Multiplication In GF Research Articles

On bit-serial multiplication and dual bases in GF(2/sup m/)

The existence of certain types of dual bases in finite fields GF(2/sup m/) is discussed. These special types of dual bases are needed for efficient implementation of (generalized) bit-serial multiplication in GF(2/sup m/). In particular, the question of choosing a polynomial basis of GF(2/sup m/), for example (1, alpha , alpha /sup 2/, alpha /sup 3/, . . ., alpha /sup m-1/), such that the change of basis matrix from the dual basis to a scalar multiple of the original basis has as few '1' entries as possible, is studied. It was previously shown by M. Wang and I. F. Blake (1990) that the optimal situation occurs when the minimal polynomial of alpha is an irreducible trinomial of degree m; then, an appropriate scalar multiple, beta , yields a change of basis matrix that is a permutation matrix. A construction is presented that often yields bases where the change of basis matrix has low weight, in the case where no irreducible trinomial of degree m exists. A simple formula can be used to compute beta and the weight of the change of basis matrix, given the minimal polynomial of alpha .< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Structure of parallel multipliers for a class of fields GF(2m)

This paper presents a configuration of parallel multipliers for GF (2 m ) based on canonical bases. The possible parallel multipliers by the proposed configuration are limited to a class of fields GF (2 m ). However they can be constructed by O(m 2 ) AND-gates and O(m 2 ) EOR-gates with the structural modularity (this is a desirable feature for the hardware implementation), and their operation time is about (log m ) T , where m is the dimension of GF (2 m ) and T is the delay time of an EOR-gate. In order to construct such parallel multipliers, we define two types of polynomials of special form over GF (2), one is called all one polynomial (denoted by AOP) and the other is called equally spaced polynomial (denoted by ESP). Furthermore, we show a necessary and sufficient condition for ESPs to be irreducible over GF (2) and the uniqueness of the irreducible ESPs over GF (2). Finally, we propose the configuration of parallel multipliers for a class of fields GF (2 m ) based on irreducible AOPs and ESPs over GF (2).

On planes of prime order with translations and homologies

An orthomorphism of Z , p an odd prime, is a bijection @:Z -~Z o ~(0)=0, such P P p that the mapping 9:Zp-~ZD defined by U(x)=@(x)-x is also a bijection. The orthomorphism graph of Zp, denoted Orth(Zp), has as its vertex set the orthomorphisms of Zp, two orthomorphisms #,# of Zp being adjacent if the mapping 6:Z -.Z defined by 6(x)=~(x) #(x) is a bijection. For the P P definition of and known results on orthomorphisms of groups in general see Johnson, Dulmage and Mendelsohn [9] or D~nes and Keedwell [4,chapter 7]. Assuming Z to be the additive group of GF(p) enables us to use P multiplication in GF(p) to define classes of orthomorphisms and to study adjacencies within these classes. As an example, if we use I to denote the a mapping x-~ax, then clearly IaeOrth(Z p) if and only if a#O, i and !a is adjacent to I b if and only if a#bo We use S 1 to denote the set {I~; a#O,l}. Another class of mappings that give rise to orthomorphisms can be defined as follows:

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.

A fast algorithm for computing multiplicative inverses in GF(2m) using normal bases

This paper proposes a fast algorithm for computing multiplicative inverses in GF(2 m ) using normal bases. Normal bases have the following useful property: In the case that an element x in GF(2 m ) is represented by normal bases, 2 k power operation of an element x in GF(2 m ) can be carried out by k times cyclic shift of its vector representation. C. C. Wang et al. proposed an algorithm for computing multiplicative inverses using normal bases, which requires ( m − 2) multiplications in GF(2 m ) and ( m − 1) cyclic shifts. The fast algorithm proposed in this paper also uses normal bases, and computes multiplicative inverses iterating multiplications in GF(2 m ). It requires at most 2[log 2 ( m − 1)] multiplications in GF(2 m ) and ( m − 1) cyclic shifts, which are much less than those required in the Wang's method. The same idea of the proposed fast algorithm is applicable to the general power operation in GF(2 m ) and the computation of multiplicative inverses in GF( q m ) ( q = 2 n ).

Algorithms for multiplication in Galois field for implementation using systolic arrays

Operations in finite fields find diverse applications. Circuits have been designed for carrying out such operations. In the paper, two circuits that carry out multiplication in GF(2p) have been presented. These circuits are suitable for implementation using VLSI techniques, and are simpler than existing circuits. The architecture used here is that of systolic arrays and consists of regular interconnection of simple cells.

Effective recursive algorithm for computing multiplicative inverses in GF(2m)

The letter presents an effective recursive algorithm for computing multiplicative inverses in GF(2m), where m = 2k, employing normal bases. The proposed algorithm requires m − 1 cyclic shifts and two multiplications in GF(2m) and in each subfield of GF(2m): GF(2m/2), GF(2m/4), …, GF(28) and GF(24).

A Fast VLSI Multiplier for GF(2&amp;lt;sup&amp;gt;m&amp;lt;/sup&amp;gt;)

Multiplication in the finite field GF(2^{m} ) has particular computational advantages in data encryption systems. This paper presents a new algorithm for performing fast multiplication in GF(2^{m} ), which is O(m) in computation time and implementation area. The bit-slice architecture of a serial-in-serial-out modulo multiplier is described and the circuit details given. The design is highly regular, modular, and well-suited for VLSI implementation. The resulting multiplier will have application in algorithms based on arithmetic in large finite fields of characteristic 2, and which require high throughput.

The multiplications in GF( q)[ x]⧸( a( x)) considered as linear transformations

We study in this paper the ring V( a)= GF( q)[ x⧸( a( x)), which also can be considered as a linear space over GF( q). In this context the multiplications are considered as linear mappings of V( a) into V( a). The ideals (invariant subspaces under multiplications) are used to formulate properties of subspaces of V( a).

Synthesis procedures for switching circuits represented Reed-Muller form over a finite field

The synthesis of economical multilevel circuits for binary and multiple-valued switching circuits is described. The mode of function description is that provided by the algebra of finite fields and this leads to a highly modular form of circuit representation. A universal-logic tree composed of GF(q) adders and multipliers is used as a template on which to construct specific multilevel circuits. The paper describes methods for assigning the input variables to the network so as to reduce the complexity of the general tree by removing the redundant circuit elements. The resulting networks are invariably less costly than those from the direct synthesis of two-level sum-of-products expressions and they use only a restricted set of circuit elements.

On the complexity of decoding Reed-Solomon codes (Corresp.)

Certain q -ary Reed-Solomon codes can be decoded by an algorithm requiring only O(q \log^{2} q) additions and multiplications in GF(q) .

Elliptic Curve Scalar Point Multiplication Algorithm Using Radix-4 Booth’s Algorithm

The main back-bone operation in elliptic curve cryptosystems is scalar point multiplication. The most frequently used method implementing the scalar point multiplication, which is performed in the topmost level of GF multiplication and GF division, has been the double-and-add algorithm, which is being recently challenged by NAF (Non-Adjacent Format) algorithm. In this paper, we propose a more efficient and novel approach of a scalar multiplication method than the double-and-add by applying redundant recoding which originates from the radix-4 modified Booth’s algorithm. We call the novel algorithm quad-and-add. After deriving the algorithm, we created a new GF operation, named point quadruple, and verified with calculations of a real-world application to utilize it. Derived numerical expressions were verified using both C programs and HDL (Hardware Description Language). Proposed method of elliptic curve scalar point multiplication can be utilized in many elliptic curve security applications for handling efficient and fast calculations.