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In this work, we present a survey of efficient techniques for software implementation of finite field arithmetic especially suitable for cryptographic applications. We discuss different algorithms for three types of finite fields and their special versions popularly used in cryptography: Binary...
In this paper, we describe, analyze and compare various
multipliers. Particularly, we investigate the standard modular multiplication, the Montgomery multiplication, and the matrix–vector multiplication techniques.
Inversion in finite fields
is a critical operation for many applications. A well-known representation basis, i.e., normal basis, provides an efficient squaring operation realized as a simple rotation of the operand coefficients. Inversion in normal basis is computed using methods...
The paper presents a survey of most common hardware architectures for finite field arithmetic especially suitable for cryptographic applications. We discuss architectures for three types of finite fields and their special versions popularly used in cryptography: binary fields, prime fields and...
Algorithms for performing divisions over Z
) are described, the corresponding digital circuits are synthesized and conclusions about their computation times are drawn. The results of their implementation within field-programmable devices are given in the case of the most efficient...
Inversion over a finite field
is usually an expensive operation which is a crucial issue in many applications, such as cryptography and error-control codes. In this paper, three different strategies for computing the inverse over binary finite fields
, called Eulerian,...
Double-exponentiation is a crucial arithmetic operation for many cryptographic protocols. Several efficient double-exponentiation algorithms based on systolic architecture have been proposed. However, systolic architectures require large circuit space, thus increasing the cost of the protocol....
We survey some applications of finite fields to finite geometries in part A and to combinatorics and error-correcting codes in parts B and C.
Quantum systems in which the position and momentum take values in the ring
and which are described with
-dimensional Hilbert space, are considered. When
is the power of a prime, the position and momentum take values in the Galois field
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