| dc.description.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. | en_US |
