Nonlinear functions, codes and quantum computation

本文的研究分为三个部分。在第一部分中，我将研究几个加密函数族，它们是分组密码和流密码的重要组成部分。这些密码保护互联网、无线网络、移动电话和其他网络上通信的隐私和安全。它们还保护数字存储的数据。因此，现代社会的大多数公民每天都在使用这些密码。我预期的结果是通过找到这些函数的公式，或者通过开发有效测试任何给定函数是否具有所需属性的计算方法，构建新的最优加密函数。****在第二部分中，我将学习线性代码。当信号在有噪声的信道上传输时，或者当数据存储在计算机存储器或磁盘中或从磁盘中检索时，错误控制代码检测并纠正由于噪声而产生的错误。例如，噪声源是对其他设备的干扰，大气中的电力或硬件制造缺陷加上其折旧。错误控制码对于Internet、无线网络、移动电话和计算机的正常运行至关重要。我研究的预期结果是精确确定任何给定代码的错误检测和纠正能力的算法；我的算法将比目前已知的算法快得多。我的算法不仅对错误控制码的实际构造很重要，而且对其他领域的研究人员也很重要，因为许多理论问题可以简化为某个线性码的存在。我还将研究我的算法是否与攻击McEliece密码系统有关。******我研究的第三部分涉及量子计算。预计量子物理的某些独特特征将使量子计算机比经典计算机具有显着的加速。其中一个特征是上下文性，Kochen-Specker集证明了这一点。我将通过为它们寻找公式或设计找到它们的算法来构建新的Kochen-Specker集合族。我的算法也将证明在某些情况下这些集合不存在。可扩展量子计算机的实际工程存在重大挑战；预计量子纠错码将在这一过程中发挥重要作用。我将构建比迄今为止已知的代码更好的量子代码。********

The proposed research is divided into three parts.***In the first part I will study several families of cryptographic functions, which are important components of block ciphers and stream ciphers. These ciphers protect privacy and security of communications on Internet, wireless networks, mobile telephony, and other networks. They also protect data that is stored digitally. Hence, these ciphers are used every day by most citizens of a modern society. My anticipated outcome is the construction of new optimal cryptographic functions, by finding formulas for these functions, or by developing computational methods that efficiently test whether any given function has the desired properties.****In the second part I will study linear codes. Error control codes detect and correct errors that occur due to noise when a signal is transmitted over a noisy channel, or when data is stored in, and retrieved from, computer memory or disks. Sources of noise are for example interference with other devices, atmospheric electricity or manufacturing imperfections in hardware compounded with its depreciation. Error control codes are critical for the proper function of Internet, wireless networks, mobile telephony and computers. The anticipated outcome of my research are algorithms that exactly determine the error detection and correction capacity of any given code; my algorithms will be significantly faster than hitherto known algorithms. My algorithms will be important not only for practical construction of error control codes but also to researchers in other areas, since many theoretical problems can be reduced to the existence of a certain linear code. I will also examine whether my algorithms may be relevant to attacking McEliece cryptosystem.******The third part of my research deals with quantum computing. It is anticipated that certain unique features of quantum physics will enable a dramatic speed-up of quantum computers over classical computers. One such feature is contextuality, and it is demonstrated by Kochen-Specker sets. I will construct new families of Kochen-Specker sets by finding formulas for them or designing algorithms that find them. My algorithms will also prove non-existence of these sets in certain cases. Significant challenges exist with practical engineering of scalable quantum computers; it is anticipated that quantum error-correcting codes will be important in this process. I will construct quantum codes that are better than hitherto known codes.********