. Let the codebook C be randomly generated, such that the size of the codebook is about . Then, partition the codebook C into about subcodes, each of which is related to a unique source message. Since the receiver is able to obtain a -subsequence of the transmitted codeword, which is probably distinct from those corresponding subsequences of other codewords, the receiver is able to decode the source message with a vanishing average decoding error probability. On the other hand, receiving a -subsequence of the transmitted codeword, the eavesdropper concludes that the transmitted codeword comes from a collection of about codewords. If those codewords are uniformly distributed on every subcode, the eavesdropper is unable to have any information on the source message.
The proof is organized as follows.
Section 4.1 firstly gives the coding scheme achieving the capacity. Then,
Section 4.2 establishes that, using the scheme of generating codebook randomly, we can obtain the codebook satisfying Equation (
10) with probability
when
. Finally,
Section 4.3 shows that when
N is sufficiently large, there exists a desired “good” partition on every random generated sample codebook such that Equation (
14) holds. Equation (
11) is an immediate consequence of (
14) and Remark 6. Equation (
7) is obtained directly from (
13). Therefore, the direct half of Theorem 1 is totally established.
4.1. Code Construction
Codebook generation. Let
be the ordered set of
i.i.d. random vectors with mass function
for all
and
, where
Codebook partition. Suppose that
is a specific sample value of
randomly generated codewords. Let
be a random variable uniformly distributed on
and
be the random sequence uniformly distributed on
. Set
and partition
into
subsets
with equal cardinality, i.e.,
. Let
be the index of subcode containing
, i.e.,
. We need to find a partition of the codebook
satisfying
The partition satisfying Inequality (
14) is called a “good” partition. It will be proved in
Section 4.3 that there exists desired “good” partition on every given sample codebook when the block length
N is sufficiently large.
Encoder. Suppose that a desired partition on a specific codebook is given. When the source message W is to be transmitted, the encoder uniformly randomly chooses a codeword from the subcode and emits it to the network.
Remark 6. For a given codebook and a desired partition applied on it, denote by the output of the encoder, when the source message W is transmitted. It is clear that and share the same joint distribution.
Decoder. Suppose that a desired partition
on a deterministic codebook
is given. Receiving digital sequence
from the sink nodes, the decoder tries to find the minimal number of
such that
, and decodes
as the estimation of the transmitted source message, where
is the index of subcode containing
, i.e.,
, and
4.2. Proof of Inequality (10)
This subsection establishes that using the coding scheme introduced in
Section 4.1, one can generate a codebook satisfying Equation (
10) with probability
when the block length
.
Let
be a fixed codebook applied by the encoder. For any
and
, denote
and
Then, it follows from the decoding scheme introduced in
Section 4.1 that
Therefore, Equation (
5) is finally established by the following lemma, whose proof is given in
Appendix A.
Lemma 1. Let be the codebook randomly generated via the scheme introduced in Section 4.1. It holds thatwhere Remark 7. It is clear that
as
, which concludes from Equations (
15) and (
16) that we can obtain the codebook satisfying (
10) with probability
when
.
Remark 8. The idea of generating codebook randomly comes from the random code for AVCs, which was firstly established by Blackwell et al. [
16] and further developed by Ahlswede and Wolfowitz [
17] (see also Lemma 12.10 in [
18]). The coding scheme for AVCs is based on the following results. Let
be a random codebook with
being smaller than the capacity. If the decoding scheme of maximal mutual information (MMI) is applied by the decoder, it follows that the expected average decoding error probability under each state sequence is
, when
N is sufficiently large. To make the random coding scheme work, for each transmission, we need a separate channel sharing the exact sample value of the random codebook, which is called the common randomness (CR). However, that would occupy a large amount of bandwidth. To solve this problem, Ahlswede developed an elimination technique [
19] and claimed that it sufficed to let the random codebook
C be uniformly selected from a collection of
deterministic codebooks. Moreover, if the capacity of an AVC was positive, the encoder could send the index of selected codebook before each transmission, and no extra CR was needed.
