Design and FPGA implementation of modular multiplication methods using cellular automata

Cellular automata (CA) have been accepted as a good evolutionary computational model for the simulation of complex physical systems. Meanwhile delegation of role is necessary for scalability of general computing environments. However, this practical principle can lead to conflicts at compile time and run time. Thus, we propose a policy for Separation of Duty (SoD) in Role-based Access Control (RBAC), and demonstrate how conflicts between specified SoD constraints and delegation activities can be detected based on evolutionary computation using Multiple-Attractor CA (MACA).

This thesis discusses two sets of fully specified algorithms which, given a univariate polynomial with integer coefficients (with possible multiple roots) and a positive rational error bound, uses infinite-precision arithmetic and Sturm's Theorem to compute intervals containing the real roots of the polynomial and whose lengths are less than the given error bound, The algorithms also provide a simple means of determining the number of real roots in any interval.The first set of algorithms uses infinite-precision integer arithmetic to compute the sequence of polynomials required by Sturm's Theorem and also to do all the required polynomial evaluations. The second set uses congruence arithmetic to compute the images of the sequence of polynomials over the finite fields GF(p i ) for several primes p i and then uses the natural homomorphisms from the integers onto the finite fields GF(p i ) and the Chinese Remainder Theorem to do the evaluations. The primary advantage of the use of congruence arithmetic in the second set of algorithms is that the computing time to multiply any two elements of GF(p i ) is a constant whereas the time to multiply any two integers is proportional to the square of their lengths.The computing times of both sets of algorithms are analyzed in detail using the techniques of computing time analysis developed in the past few years by G. E. Collins and D. Knuth. It is shown that if P is a primitive, positive, univariate polynomial over the integers of positive degree, m, whose coefficients are bounded in magnitude by d, and which has L real roots, and ε is some positive rational number, 0 < ε ≤ 1, then the computing time to approximate the roots of P to within is:[EQUATION]using either set of algorithms.The empirical results of applying the two sets of algorithms to random polynomials and the first ten Legendre and Chebyshev polynomials are reported. Neither set of algorithms proved to be faster for all the polynomials tested.

Modular exponentiation in a finite field is the basic computation involved in most public key crypto systems, such as Diffie-Hellman key exchange, ElGamal, etc. The current paper presents a new parallel architecture whereby the modular multiplication and squaring can be processed simultaneously in GF(2m) in m clock cycles using a cellular automata. Since the proposed cellular automata architecture is simple, regular, modular, cascadable, it can also be utilized efficiently for the implementation of VLSI.

Many computational problems that turn up in industry, operations research, network design, artificial intelligence, simulation of physical systems, logic, number theory, combinatorics, algebra, and computational biology lack a fast or feasible algorithmic solution. The best known algorithms for these problems are horrendously slow. One of the central open problems in computer science is the question of whether this slowness is inherent in these problems or that we simply lack good programming techniques. This question is known as the P versus NP question. The hardest computational problems of the above type are called NP-complete problems. It is widely believed that there does not exist a feasible algorithmic solution for these NP-complete problems.

Secure communication is a challenging issue in modern industries and critical infrastructures. The core technology used for securing the communication is cryptography. Modular multiplication is an important operation in cryptosystems. This paper investigates a novel modular multiplication algorithm and architecture. In the proposed algorithm and architecture, the parallel architecture and compact SD technique are utilized to improve the performance of modular multiplication operation and cryptosystems. The proposed architecture is implemented on Xilinx Virtex 5 FPGA. The complexity analysis results and FPGA implementation results show that the proposed modular multiplication algorithm and architecture provide improvement on the total computation time and area×time complexity compared to other modified modular multiplication algorithms and architectures. Keyword: Security, modular multiplication architecture, FPGA, parallel computation, compact SD

Matrix algebraic formulae concerning some exceptional rules of two-dimensional cellular automata

Parallel dynamic process simulation of a distillation column on the BBN Butterfly Parallel Processor computer

