Finite field accumulation is the simplest of all the finite field operations, but at the same time, it is one of the most frequently encountered operations in finite field arithmetic. In this paper, we present a simple but highly useful modification of the conventional hardware implementation of accumulation in finite field over GF(2m) . The critical path, as well as, the hardware-complexity are reduced in the proposed design by performing the accumulation operation using m number of T flip-flops instead of using a combination of m number of XOR gates with equal number of D flip-flops in dependent loop structures. The conventional design is found to involve nearly 39% more area, 53% more delay, and 40% more maximum ac power consumption compared with the proposed accumulator. The proposed finite field accumulator is used further for the implementation of serial/parallel polynomial-basis finite field multiplication and bit-serial inter-conversion between polynomial basis representation and normal basis representation over GF(2m). The area-time complexity of the proposed bit-serial/parallel multiplier is less than half of the best of the corresponding existing structures. The structure proposed for digit-serial/parallel multiplication for trinomials is found to involve nearly 56% less area-time complexity compared with the best of the corresponding existing multipliers; and the existing design of bit-serial basis conversion is found to involve nearly twice area-time complexity compared with the proposed design using the proposed finite field accumulator.

/pdf/on-efficient-implementation-of-accumulation-in-finite-field-3yhifqvlou.pdf

On Efficient Implementation of Accumulation in Finite Field Over $GF(2^{m})$ and its Applications

Finite field multiplication is central in the implementation of some error-correcting coders. Massey and Omura have presented a revolutionary design for multiplication in a finite field. In their design, a normal base is utilized to represent the elements of the field. The concept of using a self-dual normal basis to design the Massey-Omura finite field multiplier is presented. Presented first is an algorithm to locate a self-dual normal basis for GF(2 sup m) for odd m. Then a method to construct the product function for designing the Massey-Omura multiplier is developed. It is shown that the construction of the product function base on a self-dual basis is simpler than that based on an arbitrary normal base.

/pdf/an-algorithm-to-design-finite-field-multipliers-using-a-self-mvp9mh7ff7.pdf

An Algorithm to Design Finite Field Multipliers Using a Self-Dual Normal Basis

Recently, the k-normal elements over finite fields are defined and characterized by Huczynska et al. In this paper, we give a new characterization of k-normal elements and define k-normal polynomials over finite fields. In what follows, we show that the problem of existence of a primitive 1-normal element in &#x1d53d;qn over &#x1d53d;q, for all q and n, which has been stated by Huczynska et al., is not satisfied. Furthermore, we extend a recursive method given by Kyuregyan for constructing an infinite family of N-polynomials, to constructing an infinite sequence of 1-normal polynomials over &#x1d53d;2m.

Some notes on the k-normal elements and k-normal polynomials over finite fields

Specific irreducible polynomials with linearly independent roots over finite fields

We present a soft-decision decoding method for Reed-Solomon codes (RS codes) using both cyclic and squaring permutations. These permutations are used to provide a convenient sequence which is predicted to have relatively low complex error pattern with respect to a modified Fano sequential algorithm[1]. In order to preserve bit-level soft-decision values, each sequence of those permutation groups must keep equal weight distribution in symbol and bit level. Trellis construction is based on Wolf's method[2] and a binary systematic parity check matrix of RS codes is used for bit-level decoding[9]. In simulation results, it is shown that a hybrid of those two permutations can be used for low complexity decoding approaching maximum likelihood performance.

Low Complexity Soft-Decision Sequential Decoding Using Hybrid Permutation for Reed-Solomon Codes

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 (2^{m}) , where m = 2^{k}p^{n} and p^{n} has two as a primitive root. This result provides a way to find a normal basis in the field.

Normal basis of finite field GF(2^m) (Corresp.)

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Normal basis of finite field GF(2^m) (Corresp.)