In fact, the network model in this paper can be regarded as a special case of AVCs with state sequences known at the receiver, if we ignore the participation of eavesdroppers. The capacity of the current network model is obviously positive since each receiver has access to at least one noiseless channel. Therefore, the coding scheme for AVCs works on the current network with no need of extra CR. Nevertheless, we should point out that the communication model of AVCs with state sequences known at the receiver is essentially different from the classic AVCs. In the former model, the decoder knows exactly the probability distribution of the channel input, and this would reduce the degree of difficulty on the coding scheme. In particular, it is proved in
Appendix A that a single deterministic codebook is sufficient for the current network model.
4.3. Proof of the Existence of “Good” Partition for Every Given Sample Codebook
This subsection proves the existence of “good” partition satisfying Equation (
14) for every codebook generated via the scheme in
Section 4.1, when
N is sufficiently large. The result in this subsection can establish Equation (
9) immediately on account of Remark 6. Notations in
Section 4.1 will continue to be used in this Subsection.
The main result of this subsection is given in the following lemma.
Lemma 2. For any generated sample codebook of length N satisfyingthere exists a partition on it such thatfor all . Remark 9. Equation (
14) is finally established from the fact that the right-hand side of Equation (
17) converges to 0 as
.
Proof of Lemma 2.. The main idea of the proof is firstly pointed out here. For any
, to satisfy
, we need
. On account of the following obvious equality
it suffices to construct a partition satisfying
for almost all the
. In the following proof, we will construct a collection of subsets
of
, namely Equation (
21), and prove that there exists a partition on
such that
for all
. Then
is proved on account of Equation (
23).
The proof, based on Csiszár’s almost independent coloring scheme, is divided into the following three steps. Step 1 constructs a mapping
satisfying Equations (
27) and (
28) with the help of Lemma 2. Step 2 establishes Equation (
29) from (
28). Step 3 constructs a “good” partition satisfying Equation (
14) from the mapping
f with the help of Lemma 4.
Proof of Step 1. The following lemma plays an important role in the proof of step 1.
Lemma 3. (Lemma 3.1 in [20]) Let be a set of distributions on . If there existand , such thatholds for all , then for any positive integer,there exists a function , such thatholds for all . To apply Lemma 3, the main task is to construct the parameter
. In our proof, each element
P in
is a conditional probability distribution of
for a given
for
and
. The set
is defined as
where
The useful properties of
are given in the following proposition.
Proposition 2. For any , and , it follows thatand The proof of Proposition 2 will be given later in this subsection. With the help of
, the parameters introduced in Lemma 3 are introduced as
where
and
The verification that parameters given in Formula (
24) satisfy the requirements of Equations (
18)–(
20) is given in
Appendix B.
Remark 10. Since for every , where , it follows that
Applying Lemma 3 with Formula (
24), there exists
satisfying that
and
for all
and
, where
.
Remark 11. It is clear that the function
f will produce a partition on the codebook
. Equation (
27) ensures that every subcode in the partition has almost the same cardinality. Equation (
28) ensures that
for
.
The proof of Step 1 is completed.
Proof of Step 2. Set
. On account of Equation (
28) and the uniformly continuity of entropy (cf. Lemma 2.7 in [
18]), it follows that
for all
and
. Combining Equation (
23) and the equation above,
for every
. Since
, we arrive at
for every
. The proof of Step 2 is completed.
Proof of Step 3. The proof depends on the following lemma.
Lemma 4. For any given codebook , if the function satisfies Equation (27), there exists a partition on such that- 1.
for all ,
- 2.
,
where is the index of bin containing , i.e., . The proof of Lemma 4 is discussed in
Appendix C. In fact, Equation (
27) indicates that the random variable
is almost uniformly distributed on
. This implies the cardinalities of the sets
are quite close. Therefore, a desired partition with the same cardinality can be constructed through slight adjustments.
From Lemma 4 and Equation (
29),
for all
, where
M and
ε are given by Equations (
13) and (
24), respectively. This completes the proof of Step 3.
The proof of Lemma 2 is completed. ☐
Proof of Proposition 2. Equation (
22) follows because
for every
and
with
. Equation (
23) follows because
where (
a) follows from the fact that
is uniformly distributed on
, (
b) follows from Equation (
12) and the fact that
when
(cf. Equation (
21)), and (
c) follows because
. The proof of Proposition 2 is completed. ☐
6. Examples
This section gives two simple examples, showing how the coding scheme introduced in
Section 4.1 works.
