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Normal Bases over Finite Fields
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
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Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...
Efficient algorithms and architectures for field multiplication using Gaussian normal bases
 IEEE Transactions on Computers
, 2004
"... Abstract—Recently, implementations of normal basis multiplication over the extended binary field GFð2 m Þ have received considerable attention. A class of low complexity normal bases called Gaussian normal bases has been included in a number of standards, such as IEEE [1] and NIST [2] for an ellipti ..."
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Abstract—Recently, implementations of normal basis multiplication over the extended binary field GFð2 m Þ have received considerable attention. A class of low complexity normal bases called Gaussian normal bases has been included in a number of standards, such as IEEE [1] and NIST [2] for an elliptic curve digital signature algorithm. The multiplication algorithms presented there are slow in software since they rely on bitwise inner product operations. In this paper, we present two vectorlevel software algorithms which essentially eliminate such bitwise operations for Gaussian normal bases. Our analysis and timing results show that the software implementation of the proposed algorithm is faster than previously reported normal basis multiplication algorithms. The proposed algorithm is also more memory efficient compared with its lookup tablebased counterpart. Moreover, two new digitlevel multiplier architectures are proposed and it is shown that they outperform the existing normal basis multiplier structures. As compared with similar digitlevel normal basis multipliers, the proposed multiplier with serial output requires the fewest number of XOR gates and the one with parallel output is the fastest multiplier. Index Terms—Finite field multiplication, normal basis, Gaussian normal basis, software algorithms, ECDSA. 1