New Classes of Random Sequences for Coding and Cryptography Applications

Cryptography is required for securing data in a digital or analog medium and there exists a variety of protocols to encode the data and decrypt them without third party interference. Random numbers must be used to generate keys so that they cannot be guessed easily. This thesis investigates new classes of random numbers, including Gopala-Hemachandra (GH) and Narayana sequences, which are variants of the well-known Fibonacci sequences. Various mathematical properties of GH and Narayana sequences modulo prime have been found including their periods. Considering GH sequences modulo prime p, the periods are shown to be either (p-1) (or a divisor) or (2p+2) (or a divisor) while the Narayana sequence for prime modulo have either p2+p+1 (or a divisor) or p2-1 (or a divisor) as their periods. New results on the use of the Narayana sequence as a universal code have been obtained.It is shown that the autocorrelation and cross correlation properties of GH and Narayana sequences justify their use as random sequences. The signal to noise ratio values are calculated based on the use of delayed sequences to carry different sets of data in wireless applications. The thesis shows that GH and Narayana sequences are suitable for many encoding and decoding applications including key generation and securing transmission of data.Electrical Engineerin

Studies on Gopala-Hemachandra Codes and their Applications

Gopala-Hemachandra codes are a variation of the Fibonacci universal code and have applications in data compression and cryptography. We study a specific parameterization of Gopala-Hemachandra codes and present several results pertaining to these codes. We show that GH_{a}(n) always exists for any n >= 1, when -2 >= a >= -4, meaning that these are universal codes. We develop two new algorithms to determine whether a GH code exists for a given a and n, and to construct them if they exist. We also prove that when a = -(4+k), where k >= 1, that there are at most k consecutive integers for which GH codes do not exist. In 2014, Nalli and Ozyilmaz proposed a stream cipher based on GH codes. We show that this cipher is insecure and provide experimental results on the performance of our program that cracks this cipher