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Secure Network Coding against Wiretapping and Byzantine Attacks
EURASIP Journal on Wireless Communications and Networking volumeÂ 2010, ArticleÂ number:Â 216524 (2010)
Abstract
In wireless networks, an attacker can tune a receiver and tap the communication between two nodes. Whether or not some meaningful information is obtained by tapping a wireless connection depends on the transmission scheme. In this paper, we design some secure network coding by combining informationtheoretic approaches with cryptographic approaches. It ensures that the wiretapper cannot get any meaningful information no matter how many channels are wiretapped. In addition, if each source packet is augmented with a hash symbol which is computed from a simple nonlinear polynomial function of the data symbols, then the probability of detecting the modification is very high.
1. Introduction
Network coding is a packetlevel coding technique that generalizes the classical routing paradigm [1]. Based on linear superposition of incoming packets at intermediate nodes, linear network coding achieves multicast capacity in singlesource wired networks [2]. Recently, wireless network coding has gained much attention as one skill to enhance the overall throughput in a wireless multihop network that supports multiple communication flows [1â€“4]. Like in wired networks, the basic idea is also that a relay node can combine several incoming packets. The wireless communication medium has inherent particularities, such as the broadcast nature, high error rates, and unpredictable signal strength, which create some opportunities for attackers. Thus, secure network coding [5â€“8] is a hot topic in wireless networks.
In [9], Cai and Yeung proposed a model which incorporates network coding and information security. This is the first time that network coding was used for secure transmission. Later, Jain [5] widened the sufficient condition in [9] and generalized these results to wireless networks. In general, secure network coding is designed against two kinds of attacks: wiretapping and Byzantine modification. And we call them type I and type II secure network coding, respectively.
Type I Secure Network Coding
The wiretapping attack means that some adversary can wiretap some communication signals with the purposes of curiosity or recovering the messages. In traditional transmission, packets are generally encrypted against wiretapping. However, Cai and Yeung [9] found that without cryptographic approaches one can securely transmit the message by using network coding. With the similar model for wireless networks, Jain [5] explored widening the sufficient condition in [9] so that it becomes necessary, too. Under the transmission rate of one unit, the sender can send a message to the receiver without leaking any information to a wiretapper. In [10], Feldman et al. showed that if a small amount of overall capacity is given up, then a random code achieves security by using a much smaller base field than that in [9]. Furthermore, they pointed out that a large field size may sometimes be required to achieve security without giving up any capacity. In [11], Bhattad and Narayanan generalized the model in [9] and gave a new information theoretic model for security which accommodates a lot more practical requirements on security.
In general, this previous secure network coding by informationtheoretic approaches has to restrict the eavesdropping set or give up some capacities. To address this problem, in this paper, we present a new secure network coding scheme by combining informationtheoretic and cryptographic approaches. In our scheme, we do not restrict the eavesdropping set, that is, the wiretapper can eavesdrop any communication signals. Moreover, we do not give up any capacity. Based on these superiorities, our scheme is more suitable for wireless multicast compared with the previous schemes [5, 9, 10].
Type II Secure Network Coding
Byzantine attack means that attackers may modify the coded packets. For this case, since some important packets that are modified by an adversary will mislead the receivers and maybe cause the receivers to make wrong decisions, the modification detection of packets transmitted is also very important compared with the modification correction. How to detect the modification is a hot topic in network coding theory. With cryptographic approaches, Charles et al. [6] proposed a signature scheme for network coding based on Weil pairing on elliptic curves. Later, in [7, 8] the authors proposed some signature schemes by the linearity property of the packets in a coded system for network coding. In these schemes, one detects the modifications at intermediate nodes, so they are computationally expensive. With informationtheoretic approaches, Ho et al. [12] showed a scheme in which Byzantine modification detections are done at sink nodes. In this scheme, he used random network coding by incorporating a polynomial hash value in each packet. By this way, the computing complexity is much less.
In this paper, to optimize the capacity loss and computation complexity, we propose a new scheme with Byzantine modification detection. Our scheme only needs one hash symbol which is much less than previous results in [12] and can achieve higher detection probability. Moreover, its computation complexity is lower than that in [12]. Furthermore, by combining cryptographic and informationtheoretic approaches, we present secure network coding against wiretapping and Byzantine attacks.
The rest of this paper is organized as follows. In Section 2, we mainly give some necessary notations and definitions. In Section 3, we show some secure network coding against wiretapping or detecting Byzantine modification. One example is given at the end of this section. Some conclusions are presented in Section 4.
