Squaring Algorithms with Delayed Carry Method and Efficient Parallelization

Increasing amounts of information that needs to be protecting put in claims specific requirements for information security systems. The main goal of this paper is to find ways to increase performance of cryptographic transformation with public key by increasing performance of integers squaring. Authors use delayed carry mechanism and approaches of effective parallelization for Comba multiplication algorithm, which was previously proposing by authors. They use the idea of carries accumulation by addition products of multiplying the relevant machine words in columns. As a result, it became possible to perform addition of such products in the column independently of each other. However, independent accumulation of products and carries require correction of the intermediate results to account for the accumulated carries. Due to the independence of accumulation in the columns, it became possible to parallelize the process of products accumulation that allowed formulating several approaches. In this paper received theoretical estimates of the computational complexity for proposed squaring algorithms. Software implementations of algorithms in C++ allowed receiving practical results of the performance, which are not contrary to theoretical estimates. The authors first proposed applying the method of delayed carry and parallelization techniques for squaring algorithms, which was previously proposing for integer multiplication

Performance Increasing Approaches For Binary Field Inversion

Authors propose several approaches for increasing performance of multiplicative inversion algorithm in binary fields based on Extended Euclidean Algorithm (EEA). First approach is based on Extended Euclidean Algorithm specificity: either invariant polynomial u remains intact or swaps with invariant polynomial v. It makes it possible to avoid necessity of polynomial v degree computing. The second approach is based on searching the next matching index when calculating the degree of the polynomial, since degree polynomial invariant u at least decreases by 1, then it is possible to use current value while further calculation the degree of the polynomial

Approaches for the performance increasing of software implementation of integer multiplication in prime fields

Authors have proposed the approach to increase performance of software implementation of finite field multiplication algorithm, for 32-bit and 64-bit platforms. The approach is based on delayed carry mechanism of significant bit in sum accumulating. This allows to avoid the requirement of taking into account the significant bit carry at the each iteration of the sum accumulation loop. The delayed carry mechanism reduces the total number of additions and gives the opportunity to apply the modern parallelization technologies

Approaches for the Parallelization of Software Implementation of Integer Multiplication

In this paper there are considered several approaches for the increasing performance of software implementation of integer multiplication algorithm for the 32-bit & 64-bit platforms via parallelization. The main idea of algorithm parallelization consists in delayed carry mechanism using which authors have proposed earlier [11]. The delayed carry allows to get rid of connectivity in loop iterations for sums accumulation of products, which allows parallel execution of loops iterations in separate threads. Upon completion of sum accumulation threads, it is necessary to make corrections in final result via assimilation of carries. First approach consists in optimization of parallelization for the two execution threads and second approach is an evolution of the first approach and is oriented on three and more execution threads. Proposed approaches for parallelization allow increasing the total algorithm computational complexity, as for one execution thread, but decrease total execution time on multi-core CPU

Co-Z Divisor Addition Formulae in Jacobian of Genus 2 Hyperelliptic Curves over Prime Fields

In this paper we proposed a new approach to divisor scalar multiplication in Jacobian of genus 2 hyperelliptic curves over fields with odd characteristic, without field inversion. It is based on improved addition formulae of the weight 2 divisors in projective divisor representation in most frequent case that suit very well to scalar multiplication algorithms based on Euclidean addition chains

Techniques for Performance Improvement of Integer Multiplication in Cryptographic Applications

The problem of arithmetic operations performance in number fields is actively researched by many scientists, as evidenced by significant publications in this field. In this work, we offer some techniques to increase performance of software implementation of finite field multiplication algorithm, for both 32-bit and 64-bit platforms. The developed technique, called “delayed carry mechanism,” allows to preventing necessity to consider a significant bit carry at each iteration of the sum accumulation loop. This mechanism enables reducing the total number of additions and applies the modern parallelization technologies effectively

On the Prospects of Using Metallic Glasses for In-vessel Mirrors for Plasma Diagnostics in ITER

This chapter reviews main results obtained on mirror-like samples made of several grades of bulk metallic glasses (BMG). Experiments were carried out under simulated conditions typical for the operation of plasma facing in-vessel mirrors of optical plasma diagnostics in fusion reactor ITER. Bombardment with D0 and T0 atoms radiated from burning plasma was predicted to be the main reason for the degradation of optical properties of such mirrors. Therefore, to simulate the behavior of mirrors in ITER, mirror-like samples were subjected to bombardment by ions of deuterium plasma with fixed or wide energy distribution. The effects of ion bombardment on optical properties, development of roughness, uptake of deuterium, appearance of blisters, and manifestation of some chemical processes are presented and discussed