Showing papers on "GF(2) published in 1986"
TL;DR: A new algorithm for performing fast multiplication in GF(2^{m} ), which is O(m) in computation time and implementation area is presented and the bit-slice architecture of a serial-in-serial-out modulo multiplier is described.
Abstract: 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.
101 citations
21 citations
TL;DR: The necessary and sufficient conditions are presented for an element to generate a normal basis in the field GF(2^{m}), wherem = 2k p n and p n has two as a primitive root.
Abstract: 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.
16 citations
TL;DR: In this paper, the construction of sum and product tables for Galois fields GF(q) in which q is a power of a prime number p is studied. But the methods are oriented towards the use of operations from the prime field GF (p) that are easy to define and implement in terms of modulo-P arithmetic.
Abstract: This paper is concerned with the construction of sum and product tables for Galois fields GF(q) in which q is a power of a prime number p. Sufficient elementary theory is presented to provide a basis for the development of methods of representation, addition and multiplication for the field elements of GF (q). The methods are oriented towards the use of operations from the prime field GF (p) that are easy to define and implement in terms of modulo‐P arithmetic. They lead to a compact form of representation for the field elements and to simply applied procedures for the construction of the sum and product tables. Examples of the methods and procedures proposed are given throughout, and sum and product tables are given for Galois fields GF(q) when q has values up to 27.
16 citations
1 Jul 1986
TL;DR: In this paper, an upper bound on the number of polynomials over GF(2) that divide a polynomial of degree n over GF (2) is obtained.
Abstract: An upper bound is obtained on the number of polynomials over GF(2) that divide a polynomial of degree n over GF(2). This bound is the solution of a maximisation problem under constraints. It is used to show that most binary shortened cyclic codes (irreducible or not) satisfy the Gilbert bound.
8 citations
TL;DR: In this article, a complete algorithm to construct every possible bordered SDMC codes with coefficients in GF(2) or GF(3) was given, and several new proofs were developed too.
Abstract: We give a complete algorithm to construct every possible bordered SDMC codes with coefficients in GF(2) or GF(3). Several new proofs are developed too.
1 citations