Nonlinear functions, codes and quantum computation

非线性函数、代码和量子计算

• 批准号: RGPIN-2015-06250
• 负责人: Lisonek, Petr
• 金额: $ 2.11万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=683198
• 关键词: Nonlinear; functions; codes; quantum; computation

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.********

本研究分为三个部分。*在第一部分中，我将研究几类密码函数族，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。它们还保护以数字方式存储的数据。因此，现代社会的大多数公民每天都在使用这些密码。我期望的结果是构造新的最优密码函数，通过找到这些函数的公式，或者通过开发有效地测试任何给定函数是否具有所需性质的计算方法。*在第二部分中，我将学习线性码。当通过噪声信道传输信号时，或者当数据存储在计算机存储器或磁盘中以及从计算机存储器或磁盘中检索数据时，差错控制码会检测并纠正由于噪声而发生的差错。例如，噪音的来源是与其他设备的干扰、大气中的电力或硬件的制造缺陷，再加上它的折旧。差错控制码对于互联网、无线网络、移动电话和计算机的正常运行至关重要。我的研究的预期结果是精确确定任何给定代码的检错和纠错能力的算法；我的算法将比迄今已知的算法快得多。我的算法不仅对差错控制码的实际构造具有重要意义，而且对其他领域的研究人员也很重要，因为许多理论问题都可以归结为某一线性码的存在。我还将检查我的算法是否与攻击McEliess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功能将使量子计算机的速度大大超过经典计算机。一个这样的特征是语境性，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造新的Kochen-specker集族。我的算法还将在某些情况下证明这些集合不存在。可伸缩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。我将构建比迄今已知的代码更好的量子代码。