Example 3. Let
,
and
. We obtain a network with two sink nodes depicted in
Figure 4. In this network, the legitimate receivers are able to access both of the sink nodes, while the eavesdroppers are able to access only one sink node. It is easy to construct a code satisfying
,
and
. The coding scheme goes as the following.
Codebook generation and partition. Let the codebook be partitioned as and .
Encoder. The source message
W is uniformly distributed on the message set
in this example. To transmit
W, a random key
K, which is uniformly distributed on
and independent of
W, is firstly generated. Then, the encoder emits a codeword
into the network.
Figure 5 shows the digital bits emitted into the sink nodes with respect to different values of
W and
K.
Example 4. Let
,
and
. We obtain a network with three sink nodes, which is similar to that depicted in Example 2 (see also
Figure 3). The only difference is that the block length
in this example. Therefore, we have
Suppose that the digital sequence emitted to the network is
. The digital sequences received by Receiver 1, Receiver 2 and Receiver 3 are denoted by
,
and
, respectively. The digital sequences received by Eavesdropper 1, Eavesdropper 2 and Eavesdropper 3 are denoted by
,
and
, respectively.
It can be verified by enumerating all the possible coding schems that constructing a code satisfying
,
and
is impossible for this example. A coding scheme, achieving that
is given as the following. The codebook is defined and partitioned as
and
. The encoding scheme is similar to that introduced in Example 3, and hence omitted. The decoding scheme and the calculation of
and Δ are detailed below.
Decoding scheme and calculation of .
This part calculates the average decoding error probability of all the three receivers.
Combing the discussions above, it is concluded that the average decoding error probability of the coding scheme is
Calculation of Δ.
This part calculates the amount of the source information exposed to the eavesdroppers. We only take Eavesdropper 1 as an example for the sake of simplicity. The received digital sequences with respect to different values of W and K are given in the following table.
W and K | Received Sequence |
W = 1, K = 1 | 0?? |
W = 1, K = 2 | 0?? |
W = 2, K = 1 | 0?? |
W = 2, K = 2 | 1?? |
According to the table above, it follows that
This indicates that
Moreover, it can be obtained similarly that
Therefore, we conclude that
Remark 12. One may find that the eavesdroppers can also decode the source message with the average decoding error probability in Example 4. This indicates that the eavesdroppers have the same decoding ability as Receiver 1 and Receiver 2. The coding scheme is not a desired one. In fact, a sufficiently large block length N is necessary to design a desired coding scheme. The examples in this section do not focus on the construction of optimal coding scheme. They just show the encoding and decoding processes when the codebook is given.
7. Discussions on Field Size
In [
5], Cai and Yeung have constructed an admissible linear block code over a more general wiretap network with intermediate nodes. However, the construction was working on
with
q being sufficiently large. Applying coding scheme in [
5] to the wiretap network of this paper, we get the following linear network encoder and decoder:
The source message W is uniformly distributed on the message set of size M;
The stochastic encoder is a matrix of conditional probability for and , indicating the probability that the message w is encoded into the digital sequence , where is actually the . Supposing that the source message W is encoded into , the encoder emits the random symbol to the sink node ;
For every , the decoder is a mapping from to , where (see Definition 3);
The decoding error probability is defined as
and the quantity of source information exposed to the eavesdroppers is defined as
Remark 13. The decoding error probability and the quantity of exposed information formulated in Equations (
38) and (
39), respectively, are similar to those formulated in Equations (
2) and (
3), except that
is distributed on
while
is distributed on
, where
is binary,
is
and
.
Remark 14. The coding scheme constructed above is called linear, because the encoder
can be interpreted as a generating matrix from
. Please refer to [
5] for more details.
The following theorem is actually a direct consequence of Theorem 3 in [
5].
