Cryptographic functions, codes and quantum computation

这项拟议的研究涉及使现代数字社会和经济得以实现的一些最关键的技术。将解决对安全和保密通信的需求，以及在不完美的嘈杂通道上进行有效通信的需求。第二个研究领域是量子计算理论。人们普遍预计，量子计算机将比我们今天使用的经典计算机强大得多。这种预期加速的确切来源，以及工程师在建造量子计算机时面临的许多实际问题，仍然是密集研究的主题。我的研究计划的长期目标是数学和计算机科学在数字通信和量子计算中的应用。我运用离散数学、代数、计算机代数和算法的知识来解决上述领域的问题。我的长期目标包括为对称(私钥)密码的设计做出贡献，为噪声信道(经典和量子)提供差错控制码，以及为量子计算提供资源。我的贡献既是理论上的(例如描述新的密码函数类)，也是算法上的(例如设计搜索最佳差错控制码的新算法，或者评估新码质量的算法)。我的算法研究的输出可以作为独立的结果使用；此外，它们也为我的理论研究提供了参考。该研究的短期目标是：(1)设计新的密码函数类；(2)设计和分类用于经典、光学和量子信道的新的最优差错控制码；(3)研究用于量子计算的物理资源。目标(1)将用有限域和布尔函数的代数方法来逼近。将使用符号计算工具，允许在计算机上处理复杂的代数表达式。在(2)中，将使用先前施加的一些理论要求的新放松来发现和研究新的规范，从而为建筑提供新的成分。最优码的分类将由新开发的算法来实现。目标(3)将研究被称为情境性的量子力学现象，该现象假定测量结果取决于执行测量的环境。其结果将是：(1)可用作分组密码组件的新密码函数，以及将减少研究大量可用密码函数所需工作量的新分类方法。(2)适用于各种类型信道的新的最优码及其分类算法。(3)具有更高容错性的新的量子可观测性上下文配置，使设计更复杂、更强大的量子设备成为可能。受这项研究影响的用户将是科学家、工程师和数字社会的公民。

The proposed research concerns some of the most crucial technologies that enable the modern digital society and economy. The needs for secure and confidential communications, and for efficient communications over imperfect noisy channels will be addressed. The second research area is in the theory of quantum computation. It is widely anticipated that quantum computers will be significantly more powerful than the classical computers that we use today. The exact source of this anticipated speed-up, as well as many practical issues that engineers face in building quantum computers, are still subjects of intense research. The long-term goal of my research program are applications of mathematics and computer science in digital communications and in quantum computing. I apply knowledge of discrete mathematics, algebra, computer algebra and algorithms to address problems in the areas listed above. My long-term goals include contributions to design of symmetric (private key) ciphers, error control codes for noisy channels (classical and quantum), and resources for quantum computation. My contributions are both theoretical (such as describing new classes of cryptographic functions) and algorithmic (such as design of new algorithms that search for optimal error control codes, or algorithms that assess quality of new codes). The outputs of my algorithmic research can be used as standalone results; moreover they also inform my theoretical research. Short-term objectives of the proposed research are (1) Design of new classes of cryptographic functions, (2) Design and classification of new optimal error control codes for the classical, optical and quantum channels, and (3) Investigation of physical resources for quantum computation. Objective (1) will be approached by algebraic methods of finite fields and Boolean functions. Symbolic computation tools permitting manipulations of complex algebraic expressions on computers will be used. In (2) new codes will be found and investigated using new relaxations of some theoretical requirements imposed previously, thereby enabling new ingredients for constructions. Classification of optimal codes will be effected by newly developed algorithms. Objective (3) will investigate quantum mechanics phenomenon known as contextuality, which postulates that measurement outcomes depend on the contexts in which measurements are performed. The outcomes will be: (1) New cryptographic functions that can serve as components of block ciphers, and new classification methods that will reduce the amount of work necessary to investigate the large body of cryptographic functions available. (2) New optimal codes for various types of channels, and algorithms that classify them. (3) New contextual configurations of quantum observables with increased fault tolerance that enable engineering of more complex and more robust quantum devices. The users impacted by the research will be scientists, engineers and citizens of the digital society.