Mathematical Study on Information Transform for Data Compression and Information Security

本课题的研究主题是研究一种新的信息变换，以实现高效的数据压缩和安全的数据传输。关于用于数据压缩的信息变换，我们首先提供了固定无旋塞的一些有趣的性质。接下来，我们需要一个有效的有序树表达式来进行日期压缩。因此，我们分析了有序树编码的性能，并建立了最优码字长度函数。进一步研究了带偏概率的Bernoulli分裂过程，并利用Mellin变换渐近地给出了期望分支数和树的深度。在这里，我们遇到了相关生成函数残差中的熵函数。这种现象可以在广义谐波级数中，广义振幅和频率分别是多项式系数和概率乘积的关系中发现。在此基础上，结合二叉树的预序编码和枚举编码，提出了整数的Catalan码，并与Elias omega码的性能进行了比较。从纠错编码和数据安全两个方面研究了数据安全传输的转换。具体来说，通过引入复合错误（即随机错误和突发错误混合）数据保护的理论框架，给出了一种新的有效的迭代码译码方法来进行复合错误校正。另一方面，对于数据安全性的研究，我们着重于综合一种新的方法来抵御去水印攻击。在这里，我们考虑了与错误控制代码的关系，并研究了各种类型的攻击。我们实现了一种既方便用户使用，又具有抗碰撞和抗重复攻击能力的方法。少

The theme of this research project is on a study of novel transform of information for the efficient data compression and the secure data transmission.With respect to the information transform for data compression, we have first provided some interesting properties of the fix-free cock. Next, we need an efficient expression of the ordered tree for the date compression .and storage/ Therefore, we analyzed the performance of coding of ordered trees, and established the optimal codeword length function. Furthermore, we had studied the Bernoulli splitting process with biased probabilities, and gave asymptotically the expected branch number and depth of the trie by using the Mellin transform. Here, we encounter the entropy function in the residue of the related generating function. This phenomenon could be found in a case where the generalized amplitudes and frequencies are in the relation of multinomial coefficients and products of probabilities, respectively, in the generalized harmonic s … More eries. Furthermore, we proposed the Catalan code for integers by a combination of the pre-order coding and the enumerative coding of binary trees, and compared the performance of the code with that of Elias omega code.The transform for secure data transmission is studied from two aspects, that is, from the error-correcting coding and the data security points of views. Specifically, by introducing the theoretical framework of protection of data suffered from the compound errors, that is, the mixture of random errors and burst errors, we gave a new efficient decoding method of iterated codes for the compound error correction.On the other hand, for the study of data security we concentrated on synthesizing a new method which can protect against the attack of removing the watermarks. Here, we considered the relation to the error control code, and investigated the various kind of attacks. We have realized a method having both of the convenience for the user and of the strength against the collision and repeat attack. Less