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Title: | Computational Design of BCH-Codes and Their Applications in the Data Security |

Authors: | Asif, Muhammad |

Keywords: | Physical Sciences Mathematics |

Issue Date: | 2020 |

Publisher: | Quaid-i-Azam University, Islamabad. |

Abstract: | Due to innovations in communication technologies, digital medium is extensively used across the World. Large amount of information in digital form is stockpiled in digital libraries. The error free transportation of digital data through untrustworthy channels is a great challenge. Accordingly, in information theory, telecommunication, computer science and algebraic coding theory, an error correction code or error correcting code is used for controlling errors in data over unreliable or noisy communication channels. Because of error correcting codes, the communication is made over the short and long distances without any obstacle. Thus, it made possible the gigabit data transmission over the wireless communication mediums. Indeed, it is the fundamental part of the modern communication systems and essentially utilized in hardware level implementations of intelligent and smart machines like telecom equipment, highly sensitive video cameras, optical devices, and scanners. Development of data transferring codes were started with the first article [31] of Claude Shannon in 1948. He explained that, every communication channel has some capacity. If the rate of data transmission is smaller than capacity, then design of communication system for the channel is possible with the help of data transmission codes. This system has least probability of output errors, but Shannon did not give the method for the construction of such type of codes. In 1950, for this purpose Hamming [14] and Golay [9] introduced cyclic block codes known as binary hamming and Golay codes respectively. These classes of codes have the capability to detect up to two errors and correct one error. Furthermore, these codes have fascinating features and can be easily encoded and decoded but are not suitable for multiple errors. In 1953, Muller [18] introduced a multiple error correcting codes technique and Reed unresolved. Cyclic codes are one of the dynamic class of error correcting codes. In 1957, Prange [33] initiated an idea of cyclic codes in two symbols. In addition, Prange [24] used the coset equivalence for decoding the group codes in 1959. After that, a big development in the theory of cyclic codes was made to correct burst along with random errors initiated by various researchers. The cyclic codes were initially developed over binary field 2 and into its Galois field extension . Though, it was further extended over the prime field p and into its Galois field extension . The remarkable development in coding theory began when Hocquenghem [10], Bose and Chaudhuri [3] explained the large class of codes which correct multiple errors known as BCH codes in 1960. They explained the BCH codes over Galois field. These codes are generalization of binary Hamming codes. The advantage of BCH code is that, there is a precise control over the number of symbol errors correctable during code design and these codes are easy to decode because the polynomial algebra is used for this purpose. In 1960, Peterson [25] gave error correction procedure for BCH codes over finite field. Forney [7] developed the decoding technique of BCH codes by using Barlekamp Massey algorithm in 1965. In 1972, Peterson et.al [2 - In , Blake [1] initiated the notion of linear codes over the ring of integers modulo where is product of primes. But he did not explain the codes over local ring , for . Later, in 1975, Blake [2] discussed the linear codes over the ring where , is a prime and + . In 1977 and 1978, Spiegal [32-33] showed that codes over can be defined in terms of codes over and thus enables us to define codes over for any positive integer . Shankar [34] constructed the BCH codes over in and for this purpose used the polynomial ring and she also constructed BCH code for arbitrary integer , where . She developed BCH codes over Galois ring by using maximal cyclic subgroup of group of units of Galois ring and connected with BCH codes over Galois field by using reduction map. In 1997, Interlando et. al [15] explained powerful decoding method which was based on Barlekamp-Massey algorithm for RS and BCH codes over integer residue local ring . De Andrade et. al [5] constructed BCH codes over finite unitary commutative rings in 1999. In the construction techniques of both [5] and [34] the cyclic subgroup of the group of units of a Galois ring was specified. In [5], decoding procedure of BCH code with same algorithm that can correct all errors up to hamming weight was explained. In 2012, Shah et. al [35] defined decoding procedure which improves code rate. Further, Shah et. al [37] explained decoding method of length binary BCH code through cyclic code in . However, computationally the decoding of BCH codes over Galois ring by using modified Barlekamp Massey algorithm is still a topic of interest. In 2017, Shah et. al [38] developed an algorithm by which one can construct the maximal cyclic subgroup of group of units of any arbitrary Galois ring. In this study by using this digital library we introduce the computational technique to construct the BCH codes over Galois ring of desired order. Recently, the dimensions of two classes of primitive BCH codes with some designed distance was constructed by Ding et. al [4] in 2017, but they did not explain the dimensions of primitive BCH codes for large length corresponding to any designed distance. So, the issue of dimensions of primitive BCH codes in general is still unanswered. In this work we resolved the issue of dimensions of primitive BCH codes. Fundamentally the BCH codes are utilized for only data transmission, but not for data security. In this study we have given the idea that, BCH codes can be used for data security. Accordingly, by BCH codes over Galois field and Galois ring a couple of techniques are devised to modify AES algorithm. Accordingly, this modified AES algorithm tested on text and image data, the results assured the appropriate level of security. This thesis consists of seven chapters. In Chapter one, some important notions of algebraic structures and error correcting codes are explained which are necessary for understanding further chapters. In Chapter two, initially we have given details on obtaining the maximal cyclic subgroup of group of units of a Galois ring through computational method. Afterword the new computational encoding scheme of BCH code over Galois ring is introduced. This novel computational approach of encoding of BCH codes provides generator polynomial for any length corresponding to each designed distance . Furthermore, the encoding of BCH codes over Galois field has also been explained with the help of reduction map. Another outcome of this study is that one can find the dimensions of primitive BCH codes for any length and designed distance. In Chapter three, using C# computer language a computational decoding scheme for BCH codes over Galois ring has been designed by which Barlekamp Massey decoding algorithm of BCH codes over Galois field is employed to correct the errors. Indeed, this modified Barlekamp Massey decoding algorithm is designed for large length BCH codes over Galois field. The special feature of this study is the syndrome calculation with computational approach. Thus, decoding of BCH and RS codes over Galois ring by using modified Barlekamp Massey algorithm has been ensured. In Chapter four, BCH codes have been utilized to improve the AES algorithm. BCH codes have been utilized as a secret key in round key addition step of AES algorithm. In addition, using BCH codes, the maximum distance separable matrix has been constructed and applied in mixed column matrix step in modified AES algorithm. Thus, this modified AES algorithm has been applied in image encryption and different analyses on encrypted image have been performed. The comparison of results of encrypted image by using original and modified AES algorithms have been discussed. In Chapter five, The AES algorithm is modified. Initially we use BCH codes and calculated secret keys for each round in AES algorithm. In second step, mixed column matrices have been computed by using BCH codes for each round. This modified AES algorithm has been used for text encryption and then applied avalanche effect to cipher text. NIST statistical test have been applied on proposed text encryption scheme. In Chapter six, S-box, the only nonlinear component of block cipher in symmetric key cryptosystem has been constructed by using Galois cyclic group and maximal cyclic subgroup of group of units of the Galois ring. BCH codes have been used for secret keys and MDS matrices. An image encryption scheme has been designed in such a way that in each round different key, different matrix and different S-box have been utilized. Image encryption scheme has been employed on different images and for authenticated results diverse statistical tests have been applied to encrypted images. In Chapter seven, we have concluded our thesis and have given future directions and motivations of this work. |

Gov't Doc #: | 20903 |

URI: | http://prr.hec.gov.pk/jspui/handle/123456789/15758 |

Appears in Collections: | PhD Thesis of All Public / Private Sector Universities / DAIs. |

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Muhammad asif maths 2020 qau isb.pdf | phd.Thesis | 85.87 MB | Adobe PDF | View/Open |

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