Cryptographic functions, codes and quantum computation

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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.