| Summary: | 碩士 === 國立清華大學 === 電機工程學系 === 88 === Finite fields play an important role in areas such as error correcting codes and cryptography. Three fundamental operations: additions, multiplications, and inversions must be provided. Since additions are just binary additions without carry, multiplications and inversions are the operations of the most concern. In this thesis, we first review some results given by He, in which a recursive method similar to that developed by Fan and Paar was used to develop multiplications and inversions in GF(2^{(p-1)p^k}), where p satisfies certain conditions. By combining the recursive methods for construction of tower fields GF(2^{2^k}) and GF(2^{(p-1)p^k}), we derive more general finite fields of characteristic 2. A more generalized definition of duality is used to analyze the serial multipliers explored by He, and the newly developed multiplier has lower space complexity than He's approach. A regular architecture for the parallel multiplier in GF(2^{(p-1)p^k}) developed by He is proposed, and its structure is regular, modular, and suitable for VLSI implementation. We also develop a normal basis multiplier for GF(2^{2^k}) and compare it with the multiplier recently proposed by Koc and Sunar. The results show that our approach has a lower time-space product than Koc and Sunar's approach.
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