We present a new information-theoretic definition and associated results, based on list decoding in a source coding setting. We begin by presenting list-source codes, which naturally map a key length (entropy) to list size. We then show that such codes can be analyzed in the context of a novel information-theoretic metric, -symbol secrecy, that encompasses both the one-time pad and traditional rate-based asymptotic metrics, but, like most cryptographic constructs, can be applied in non-asymptotic settings. We derive fundamental bounds for -symbol secrecy and demonstrate how these bounds can be achieved with MDS codes when the source is uniformly distributed. We discuss applications and implementation issues of our codes.
@cite_7: We consider the level of information security provided by random linear network coding in network scenarios in which all nodes comply with the communication protocols yet are assumed to be potential eavesdroppers (i.e. "nice but curious"). For this setup, which differs from wiretapping scenarios considered previously, we develop a natural algebraic security criterion, and prove several of its key properties. A preliminary analysis of the impact of network topology on the overall network coding security, in particular for complete directed acyclic graphs, is also included.
@cite_21: In this tutorial paper, we focus on the basic theory of linear secure network coding. Our goal is to present fundamental results and provide preliminary knowledge for anyone interested in the area. We first present a model for secure network coding and then a necessary and sufficient condition for a linear network code to be secure. Optimal methods to construct linear secure network codes are also provided. For further investigation of the secure properties of linear network codes, we illuminate different secure criteria and requirements, with a few alternative models.
@cite_1: The problem of securing a network coding communication system against an eavesdropper is considered. The network implements linear network coding to deliver n packets from source to each receiver, and the adversary can eavesdrop on μ arbitrarily chosen links. The objective is to provide reliable communication to all receivers, while guaranteeing that the source information remains information-theoretically secure from the adversary. A coding scheme is proposed that can achieve the maximum possible rate of n - μ packets. The scheme, which is based on rank-metric codes, has the distinctive property of being universal: it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying linear network code. The only requirement of the scheme is that the packet length be at least n, which is shown to be strictly necessary for universal communication at the maximum rate. A further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network, and the network may suffer from a rank deficiency of at most ρ. In this case, the proposed scheme can be extended to achieve the rate of n - ρ - 2t - μ packets. This rate is shown to be optimal under the assumption of zero-error communication.
@cite_19: Recent work on network coding renders a new view on multicasting in a network. In the paradigm of network coding, the nodes in a network are allowed to encode the information received from the input links. The usual function of switching at a node is a special case of network coding. The advantage of network coding is that the full capacity of the network can be utilized. In this paper, we propose a new model which incorporates network coding and information security. Specifically, a collection of subsets of links is given, and a wiretapper is allowed to access any one (but not more than one) of these subsets without being able to obtain any information about the message transmitted. Our model includes secret sharing as a special case. We present a construction of secure linear network codes provided a certain graph-theoretic sufficient condition is satisfied.
@cite_2: 
@cite_15: We consider the problem of using a multicast network code to transmit information securely in the presence of a “wire-tap” adversary who can eavesdrop on a bounded number of network edges. Cai & Yeung (ISIT, 2002) gave a method to alter any given linear network code into a new code that is secure. However, their construction is in general inefficient, and requires a very large field size; in many cases this is much greater than the field size required by standard network code construction algorithms to achieve the min-cut capacity (without a security guarantee). In this paper we generalize and simplify the method of Cai & Yeung, and show that the problem of making a linear network code secure is equivalent to the problem of finding a linear code with certain generalized distance properties. We show that if we give up a small amount of overall capacity, then a random code achieves these properties using a much smaller field size — in some cases a field of constant size suffices — than the construction of Cai & Yeung. We add further support to this approach by showing that if we are not willing to give up any capacity, then a large field size may sometimes be required to achieve security.
@cite_16: We consider the problem of securing a multicast network against a wiretapper that can eavesdrop on the packets on a limited number of network edges of its choice. We assume that the network employs network coding to simultaneously deliver the packets available at the source to all the destinations. We show that this problem can be looked at as a network generalization of the wiretap channel of type II introduced in a seminal paper by Ozarow and Wyner. In particular, we show that the transmitted information can be secured by using the Ozarow-Wyner approach of coset coding at the source on top of the existing network code. This way, we quickly and transparently recover some of the results available in the literature on secure network coding for wiretap networks. Moreover, we use this framework to derive new bounds on the code alphabet size that are independent of the network size, and provide algorithms for explicit construction of secure network codes. We also analyze the amount of information that can be leaked to the wiretapper as a function of the number of wiretapped edges.
@cite_13: 
@cite_17: This paper studies the secrecy capacity of unicast communication in a wireless erasure network setting in the presence of a wire-tapper. From an information-theoretic setting of perfect secrecy, both upper bounds and achievable secrecy rates are presented. Secrecy capacity is determined in closed form for a class of broadcast constrained erasure networks.
Tools from algebraic coding theory have been widely used for constructing secrecy schemes @cite_13 . In addition, the notion of providing security by exploiting the fact that the adversary has incomplete access to information is also central to several secure network coding schemes and wiretap models. Ozarow and Wyner @cite_2 introduced the wiretap channel II, where an adversary can observe a set @math of his choice out of @math transmitted symbols, and proved that there exists a code that achieves perfect secrecy. A generalized version of this model was investigated by Cai and Yeung in @cite_19 , where they introduce the related problem of designing an information-theoretically secure linear network code when an adversary can observe a certain number of edges in the network. Their results were later extended in @cite_15 @cite_17 @cite_16 @cite_1 . A more practical approach was presented by Lima al in @cite_7 . For a survey on the theory of secure network coding, we refer the reader to @cite_21 .