2. Primaries
In this section, some necessary notations and definitions such as network model, descriptions of a linear network code and allornothing transform are presented. We denote matrices and linear spaces with bold uppercase letters and vectors with bold lowercase letters. All vectors are column unless some additional illustrations.
2.1. Network Model
Network coding has been leveraged as a generic technique in several types of wireless networks, such as vehicular ad hoc networks [13], wireless sensor networks [14], and Mesh networks [15],. In this paper, our focus is on secure network coding for acyclic wired networks and acyclic wireless networks which include parts of vehicular ad hoc networks, wireless sensor networks and Mesh networks. In detail, by the broadcast nature of the wireless interface each node is possiblly connected to several other nodes, where one node connects to node means that is in the coverage of s signal. By this way, we can obtain a directed graph . Our attentions are mainly focused on the acyclic wired networks and acyclic wireless networks, both of which can be represented by a directed acyclic network , where is the set of nodes and is the set of edges. The source node is denoted by , and edges are denoted by round brackets , in which and . Let In(Out) be the set of edges that end (start) at a vertex .
2.2. Descriptions of a Linear Network Code
Now we give two kinds of descriptions of a linear network code.
Definition 1 (see [16] (Local Description of a Linear Network Code)).
An dimensional linear network code on an acyclic network over a base field consists of a scalar , called the local encoding coefficient, for every adjacent pair of channels in the network. The matrix
is called the local encoding kernel at node .
Definition 2 (see [16] (Global Description of a Linear Network Code)).
An dimensional linear network code on an acyclic network over a base field consists of a scalar for every adjacent pair of channels in the network as well as a column vector for every channel such that

(1)
for â€‰Out;

(2)
the vectors for the imaginary channels form a standard basis of .
The vector is called the global encoding kernel for channel .
For convenience of decode, during the transmission process, global encoding kernels are combined in the head of packets.
2.3. AllOrNothing Transform
In [17], Rivest presented a model of encryption for block ciphers, which is called allornothing transform (AONT in short). AONT is defined for informationtheoretic security [18]. In detail, let be a finite field and be an dimensional space over . Suppose that . is named as an AONT if satisfies the following properties:

(1)
is a bijection;

(2)
If any of the output values is fixed, then the value of any input value is completely undetermined.
From this definition, for some input vectors (a basis of the space ) and the corresponding output vectors , we have the following result.
Theorem 1.
For any , if any of output vectors is fixed, then each input vector is completely undetermined.
Proof.
Let and , where denotes the transpose of vector . For any , from the definition of AONT if any of the output is fixed, then the value of any input is completely undetermined. Therefore, when any of output vectors is fixed, any input vector is completely undetermined.
If an AONT is also linear, is called a linear allornothing transform. In fact, the linear AONT is very useful for constructing secure linear network coding because of its low computation complexity and convenience for decoding. In [18], Stinson proved that for prime power and positive integer there exists a linear ANOT. Moreover, he constructed the following linear AONT which can be implemented very efficiently. Let , where is prime and is a positive integer. such that . Then the linear function defined by is a linear ANOT, where
We call an ANOT matrix. This transform (and the inverse transform) can be implemented very efficiently. Given , we can compute as follows:
Conversely, given , we can compute as follows:
where .
3. Main Schemes
In this section, we present some schemes that achieve different securities. Suppose that is the source rate. Each packet is represented by one vector in some linear space based on . The output packets of an AONT is called pseudopackets. In our schemes, AONT, the hash function and cryptosystem are public. The only shared secret is the key of the encryption when we use symmetric cryptsystem.
3.1. Against Wiretapping Attack
In wireless networks, because of the broadcast nature of the wireless interface, we canot determine which edges can be eavesdropped. So we canot obtain the same secure communication on the wireless networks if we made use of the scheme in [9] against wiretapping attacks. For example, consider the wireless network shown in Figure 1(a). From the presentation of the wireless network model in Section 2.1, we can get its equivalent graph model shown in Figure 1(b).
