On Application Of Operator And Group-Theoretic Concepts In Signal Processing And Cryptography

Abstract

In this thesis we have studied how operator theory is applied in signal processing and how some concepts in group theory are crucial in the design of cryptosystems and their use in hiding information. A frame is a redundant (i.e. not linearly independent) coordinate system for a vector space that satis es a certain Parseval-type norm inequality. Frames provide a means for transmitting data and, when a certain loss is anticipated, their redundancy allows for better signal reconstruction. We have started with the basics of frame theory and given examples of frames and applications that illustrate how this redundancy can be exploited to achieve better signal reconstruction. The key idea is that in order to protect against a noise, we should encode the message by adding some redundant information to the message. In such a case, even if the message is corrupted by noise, there will be enough redundancy in the encoded message to recover, or to decode the message completely Cryptography is the science of information security, it is the practice of defending information from unauthorized access, use, disclosure,disruption, modi cation, perusal, inspection, record- ing or destruction. It is a general term that can be used regardless of the form the data may take(electronic, physical, etc). We have explored and demonstrated how simple concepts like divisibility of integers, primes and other concepts in number theory come in handy in cryptog- raphy. We have demonstrated how to use group theory concepts to send messages(plaintext) in disguised form so that only the intended recipients can remove the disguise and read the message(ciphertext). To be able to achieve all this, we have spent a bit of time developing the notion of Hilbert space frames, some groups, number systems and their their properties. We have chosen the most optimal frames(tight frames) and groups(cyclic) for use in sending signal and also reconstructing the sent signal and enciphering and deciphering a message.