Normal Basis Representation Research Articles

Efficient parallel exponentiation in [formula omitted] using normal basis representations

Von zur Gathen proposed an efficient parallel exponentiation algorithm in finite fields using normal basis representations. In this paper we present a processor-efficient parallel exponentiation algorithm in GF ( q n ) which improves upon von zur Gathen's algorithm. We also show that exponentiation in GF ( q n ) can be done in O ( ( log 2 n ) 2 / log q n ) time using n / ( log 2 n ) 2 processors. Hence we get a processor-time bound of O ( n / log q n ) , which matches the best known sequential algorithm. Finally, we present an efficient on-line processor assignment scheme which was missing in von zur Gathen's algorithm.

Improved Scalar Multiplication on Elliptic Curves Defined over F2mn

We propose two improved scalar multiplication methods on elliptic curves over F q n where q = 2 using Frobenius expansion. The scalar multiplication of elliptic curves defined over subfield F q can be sped up by Frobeniusexpansion. Previous methods are restricted to the case of a small m. However, when m is small, it is hard to find curves having good cryptographic properties. Our methods are suitable for curves defined over medium-sized fields, that is, 10 ≤ m ≤ 20. These methods are variants of the conventional multiple-base binary (MBB) method combined with the window method. One of our methods is for a polynomial basis representation with software implementation, and the other is for a normal basis representation with hardware implementation. Our software experiment shows that it is about 10% faster than the MBB method, which also uses Frobenius expansion, and about 20% faster than the Montgomery method, which is the fastest general method in polynomial basis implementation.

Efficient Implementation of Elliptic Curve Cryptography (ECC) for Personal Digital Assistants (PDAs)

Wireless devices are characterized by low computational power and memory. In addition to this wireless environment are inherently less secure than their wired counterparts, as anyone can intercept the communication. Hence they require more security. One way to provide more security without adding to the computational load is to use elliptic curve cryptography (ECC) in place of the more traditional cryptosystems such as RSA. As ECC provides the same level of security for far less key sizes, as compared to the traditional cryptosystems, it is ideal for wireless security. In this thesis we will investigate the different ways of implementing ECC on wireless devices such as personal digital assistants (PDAs). We will present our findings and compare the different implementations. In our implementation ECC over the field Fn2 using optimal normal basis representation gives the best results.

A Scalable Structure for a Multiplier and an Inversion Unit in GF(2m)

Elliptic curve cryptography (ECC) offers the highest security per bit among the known public key cryptosystems. The operation of ECC is based on the arithmetic of the finite field. This paper presents the design of a 193-bit finite field multiplier and an inversion unit based on a normal basis representation in which the inversion and the square operation units are easy to implement. This scalable multiplier can be constructed in a variable structure depending on the performance area trade-off. We implement it using Verilog HDL and a 0.35 µm CMOS cell library and verify the operation by simulation.

A new hardware architecture for operations in GF(2/sup n/)

The efficient computation of the arithmetic operations in finite fields is closely related to the particular ways in which the field elements are presented. The common field representations are a polynomial basis representation and a normal basis representation. In this paper, we introduce a nonconventional basis and present a new bit-parallel multiplier which is as efficient as the modified Massey-Omura multiplier using the type I optimal normal basis.

Efficient normal basis multipliers in composite fields

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.

Massey-Omura type adder for elements of finite fields GF(2m) in logarithmic representation

The multiplication, inversion, division and exponentiation of elements of GF(2/sup m/) are easily implemented with conventional arithmetic and logical units when the elements are in the logarithmic representation. An electronic architecture for the addition of two elements in the logarithmic representation is presented. The architecture of the adder is similar to that of the Massey-Omura multiplier for the normal basis representation. In particular, the same combinatorial circuit is used to successively compute every bit of the sum.

Improved normal basis inversion in GF(2m)

Recently, Fenn et al. (1996) developed a fast finite field inversion algorithm in GF(2/sup m/) over the normal basis representation which uses about half the time complexity of traditional approaches. Further extensive improvement to this algorithm is presented here.

Arithmetic operations inGF(2 m )

This article is concerned with various arithmetic operations inGF(2m). In particular we discuss techniques for computing multiplicative inverses and doing exponentiation. The method used for exponentiation is highly suited to parallel computation. All methods achieve much of their efficiency from exploiting a normal basis representation in the field.