As for this wireless network, the scheme in [9] is not efficient and secure enough against wiretapping attack. In detail, the scheme in [9] for the network in Figure 1(b) is shown in Figure 2(a). The collection of sets of wiretap edges is ,,,. For secure transmission, Cai et al. added some randomly chosen key vector to the packet vector at source node . If the wiretapper only eavesdrops one channel in , he will get nothing about packet . However, the source only transmits one packet to the sink nodes in one transmission process. It means that one gets a secure transmission by giving up some capacity. In fact, if we choose an appropriate local encoding kernel for source node we can obtain the similarly secure transmission without capacity loss. In Figure 2(b), let a AONT matrix defined in (2) be the local encoding kernel of the source node , and . The network codes of the intermediate nodes in Figures 2(a) and 2(b) are the same. If the wiretapper only wiretaps one channel, he can not get any meaningful information about or even if he knows . Therefore, we can get secure transmission without giving up any capacity.
In reality, however, as for this wireless network, it is possible that the wiretapper can eavesdrop all the network linkages because of the broadcast nature of the wireless interface. Then the previous two schemes are not secure enough in practical applications. So some cryptographic approaches are required to address this problem. In fact, by combining ANOT with symmetrical cryptography, without constrictions of wiretapping sets, we can construct secure network coding in the sense of cryptographic security. That means the wiretapper cannot obtain any massage if he has not the secret key. Our secure network coding is presented as follows.
Scheme 1.
Let be packets, where . An AONT matrix is the local encoding kernel of source node .
Step 1.
Let . The source node encrypts using AES cryptsystem (the source can also choose other high speed asymmetric cryptsystem. And the only secret for this scheme is the private key owned by the sender and receiver.) and sends out , where .
Step 2.
Based on Jaggi's construction of network coding [19] for wired networks and Rajawat's [20] for wireless networks, we can construct the codes for the intermediate nodes in wired networks and wireless networks, respectively.
Step 3.
Each sink node first decodes the received packets and gets then decrypts and obtains . By the inverse of , they get the original packets .
Time Complexity Analysis
Since the orders of matrix and its inverse are both , the time complexity of multiplying or is at most . In addition, there are two operations, encryption and decryption. So the more time complexity of this construction than those of Jaggi's and Rajawat's is and the time for encryption and decryption.
Security Analysis
Since the network coding in this paper is linear, all of the network coding operations in the network are linear. The packets in the network are linear combinations of . On one hand, if the rank of the linear packets that an adversary eavesdrops is less than , he can only get some (not all) of the packets . By the definition of AONT, he can not obtain any original packet . On the other hand, even if the rank of eavesdropped packets is equal to , the wiretapper can not get the pseudopacket without the private key. So he can not obtain any original packet either by Theorem 1.
In this model, we do not need to encrypt all the transmitted packets (In [18], all pseudopackets are encrypted, because this requires an adversary to decrypt all the blocks of ciphertext to determine any block of plaintext by the definition of AONT. Then the attack will be slowed down without any change in the size of the secrete key. Therefore, AONT is used to afford a certain amount of additional security for a block cipher encryption.). Only one is enough by combing with AONT. By Theorem 1, each original packet is relative to all the pseudopackets. When we encrypt one of the pseudopackets, the wiretapper canot get all of the pseudopackets without the private key. So he canot obtain any original packet. For example in Figure 2(b), we only need to encrypt or , then the wiretapper can not get any meaningful information about and . The security here combines the informationtheoretic security with cryptographic security. However, by the wooden barrel theory the whole security of this scheme is reduced to cryptographic security.
Now, we show the advantages of AONT as the local encoding kernel of the source node. Firstly, from the informationtheoretic point: we not only increase the achievable throughput, but also get secure transmission. Secondly, from the cryptographic point: we only need to encrypt one packet out from the source node instead of encrypting all the packets which will be sent to sink nodes. Moreover, we can save lots of time consumption, explained from Table 1, where denotes the source rate, the length of each packet is 2 bytes and "clk" is the abbreviated clock.
3.2. Byzantine Modification Detection
Since some important packets that are modified by an adversary will mislead the receivers and may cause the receivers to make wrong decisions, the modification detection of packets transmitted is also very important compared with the modification correction in both wireless and wired networks. In this subsection, we present a scheme to detect the Byzantine modification combining AONT with a simple polynomial hash function.
By the definition of AONT, we find that if one of the pseudopackets from source node is damaged, then it is likely that every packet will be damaged. This is the errorpropagation property of AONT. So we can append a suitable block of redundancy to the packet before applying an AONT. And this redundancy can be used to verify the integrity of the packets and also can be removed after decode.
