Today’s society requires that transformation of information is done effectively and correctly. In other words, the received message must correspond to the message being sent. There are a lot of decoding methods to locate and correct errors. The main purpose in this degree project is to study one of these methods based on the Euclidean algorithm. Thereafter an example will be illustrated showing how the method is used when decoding a three - error correcting BCH code. To begin with, fundamental concepts about coding theory are introduced. Secondly, linear codes, cyclic codes and BCH codes - in that specific order - are explained before advancing to the decoding process. The results show that correcting one or two errors is relatively simple, but when three or more errors occur it becomes much more complicated. In that case, a specific method is required. / Dagens samhälle kräver att informationsöverföring sker på ett effektivt och korrekt sätt, det vill säga att den information som når mottagaren motsvarar den som skickades från början. Det finns många avkodningsmetoder för att lokalisera och rätta fel. Syftet i denna uppsats är att studera en av dessa, en som baseras på Euklides algoritm och därefter illustrera ett exempel på hur metoden används vid avkodning av en tre - rättande BCH - kod. Först ges en presentation av grunderna inom kodningsteorin. Sedan introduceras linjära koder, cykliska koder och BCH - koder i nämnd ordning, för att till sist presentera avkodningsprocessen. Det visar sig att det är relativt enkelt att rätta ett eller två fel, men när tre eller fler fel uppstår blir det betydligt mer komplicerat. Då krävs någon speciell metod.

The work deals with data security using BCH codes. In the work are described BCH codes in binary and non-binary form, and their most important subclass RS codes. Furthermore, this work describes the method of decoding Peterson-Gorenstein-Zierl, Berlekamp- Massey and Euclidean algorithm. For the presentation of encoding and decoding process, the application was created in Matlab, which has two parts – Learning BCH codes and Simulation of BCH codes. Using the generated application performance of BCH codes was compared at the last part of the work.

Data security will play a central role in the design of future IT systems. The PC has been a major driver of the digital economy. Recently, there has been a shift towards IT applications realized as embedded systems, because they have proved to be good solutions for many applications, especially those which require data processing in real time. Examples include security for wireless phones, wireless computing, pay-TV, and copy protection schemes for audio/video consumer products and digital cinemas. Most of these embedded applications will be wireless, which makes the communication channel vulnerable. The implementation of cryptographic systems presents several requirements and challenges. For example, the performance of algorithms is often crucial, and guaranteeing security is a formidable challenge. One needs encryption algorithms to run at the transmission rates of the communication links at speeds that are achieved through custom hardware devices. Public-key cryptosystems such as RSA, DSA and DSS have traditionally been used to accomplish secure communication via insecure channels. Elliptic curves are the basis for a relatively new class of public-key schemes. It is predicted that elliptic curve cryptosystems (ECCs) will replace many existing schemes in the near future. The main reason for the attractiveness of ECC is the fact that significantly smaller parameters can be used in ECC than in other competitive system, but with equivalent levels of security. The benefits of having smaller key size include faster computations, and reduction in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments where resources such as power, processing time and memory are limited. The implementation of ECC requires several choices, such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic, the type of the elliptic curve, algorithms for implementing the elliptic curve group operation, and elliptic curve protocols. Many of these selections may have a major impact on overall performance. In this dissertation a finite field from a special class called the Optimal Extension Field (OEF) is chosen as the underlying finite field of implementing ECC. OEFs utilize the fast integer arithmetic available on modern microcontrollers to produce very efficient results without resorting to multiprecision operations or arithmetic using polynomials of large degree. This dissertation discusses the theoretical and implementation issues associated with the development of this finite field in a low end embedded system. It also presents various improvement techniques for OEF arithmetic. The main objectives of this dissertation are to --Implement the functions required to perform the finite field arithmetic operations. -- Implement the functions required to generate an elliptic curve and to embed data on that elliptic curve. -- Implement the functions required to perform the elliptic curve group operation. All of these functions constitute a library that could be used to implement any elliptic curve cryptosystem. In this dissertation this library is implemented in an 8-bit AVR Atmel microcontroller. / Dissertation (MEng (Computer Engineering))--University of Pretoria, 2006. / Electrical, Electronic and Computer Engineering / unrestricted