Trace- and norm-compatible extensions of finite fields

J. H. Conway suggested to use norm-compatible polynomials when constructing extension fields in order to get finite field embeddings, which are computationally easy to handle. Analogous to norm-compatibility for the multiplicative group of ¯Fq we exhibit the notion of trace-compatibility for its additive group. By this we get computationally simple embeddings in the case of normal basis representations of finite fields. We give an algorithm for computing tracecompatible polynomials and count the number of distinct compatible polynomials of a fixed degree of both kinds.

Some Observations on Parallel Algorithms for Fast Exponentiation in $\operatorname{GF}(2^n)$

A normal basis representation of $\operatorname{GF}(2^{n})$ allows squaring to be accomplished by a cyclic shift. Algorithms for multiplication in $\operatorname{GF}(2^{n})$ using a normal basis have been studied by several researchers. In this paper, algorithms for performing exponentiation in $\operatorname{GF}(2^{n})$ using a normal basis, and how they can be speeded up by using parallelization, are investigated.

Low complexity normal bases

A normal basis in GF( q m ) is a basis of the form { β, β q , β q 2,…, β q m−1 }, i.e., a basis of conjugate elements in the field. In GF(2 m ) squaring with respect to a normal basis representation becomes simply a cyclic shift of the vector. For hardware design this is one of the very attractive features of these bases. Multiplication with respect to a normal basis can be defined in terms of a certain bilinear form. Define the complexity of the normal basis to the number of nonzero terms in this form. Again, for hardware design, it is important to find normal bases with low complexity. In this paper we investigate low complexity normal bases, give a construction for such bases and apply it to a number of cases of interest.

A new algorithm for multiplication in finite fields

A new algorithm is presented for computing the product of two elements in a finite field by means of sums and products in a fixed subfield. The algorithm is based on a normal basis representation of fields and assumes that the dimension m of the finite field over the subfield is a highly composite number. A very fast parallel implementation and a considerable reduction in the number of computations are allowed, in comparison to some methods discussed in the literature.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The least complex parallel massey-omura multiplier and its LCA and VLSI designs

It is known that for m≥4 there is more than one irreducible polynomial which generates a GF(2m) represented in normal basis. Therefore, a parallel Massey-Omura multiplier in GF(2m) has more than one structure for m≥4. The number of these structures is equal to the number of irreducible polynomials which can generate that specific Galois field in a normal-basis representation. In the paper, as an illustrative example, the structure of GF(25) multipliers for different generator polynomials are compared. Based on this comparison the least complex structure for this multiplier is introduced. The implementation of this structure using LCA (logic cell array) and VLSI (very large scale integration) technologies is given. Their complexities and propagation delays are compared.

Direct hardware solution to quadratic equation Z2⊕Z⊕β=0 in Galois fields based on normal basis representation

Normal basis representation is considered to represent the elements of Galois fields. The quadratic equation, Z/sup 2/(+)Z(+) beta =0, is solved directly and a new, simple, regular and expandable hardware structure is introduced to solve this equation. The main advantage of this structure over the structures in non-normal basis representations is its independence from generating polynomial of the field.

Normal basis of finite field&amp;lt;tex&amp;gt;GF(2^m)&amp;lt;/tex&amp;gt;(Corresp.)

Massey and Omura recently developed a new multiplication algorithm for Galois fields based on the normal basis representation. This algorithm shows a much simpler way to perform multiplication in finite field than the conventional method. The necessary and sufficient conditions are presented for an element to generate a normal basis in the field GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2^{m})</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m = 2^{k}p^{n}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p^{n}</tex> has two as a primitive root. This result provides a way to find a normal basis in the field.

VLSI architectures for computing multiplications and inverses in GF(2m).

Finite field arithmetic logic is central in the implementation of Reed-Solomon coders and in some cryptographic algorithms. There is a need for good multiplication and inversion algorithms that can be easily realized on VLSI chips. Massey and Omura recently developed a new multiplication algorithm for Galois fields based on a normal basis representation. In this paper, a pipeline structure is developed to realize the Massey-Omura multiplier in the finite field GF(2m). With the simple squaring property of the normal basis representation used together with this multiplier, a pipeline architecture is developed for computing inverse elements in GF(2m). The designs developed for the Massey-Omura multiplier and the computation of inverse elements are regular, simple, expandable, and therefore, naturally suitable for VLSI implementation.