Suppose source multicast vector packets to the sink nodes. For convenience, each packet in the network is represented by a column vector of () symbols over a finite field , where the first entries are data symbols and the last one is a redundant hash symbol. The hash symbol in each augmented packet is given by a hash function of the data symbols. Of course, we can choose any nonlinear hash function. In fact, we find that the security can be ensured by a simple nonlinear function. In detail, we take the following simple nonlinear function as the secure hash function in this scheme.
Let be the function mapping to
Denote . Denote the augmented packets by
where denotes the concatenation of two vectors and , and is the hash symbol satisfying
Now we give a brief description of our scheme as follows.
Scheme 2.
Initialization: For each original packets , the source calculates the hash values , and obtains the augmented packets , by concatenating the hash value to each original packet .
Step 1.
The source takes the AONT matrix as its local encoding kernel and computes . Then sends out .
Step 2.
Based on the Jaggi's construction of network coding for wired networks and Rajawat's for wireless networks, we can construct the codes for the intermediate nodes.
Step 3.
Each sink node first decodes the received packets and gets . (Since in this scheme, we donot consider the wiretapping but the integrity of the packets, it does not need to encrypt any pseudopackets from the source. We note that is the local encoding kernel of the source , and thus we can decode directly and get the packets .) Then it verifies whether . If for all , then there does not exist modification on the transmission and . Finally, they remove the hash values and obtain the original packets .
Time Complexity Analysis
This scheme is similar to Scheme 1. The differences are additional calculations for hash symbols in Step 1 and verifications for hash symbols in Step 3. The time complexity of these two operations is . So the time complexity of Steps 1 and 3 is polynomial on the length of the packet vector and equal to . So the total time complexity of this secure network coding construction is only more than that of Jaggi's for wired networks and Rajawat's for wireless networks.
Security Analysis
Based on the model above, an adversary successfully modifies the packet that he can construct the logical hash symbol after modifying the data symbols (actually here he modifies the pseudopackets). From the following theorem we will find that an adversary can construct a logical hash symbol after modifying the data symbols with a very low probability.
Theorem 2.
In Scheme 2, the probability of not detecting an error is at most , where is the source rate.
To prove this theorem, we first prove the following two lemmas.
Lemma 3.
Given the vector and scalar value , the probability of randomly choosing a vector such that the inner product is .
Proof.
The number of points on the hyperplane is . And the cardinality of the field is . So the probability of choosing a vector such that is .
Lemma 4.
The probability of randomly choosing a vector such that
is at most , where the vectors and scalar values are fixed and independent.
Proof.
By Lemma 3, we randomly choose a vector such that with probability . Then the probability that choosing an appropriate vector such that satisfies (8) is at most .
Now we prove Theorem 2.
Let be the packet vectors transmitted, each of which consists of symbols from a finite field and is a column vector. The first entries are data symbols and the rest one is the redundant hash symbol. It can be represented as
where denotes the data. The hash symbol .
The matrix is the local encoding kernel of source node, and let
So the hash symbols satisfy
where denotes the hash symbol of . Therefore, because from (3). Notice that the hash function is not a linear function, that is, . When the adversary modifies some pseudopacket , he has to modify the hash symbol such that the sink nodes can not detect the modification. The proof can be completed by two steps.
Step 1.
We suppose that only the first pseudopacket is modified and the new pseudopacket is denoted by . Let and . is known to the adversary.
The adversary wants to know . By the representation ofâ€‰â€‰ in (4), we have
So even only one of is modified, all the packets will be changed. From (13), we have
Firstly, suppose that and . Then, by the definition of hash function in (5),
In (15), are unknown to the adversary except for and . Moreover, is unknown to the adversary, but fixed. By Lemma 4, the probability of constructing an appropriate satisfying (15) is at most .
Secondly, suppose that and are modified. Let and . Then,
From (16), the probability of randomly choosing the right and for constructing logical hash symbols is also at most by Lemma 4.
Thirdly, when the adversary modifies more data symbols of , by the similar method, we can prove that the probability of constructing the logical hash symbols is at most .
Step 2.
When the adversary modifies more pseudopackets at one time, from the similar method above, the probability of constructing the logical hash symbols is no more than .
From the proof of Theorem 2, we have the following two corollaries.
Corollary 1.
The probability of not detecting an error is not related to both the number of modified packets and the symbols of one pseudopacket, but the cardinality of and the source rate.
Corollary 2.
If the redundant hash symbol in the packet is a constant or a linear function of the data symbols, then the scheme can not defend the Byzantine modification.
Proof.
