Nonlinear functions, codes and quantum computation

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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.********