Theorem 2. Suppose that the wiretap network depicted in Figure 2 with K sink nodes is given. Let be a alphabet of size q such thatwith and for fixed (cf. Definitions 1 and 3). Then, there exists a pair of linear block encoder and decoder working on the alphabet such thatwhere and are given by Equations (38) and (39), respectively. Theorem 2 asserts that the capacity is able to be achieved absolutely with exactly no decoding error and no exposed source information, if the alphabet is sufficiently large. However, it is well known that the alphabets of most channels are binary. To make the theorem work on the binary channels, it is necessary to map the elements in on to binary digital sequences, which produces the following corollary.
Corollary 1. When with q satisfying Equation (40), there exists a pair of encoder-decoder formulated by Definitions 1 and 3 (working on binary alphabet), such thatwhere and Δ are given by Equations (2) and (3), respectively. Proof. Let
and
be a pair of linear network code working on the alphabet
such that Equation (
41) holds, where
q satisfies Equation (
40). Denote
. It is easy to construct an injection
.
Set
. The stochastic encoder
E is defined as
for every
and
, where
is given by Equation (
1) for
and the range
is the subset of
. The decoder
ϕ is defined as
for every
and
. One can easily verify that the pair of encoder and decoder
constructed above satisfies Equation (
42). The proof is completed. ☐
Remark 15. Corollary 1 requires the block length
N should satisfy that
indicating
N is an approximately quadratic function of
K, where the second inequality follows from (
40) and the third inequality follows from Lemma 2.3 in [
18].
Remark 16. Through a similar way of establishing Corollary 1, it is concluded that with the coding scheme introduced in [
6], for any
and
, there exists a pair of encoder-decoder
formulated by Definitions 1 and 3 (
with binary alphabet), such that
where
and Δ are given by Equations (
2) and (
3), respectively.
Remark 15 claims that the block length
N is an approximately quadratic function of
K if the linear network coding scheme introduced in [
5] is applied over the wiretap network and one needs to emit about
digital bits onto each edge. Remark 16 asserts that by sacrificing a small portion of transmission rate, the length of digital bits emitted on each edge can be decreased to
, if the coding scheme in [
6] is applied. Nevertheless, when the number
K of sink nodes is large, both of the encoding processes turn out to be quite complicated. To have a comparison between the coding scheme in this paper and those in [
5,
6], the following corollary claims that by sacrificing a tiny portion of transmission rate, it is possible to construct a pair of encoder and decoder with a vanishing average decoding error probability and vanishing exposed source information to the eavesdroppers such that only one digital bit is transmitted onto each edge when the number
K of sink nodes is sufficiently large.
Corollary 2. For any given and , if the block length N satisfiesandone can construct a pair of encoder and decoder formulated by Definitions 1 and 3 (with binary alphabet), such that Proof. On account of Lemma 1, Equation (
44) claims the existence of codebook
with
. Invoking Lemma 2, Equations (
45) and (
46) indicate the existence of partition on the codebook
satisfying Equation (
17). The inequality
is established from Equations (
17), (
47) and Remark 6. The corollary is proved. ☐
Remark 17. The constraints (
44)–(
47) are independent of
K. Therefore, when
K is sufficiently large, only Inequality (
43) is active. This indicates that we can set
, and hence it suffices to emit only one digital bit to each edge, when the number
K of sink nodes is sufficiently large.
When
and
, the wiretap network depicted in
Figure 2 is equivalent to the wiretap channel II [
9] of
N times of transmission. Each edge in the network is related to one time of transmission in wiretap channel II. See
Figure 1. Therefore, the coding scheme introduced in this paper also works for the communication model of wiretap channel II. However, it is clear that the coding schemes introduced in [
5,
6], which depend on the size of alphabet, do not work for wiretap channel II.
After the discussion above, it has been known that the major advantage of the coding scheme in this paper is that there exists a pair of encoder and decoder such that the source message is transmitted to the legitimate receivers with exactly one time of transmission, when the number
K of sink nodes is sufficiently large. More precisely, denoting by
the minimal value of
N satisfying Equations (
44)–(
47), when
and
ϵ are given, the number sink nodes
K should be at least
to implement the one-time transmission. The values of
versus
and
is given by
Figure 6. The figure shows that when
, the value of
is totally determined by the value of
. As a concrete example, the data points marked in
Figure 6 are those with
and they lie on the identical horizontal line and hence share the same value of
.