First, when the hash symbol is a constant. The adversary only modifies the data symbols and keeps the hash symbols unchanged. Then the receivers can not detect the modification.
Second, when the hash symbol is a linear function of the data symbols, we have
By (15), â€‰â€‰ are known by attackers. So, by the relationship between and , are easily calculated and this scheme canot defend Byzantine modification.
3.3. Against Wiretapping and Byzantine Attacks
Further, by combining with Scheme 1, we can improve Scheme 2 to against wiretapping attack. Before sending out , the source encrypts the last packet and denotes the encrypted packet by . The aim is to prevent wiretapper from recovering any original packets.
Scheme 3 provides not only security but also authenticity.
Scheme 3.
Initialization: For each packet , the source calculates the hash values , and obtains the augmented packets , by concatenating the hash value to each original packets .
Step 1.
The source takes as its local encoding kernel. Computes and encrypts using AES cryptsystem (Here we use symmetry cryptsystem. Because if we use asymmetry cryptsystem, by the public key the adversary may successfully modify all the pseudopackets at the same time when he controls edge disjoint paths.) to get . Then sends out .
Step 2.
Based on the Jaggi's construction of network coding for wired networks and Rajawat's for wireless network, we can construct the codes for the intermediate nodes.
Step 3.
Each sink node first decodes the received packets and gets , then gets by decrypting . Verify whether , . If for all , there does not exist modification on the transmission. They get the original packets by .
These three schemes are based on Jaggi's construction for wired networks and Rajawat's construction for wireless networks. Actually, we can also use Ho's random network coding [21]. In Scheme 2, the only change is to randomly choose the local encoding kernels from a large finite field. Except for the change in Scheme 2, in Schemes 1 and 3 the packets from the source will be appended with an dimensional identity vector, the global encoding kernel, before being sent out. However, random network coding for wireless networks requires a large alphabet size to render networks robust to link failures.
Example 5.
We construct a secure network code on the wireless network in Figure 1(a) to detect Byzantine modifications. Suppose the base field is . Let . Then So the AONT matrix , which is used to encode the two original packets at the source node , is
Suppose the packets and will be sent to the sink nodes and . The two encoded packets (pseudopackets) from the source are
Then the pseudopackets transmitted on the edges are shown in Figure 3. When an adversary wiretaps any one of , he can not get any meaningful information about packet or . If we encrypt the packet , then the adversary can get nothing even when he wiretaps all the channels.
Let . The last symbol is the hash symbol and calculated using the hash function in (4). The two augmented pseudopackets are
Suppose that an adversary modifies the data packet and let Then, the receiver can decode the packet vectors and get . It is easy to verify that and . So the receivers can find that the packets are modified.
4. Conclusion
For secure transmission, if only the informationtheoretic approach is used, some bandwidth has to be given up or a high computation complexity is necessary. As to cryptographic approach, all the packets have to be encrypted against wiretapping. Even if the data is hashed and appended with its hash value, one may not detect the modifications when the adversary modifies the data and its hash value simultaneously. To address these problems, we combine the informationtheoretic approach with cryptographic approach to design secure network coding. On one hand, we do not give up any network capacity to achieve the same security as that of Cai and Yeung. More importantly, our Scheme 1 does not require any restrictions on the wiretapping sets compared with that of Cai and Yeung. It means that our secure network coding is suitable for both wired networks and wireless networks. On the other hand, we decrease the resource consumptions of encryption and decryption. Furthermore, based on some simple hash function, our Scheme 2 is designed to detect the Byzantine modification. It can achieve a high detection probability with only one hash symbol and low computation complexity. In the end, combining the two schemes above we propose Scheme 3 which provides not only security but also authenticity.
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Acknowledgments
The authors would like to thank editor and all the anonymous reviewers for their helpful advices. This paper was supported by the National Natural Science Foundation of China and the Research Grants Council of Hong Kong Joint Research Scheme (no. 60731160626), the National Natural Science Foundation of China (no. 60821001), the Specialized the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (FANEDD) (Grant no. 200951), the 111 Project (no. B08004).
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Guo, Q., Luo, M., Li, L. et al. Secure Network Coding against Wiretapping and Byzantine Attacks. J Wireless Com Network 2010, 216524 (2010). https://doi.org/10.1155/2010/216524
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DOI: https://doi.org/10.1155/2010/216524
Keywords
 Wireless Network
 Source Node
 Hash Function
 Intermediate Node
 Sink Node