Gauss periods as constructions of low complexity normal bases

In this paper, we prove the following results: 1) A normal basis N over a finite field is equivalent to its dual basis if and only if the multiplication table of N is symmetric; 2) The normal basis N is self–dual if and only if its multiplication table is symmetric and Tr(α2) = 1, where α generates N; 3) An optimal normal basis N is self–dual if and only if N is a type–I optimal normal basis with q = n = 2 or N is a type–II optimal normal basis.

유한체 연산은 부호이론과 암호학에 널리 쓰이고 있으므로, 유한체 연산의 복잡도를 낮출 수 있는 연산기가 절실하게 필요하다. 그런데 연산기의 복잡도는 유한체의 원소를 표현하는 방법에 달려있다. 복잡도를 줄이기 위해서, 지금까지 알려진 원소를 표현하는 가장 좋은 방법이 최적정규기저를 사용하는 것이다. 본 논문에서는 최적정규기저로 표현된 원소의 곱셈시에 구축되는 곱셈행렬의 1의 개수를 최소화하는 알고리즘을 개발하여 시간과 공간을 최소화하는 곱셈기를 제안하고자 한다. Arithmetic operations over finite fields are widely used in coding theory and cryptography. In both of these applications, there is a need to design low complexity finite field arithmetic units. The complexity of such a unit largely depends on how the field elements are represented. Among them, representation of elements using a optimal normal basis is quite attractive. Using an algorithm minimizing the number of 1's of multiplication matrix, in this paper, we propose a multiplier which is time and area efficient over finite fields with optimal normal basis.

It is well-known that a class of finite fields GF(2/sup n/) using an optimal normal basis is most suitable for a hardware implementation of arithmetic in finite fields. In this paper, we introduce composite fields of some hardware-applicable properties resulting from the normal basis representation and the optimal condition. We also present a hardware architecture of the proposed composite fields including a bit-parallel multiplier.

This paper propose a new multiplicative inverse algorithm for Galois field GF(2n) whose elements are represented by optimal normal bases type II. The efficiency of the arithmetic algorithms depends on the basis and many foregoing papers use either polynomial or optimal normal basis. A normal basis element is always possible to rewrite canonical basis form. The proposed algorithm combines normal basis and canonical basis. It is shown that the suggested algorithm is suitable for implementation and reduces the computation time to 5–10 % of the normal basis algorithm.KeywordsCanonical BasisNormal BasisInversion AlgorithmElliptic Curve CryptographyArithmetic AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper proposes a simple pseudorandom number generator [PRNG] by using optimal normal basis. It is well known that the squaring and multiplication in finite field with optimal normal basis is very fast and the basis can be transformed to a canonical form. The suggested PRNG algorithm combines typical multiplications and exclusive-or bit operations, both operations can be easily implemented. It is shown that the algorithm passes all terms of the Diehard and the ENT tests for long sequences. This algorithm can be applied in various applications such as financial cryptography.KeywordsSeed SizeTest SuiteRandom SeedPseudorandom Number GeneratorBinary FileThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A formula on the complexity of the normal bases generated by prime Gauss period over finite fields is presented in terms of cyclotomic numbers. Then, the authors determine explicitly the complexity of such normal bases and their dual bases in several cases where the related cyclotomic numbers have been calculated. Particularly, the authors ¯nd several series of such normal bases with low complexity.

In this paper, a high-speed word level finite field multiplier in F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> using redundant representation is proposed. For the class of fields that there exists a type I optimal normal basis, the new architecture has significantly higher speed compared to previously proposed architectures using either normal basis or redundant representation at the expense of moderately higher area complexity. One of the unique features of the proposed multiplier is that the critical path delay is not a function of the field size nor the word size. It is shown that the new multiplier outperforms all the other multipliers in comparison when considering the product of area and delay as a measure of performance. VLSI implementation of the proposed multiplier in a 0.18- mum complimentary metal-oxide-semiconductor (CMOS) process is also presented.

Finite fields are widely applied to ECC and cryptographic area, so many of researchers interested in efficient finite field arithmetic. In particular, it is efficient to use the normal basis in hardware implementation. Using the fact that the finite field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ) is the subfield of GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mk</sup> ) when GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mk</sup> ) has the type-I optimal normal basis, in this paper, we propose a new multiplier. Comparing the complexity of the proposed multiplier with Reyhani-Masoleh's multiplier proposed in 2006 which is faster, and has smaller number of XOR gates than the existing multipliers, the number of XOR gates of the multiplier is equal to that of ours for k=4,6 and 10, the XOR critical path delay, however, is more than that of the proposed one by 20% for k=10.

We present an efficient exponentiation algorithm in a finite field GF(q n ) using a Gauss period of type (n,1). Though the Gauss period α of type (n,1) in GF(q n ) is never primitive, a computational evidence says that there always exists a sparse polynomial (especially, a trinomial) of α which is a primitive element in GF(q n ). Our idea is easily generalized to the field determined by a root of unity over GF(q) with redundant basis technique. Consequently, we find primitive elements which yield a fast exponentiation algorithm for many finite fields GF(q n ), where a Gauss period of type (n,k) exists only for larger values of k or the existing Gauss period is not primitive and has large index in the multiplicative group GF(q n )×.

We show that Gauss periods of special type give an explicit polynomial-time construction of elements of exponentially large multiplicative order in some finite fields. It can be considered as a step towards solving the celebrated problem of finding primitive roots in finite fields in polynomial time.

It is shown that Gauss periods of a special type give an explicit polynomial-time computation of elements of exponentially large multiplicative order in some finite fields. This can be considered as a step towards solving the celebrated problem of finding primitive roots in finite fields in polynomial time.

This paper proposes a method for generating a basis translation matrix between isomorphic extension fields. To generate a basis translation matrix, we need the equality correspondence of a basis between the isomorphic extension fields. Consider an extension field Fp m where p is characteristic As a brute force method, when p m is small, we can check the equality correspondence by using the minimal polynomial of a basis element; however, when p m is large, it becomes too difficult. The proposed methods are based on the fact that Type I and Type II optimal normal bases (ONBs) can be easily identified in each isomorphic extension field. The proposed methods efficiently use Type I and Type II ONBs and can generate a pair of basis translation matrices within 15 ms on Pentium 4 (3.6 GHz) when mlog 2 p = 160.

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

Let L be a number field and OL the ring of algebraic integers in L. For apolynomial f with coefficients in OL, the content of f in L is the ideal of OLgenerated by coefficients of f. The polynomial f is primitive in L if the contentof f in L is OL. In 2005, Arturo Magidin and David McKinnon proved the Gauss’ lemma fornumber fields, the product of two primitive polynomials is also primitive, andsome applications following from Gauss’ lemma for number fields. A function field K over a finite field k is a finite separable field extension overk(x) where x is a transcendantal element. In this research, we study Magidin and McKinnon’s work on the function fields.

This paper propose the method of constructing the highly efficiency adder and multiplier systems over finite fields. The addition arithmetic operation over finite field is simple comparatively because that addition arithmetic operation is analyzed by each digit modP summation independently. But in case of multiplication arithmetic operation, we generate maximum k = 2m − 2° of αk terms, therefore we decrease k into m − 1 using irreducible primitive polynomial. We propose two method of control signal generation for the purpose of performing above decrease process. One method is the combinational logic expression and the other method is universal signal generation.