Study on Information Security and Random Number Generation from Information Theoretical Viewpoints

本文的研究成果总结如下：1.本文的研究成果。一维映射产生的混沌二进制序列的统计特性：有几种尝试使用一维映射来产生混沌二进制序列。从工程应用的角度来看，有必要对有限长样本序列的统计性质进行评估。利用动力系统的大偏差理论框架，我们将这个问题描述为一个计算速率函数的问题。此外，对于由并矢映射和门限函数产生的混沌二进制序列，我们得到了一类门限函数的速率函数的显式形式。使用中继通道传输保密消息的问题：在中继通道中，中继帮助将消息从一个发送方传输到另一个接收方。中继器也被视为窃听者，可以获得有关所传输消息的一些知识。基于以上观点，我们对中继信道的安全性进行了讨论，并得到了一些明确的结果。噪声信道识别：研究了一般噪声信道识别的逆编码定理。我们估计了识别容量之外的两种类型的识别错误概率，导出了作为两种识别错误概率之和的下界的函数。特别地，在平稳离散无记忆信道的情况下，我们证明了两个差错概率之和以指数形式趋于1，并导出了该指数函数的一个显式下界。此外，我们还在多用户信道的两种情况下扩展了我们的论点。一个是多路接入通道，另一个是广播通道。对于这两个多用户信道，我们得到了识别容量区域的外界的结果。

Our research results are summarized in the followings.1. Statistical Properties of Chaotic Binary Seauences Generated by One-dimensional Maps :There are several attempts to generate chaotic binary sequences by using one-dimensional maps. From the standpoint of engineering applications, it is necessary to evaluate statistical properties of sample sequences of finite length. Using the framework of large deviation theory for dynamical systems, we formulated this problem as a problem of computing the rate function. Furthermore, for the chaotic binary sequence generated by the dyadic map and threshold function, we obtained an explicit form of the rate function for some class of threshold functions.2. Transmission Problem of Confidential Messages Using Relay Channels :In the relay channel a relay helps the transmission of messages from one sender to one receiver. The relay is also regarded as a wire-tapper who can obtain some knowledge about the transmitted messages. From the above viewpoints we have discussed the security of the relay channel and have derived some explicit result.3. Identification via Noisy Channels :We have studied the converse coding theorem for the identification via general noisy channels. We have estimated two types of error probabilities of identification for rates outside the identification capacity, deriving some function which serves as a lower bound of the sum of two error probabilities of identification. In particular, in the case of stationary discrete memoryless channels, we have proved that the sum of two error probabilities tends to one exponentially and derive an explicit lower bound of this exponent function. Furthermore, we have extended our arguments in two cases of multi-user channels. One is the multiple access channels and the other is the broadcast channels. For those two multi-user channels we have obtained results for outer bounds of the identification capacity regions.