Parallel computing finds its place in the world of computation long back since 1842 with the first ever design of Analytical Engine by English mathematician and computer pioneer Charles Babbage. It took lots of brainstorming by the great researchers like John Cocke, Daniel Slotnick, Gene Amdahl and the big organizations like IBM, Burroughs Corporation and Honeywell for the next century, to coin various laws and set a background for today's parallel computing platform. Since then, parallel computing has successfully replaced the notion of sequential computing in terms of instruction level, task level, thread level parallelism. From the use of SISD programs to MIMD programs, the evolution of parallelism in computing has been phenomenon. When a job is divided into several parts where each part is capable of running with others in parallel, then this not only increases the overall speed of execution of job, but also ascertains the optimal utilization of computer resources like memory, processor time and the like. On this basis, the computers can be categorized on hardware level at which they can support parallelism which include multi core/ many core computers, multi processor computers, clusters, grids and massively parallel supercomputers. With the concept of parallelization concepts like concurrency control, deadlocks, synchronization of tasks have been there. Lot of work is going on since years in addressing these concepts in virtue of parallel computing. This chapter is focusing on the history, background, types of parallel computing, the memory architectures, message passing, concurrency control, deadlocks and their possible solution in parallel computing.

Many real-life applications of processor-arrays suffer from memory bandwidth limitations. In many cases an unstructured mesh is given (computation on sensor data, simulations of physical systems — PDEs), where the vertices represent computations with dependencies represented by the edges. Utilization of processing elements (PEs) during these computations is mainly depends on the node indexing of the mesh. If the adjacent nodes are stored close to each other in main memory, the reloading of node data can be significantly decreased. In case of FPGA the memory accesses can be fully determined by the designer. The mesh and an ordering of its nodes, define the graph bandwidth, which determines the minimum size of on-chip memory to avoid reloading of the nodes from the off-chip memory. If the required on-chip memory size is higher than the available resources, the mesh must be divided into parts. In this paper a novel geometry-based method is presented, which constructs reordered parts from a given unstructured mesh, where each part meets some predefined constraints on graph bandwidth.

Quantum modular multipliers are a promising development in the field of quantum computing that aim to perform modular multiplication faster and more efficiently than classical algorithms. This is achieved by encoding the numbers being multiplied into quantum bits (qubits) and using quantum gates to manipulate them in a way that represents the calculation. Since quantum physics principles like superposition and entanglement dramatically speed up processing compared to conventional computing, many calculations may be run simultaneously. Quantum modular multipliers have the potential to have a significant impact in a range of fields, including cryptography and beyond, since they offer a faster and more efficient way to do modular multiplications. Several fields, including simulation, optimization, and cryptography, can benefit from the use of quantum computing.One of the crucial operations in many of these applications is modular multiplication, a sort of multiplication that requires obtaining the remainder of the product of two numbers when divided by a third number. In this paper, we provide a novel modular multiplication method based on a quantum algorithm. To examine its complexity and compare it to that of current adders, we employ a novel form of adder that is intended to not retain the final carry. This adder's decrease percentage in terms of gates and depth is roughly 70%. Our method takes advantage of the special qualities of qubits to complete the computation significantly more quickly than conventional algorithms. It is based on the fundamental ideas of quantum physics, such as superposition and entanglement.

A novel approach for bit-serial AB2 multiplication in finite fields GF(2 m)

Modular arithmetic has often been regarded as something of amathemat-ical curiosity, at least by those unfamiliar with its importance to both abstract algebra and number theory, and with its numerous applications. However, with the ubiquity of fast digital computers, and the need for reliable digitalsecurity systems such as RSA, this important branch of mathematics is now consideredessential knowledge for many professionals. Indeed, computer arithmetic itself is, ipso facto, modular. This chapter describes how the modern graphical spreadsheet may be used to clearly illustrate the basics of modular arithmetic, and to solve certain classes of problems. Students may then gain structural insight and the foundations laid for applications to such areas as hashing, random number generation, and public-key cryptography.

Improving the computational efficiency of modular operations for embedded systems

Fine-Grained Parallel and Distributed Spatial Stochastic Simulation of Biological Reactions

The performance of the modular multiplication is the core arithmetic of the RSA cryptosystem for e-commerce security. In this paper we present a novel technique for modular multiplication by performing complements on the multiplicand and partitioning the key size of the multiplier. By applying the modular arithmetic and complement technique to the Montgomery algorithm, we propose an efficient modular multiplication method that requires exactly 50% multiplications comparing to Lee-Jeong-Kwon's algorithm.