Information-theoretic security is considered in the paradigm of network coding in the presence of wiretappers, who can access one arbitrary edge subset up to a certain size, also referred to as the security level. Secure network coding is applied to prevent the leakage of the source information to the wiretappers. In this paper, we consider the problem of secure network coding when the information rate and the security level can change over time. To efficiently solve this problem, we put forward local-encoding-preserving secure network coding, where a family of secure linear network codes (SLNCs) is called local-encoding-preserving if all the SLNCs in this family share a common local encoding kernel at each intermediate node in the network. We first consider the design of a family of local-encoding-preserving SLNCs for a fixed security level and a flexible rate. A simple approach is presented for efficiently constructing upon an SLNC that exists a local-encoding-preserving SLNC with the same security level and the rate reduced by one. By applying this approach repeatedly, we can obtain a family of local-encoding-preserving SLNCs with a fixed security level and multiple rates. We further consider the design of a family of local-encoding-preserving SLNCs for a fixed rate and a flexible security level. We present a novel and efficient approach for constructing upon an SLNC that exists a local-encoding-preserving SLNC with the same rate and the security level increased by one. Next, we consider the design of a family of local-encoding-preserving SLNCs for a fixed dimension (equal to the sum of rate and security level) and a flexible pair of rate and security level. We propose another novel approach for designing an SLNC such that the same SLNC can be applied for all the rate and security-level pairs with the fixed dimension. Also, two polynomial-time algorithms are developed for efficient implementations of the later two proposed approaches, respectively. Furthermore, we prove that all our three approaches do not incur any penalty on the required field size for the existence of SLNCs in terms of the best known lower bound by Guang and Yeung. Finally, we consider the ultimate problem of designing a family of local-encoding-preserving SLNCs that can be applied to all possible pairs of rate and security level. By combining the constructions of the three families of local-encoding-preserving SLNCs in the paper in suitable ways, we can obtain a family of local-encoding-preserving SLNCs that can be applied for all possible pairs of rate and security level. Three possible such constructions are presented.

中文翻译：

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本地编码保留安全网络编码

在存在窃听者的情况下，网络编码范式中考虑了信息论安全性，窃听者可以访问一个任意边缘子集，达到特定大小，也称为安全级别。采用安全网络编码，防止源信息泄露给窃听者。在本文中，我们考虑了当信息速率和安全级别可以随时间变化时的安全网络编码问题。为了有效地解决这个问题，我们提出了局部编码保留的安全网络编码，如果该系列中的所有 SLNC 共享一个公共的局部编码内核，则该系列的安全线性网络代码（SLNC）被称为局部编码保留网络中的每个中间节点。我们首先考虑为固定安全级别和灵活速率设计一系列本地编码保留 SLNC。提出了一种简单的方法，用于在存在本地编码保留的 SLNC 的 SLNC 上进行有效构建，该SLNC 具有相同的安全级别且速率降低了一个。通过反复应用这种方法，我们可以获得一系列具有固定安全级别和多种速率的本地编码保留 SLNC。我们进一步考虑了为固定速率和灵活安全级别设计一系列本地编码保留 SLNC。我们提出了一种新颖有效的方法来构建 SLNC，该 SLNC 存在一个本地编码保留 SLNC，具有相同的速率和安全级别增加一个。下一个，我们考虑为固定维度（等于速率和安全级别的总和）和一对灵活的速率和安全级别设计一系列本地编码保留 SLNC。我们提出了另一种设计 SLNC 的新方法，使得相同的 SLNC 可以应用于所有具有固定维度的速率和安全级别对。此外，还开发了两种多项式时间算法，分别用于后两种建议方法的有效实现。此外，我们证明了我们的所有三种方法都不会对存在 SLNC 所需的字段大小产生任何惩罚，这取决于Guang 和Yeung 最著名的下界。最后，我们考虑设计一系列可应用于所有可能的速率和安全级别对的本地编码保留 SLNC 的最终问题。通过以适当的方式组合本文中三个本地保留编码SLNC族的构造，我们可以得到一个可以应用于所有可能的速率和安全级别对的本地保留编码SLNC族。提出了三种可能的这种结构。