Galois Fields

This paper presents a scalable and systolic Montgomery's algorithm in GF(2m) using the Hankel matrix-vector representation. The hardware architectures derived from this algorithm represents low-complexity bit-parallel systolic multipliers with trinomials. The results reveal that our proposed multiplier saves approximately 36% space complexity as compared to an existing systolic Montgomery multiplier for trinomials. Moreover, the proposed architectures have the features of regularity, modularity, and local interconnect ability. Accordingly, they are well suited for VLSI implementation

The design and implementation of lossless audio signal processing using Finite Field Transforms is discussed. Finite field signal processing techniques are described. The effects of filter length and coefficient accuracy are also discussed. Finite field transform algorithms which would be suitable for lossless signal processing are presented

A branch of mathematics commonly used in cryptography is Galois Fields GF(p n). Two basic operations performed in GF(p n) are the addition and the multiplication. While the addition is generally easy to compute, the multiplication requires a special treatment. A well-known method to compute the multiplication is based on logarithm and antilogarithm tables. A primitive element of a GF(p n) is a key part in the construction of such tables, but it is generally hard to find a primitive element for arbitrary values of p and n. This article presents a naive algorithm that can simultaneously find a primitive element of GF(p n) and construct its corresponding logarithm and antilogarithm tables. The proposed algorithm was tested in GF(p n) for several values of p and n; the results show a good performance, having an average time of 0.46 seconds to find the first primitive element of a given GF(p n) for values of n ¼ {2, 3, 4, 5, 8, 12} and prime values p between 2 and 97.

Galois appears to have been the first mathematician to realize  the impotance of the group concept in mathematics. A few hours before meeting the death in the dwell, he jotted down a summary  of his discoveries on groups of pemutations , which were the only groups he concidered ,,,,,
from my book "Selected stories in mathematics and physics, LAP    amazon

Digital Transforms have important applications on subjects such as channel coding, cryptography and digital signal processing. In this paper, two Fourier Transforms are considered, the discrete time Fourier transform (DTFT) and the finite field Fourier transform (FFFT). A finite field version of the DTFT is introduced and the FFFT is redefined with a complex kernel, which makes it a more appropriate finite field version of the Discrete Fourier Transform. These transforms can handle FIR and IIR filters defined over finite algebraic structures.

A new cellular structure for a versatile Reed-Solomon (RS) decoder is introduced based on time domain decoding algorithm. The time domain decoding algorithm is restructured to be suitable for introducing the cellular structure. The main advantages of ...

In this paper extrinsic information transfer (EXIT) charts are proposed to design non-binary low-density parity-check (LDPC) codes for the AWGN channel. A new metric is presented to describe the mutual information of the non-binary messages. The a priori information is modelled using a Gaussian mixture distribution. Analytical expressions are given for the EXIT curves of the variable and check node decoders for both regular and irregular LDPC codes. The analytical expressions are shown to agree well with simulation results. Finally, by matching the variable and check node EXIT curves, it is shown that good nonbinary LDPC can be designed for the AWGN channel.

Abstract. This paper presents an algebraic method for constructing regular low-density parity-check (LDPC) codes based on Reed–Solomon codes with two information symbols. The construction method results in a class of LDPC codes in Gallager's original form. Codes in ...

We realize square operation for quantum ternary logic using basic quantum ternary gates. With the aid of this square operation, we develop a square-multiplier unit. We further develop a cost measurement technique of the square operation and square multiplication operation through general expressions.

This paper proposes a hierarchical method for the formal hardware verification of Galois field architecture circuits. The reduced ordered functional decision diagram has been explored. The proposed method has been found to lead to significant gains in time and space, depending on the resources that are available. The theoretical claims that were made have been supported by experiments.