This article addresses an efficient hardware implementations for multiplication over finite field GF(2). Multiplication in GF(2) is very commonly used in cryptography and error correcting codes. An efficient hardware could reduce the cost and development for these applications. This work presents the hardware implementation of polynomial basis. In this case, the multipliers were designed using ...

We show that the multiplication operation c = a · b · r −1 in the field GF(2 k) can be implemented significantly faster in software than the standard multiplication, where r is a special fixed element of the field. This operation is the finite field analogue of the Montgomery multiplication for modular multiplication of integers. We give the bit-level and word-level algorithms for computing the...

The Truncated Multiplication computes a truncated product, a contiguous subsequence of the digits of the product of 2 integers. A few truncated polynomial multiplication algorithms are presented and adapted to integers. They are based on the most often used n-digit full multiplication algorithms of time complexity O(n), with 1< α ≤ 2, but a constant times faster. For example, the least signific...

The basic properties of multiplication algebras of nonassociative algebras over rings are introduced, including a discussion of multiplication algebras of tensor products of algebras. A characterization of semisimple artinian multiplication algebras is given along with a discussion of the nature of the simple factors of a multiplication algebra modulo its Jacobson radical. A criterion distingui...

This paper investigates performance and energy characteristics of software algorithms for long integer arithmetic. We analyze and compare the number of RISC-like processor instructions (e.g. singleprecision multiplication, addition, load, and store instructions) required for the execution of different algorithms such as Schoolbook multiplication, Karatsuba and Comba multiplication, as well as M...

The constant depth threshold circuit complexity of iterated addition, multiplication, and iterated multiplication in rings Z Z], where is an algebraic integer, is studied. It is shown that iterated addition is 1{approximable, multiplication is 2{approximable, and iterated multiplication is 3{approximable.

Counting is AC -reducible to the graph of multiplication. Hence the graph of multiplication is equivalent under AC reductions to majority and to the function form of multiplication.