Mathematical Study on Information Transform for Data Compression and Information Security

本研究项目的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。在数据压缩的信息变换方面，我们首先给出了免固定公鸡的一些有趣的性质。接下来，为了数据的压缩和存储，我们需要一个有效的有序树的表达式。因此，我们分析了有序树的编码性能，并建立了最优码字长度函数。此外，我们还研究了有偏概率下的Bernoulli分裂过程，并利用Mellin变换渐近地给出了Trie的期望分支数和深度。在这里，我们遇到了相关母函数的残数中的熵函数。在广义调和S…中，当广义振幅和频率分别为多项式系数和概率积的关系时，可以发现这种现象更多的剧集。在此基础上，将二叉树的预序编码和枚举码相结合，提出了整数的加泰罗尼亚码，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变换。具体而言，通过引入复合差错保护的理论框架，即随机差错和突发差错的混合，给出了一种新的用于复合差错纠正的高效迭代码译码方法；另一方面，对于数据安全性的研究，我们致力于合成一种新的方法来抵御去除水印的攻击。在这里，我们考虑了与差错控制码的关系，并对各种攻击进行了研究。我们实现了一种既方便用户又具有抗碰撞和重复攻击的强度的方法。较少

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