This document summarizes an FPGA implementation of encryption primitives including modular multiplicative inverse and Montgomery multiplication. It begins with an abstract describing the circuit design and simulation of a modular multiplicative inverse module used in the Montgomery algorithm for efficient encryption in binary arithmetic. It then provides background on modular arithmetic and the Montgomery algorithm. It describes the hardware implementation of the modular multiplicative inverse using an extended Euclid algorithm and a long division architecture in VHDL.
This document discusses the complexity of primality testing. It begins by explaining what prime and composite numbers are, and why primality testing is important for applications like public-key cryptography that rely on the assumption that factoring large composite numbers is computationally difficult. It then covers algorithms for primality testing like the Monte Carlo algorithm and discusses their runtime complexities. It shows that while testing if a number is composite can be done in polynomial time, general number factoring is believed to require exponential time, making primality testing an important problem.
Euclid's Algorithm for Greatest Common Divisor - Time Complexity AnalysisAmrinder Arora
Euclid's algorithm for finding greatest common divisor is an elegant algorithm that can be written iteratively as well as recursively. The time complexity of this algorithm is O(log^2 n) where n is the larger of the two inputs.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
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You can also call on +1 678 648 4277 for any assistance with Statistics Assignment.
I am Anthony F. I am a Math Exam Helper at liveexamhelper.com. I hold a Masters' Degree in Maths, University of Cambridge, UK. I have been helping students with their exams for the past 9 years. You can hire me to take your exam in Math.
Visit liveexamhelper.com or email info@liveexamhelper.com.
You can also call on +1 678 648 4277 for any assistance with Math Exams.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
The document discusses approximation algorithms for solving hard combinatorial optimization problems. It defines optimization problems and covers NP-hard problems like the clique, independent set, vertex cover, and traveling salesman problems. Approaches for solving NP-hard problems include exact algorithms, approximation algorithms that provide guaranteed good solutions, and heuristics without guarantees. Approximation algorithms aim to settle for good enough solutions rather than optimal ones.
This document discusses greedy algorithms and provides examples of their use. It begins by defining characteristics of greedy algorithms, such as making locally optimal choices that reduce a problem into smaller subproblems. The document then covers designing greedy algorithms, proving their optimality, and analyzing examples like the fractional knapsack problem and minimum spanning tree algorithms. Specific greedy algorithms covered in more depth include Kruskal's and Prim's minimum spanning tree algorithms and Huffman coding.
Giáo trình Phân tích và thiết kế giải thuật - CHAP 8Nguyễn Công Hoàng
The document outlines approximation algorithms for NP-hard problems including vertex cover, set cover, and traveling salesman problem (TSP). It discusses why approximation algorithms are useful for intractable but important problems. For vertex cover, it presents a 2-approximation algorithm and proves its performance ratio. For set cover, it presents the greedy algorithm and proves its bound of O(ln|X|). For TSP, it presents an algorithm that returns a tour within a factor of 2 the optimal using minimum spanning trees.
This document discusses the complexity of primality testing. It begins by explaining what prime and composite numbers are, and why primality testing is important for applications like public-key cryptography that rely on the assumption that factoring large composite numbers is computationally difficult. It then covers algorithms for primality testing like the Monte Carlo algorithm and discusses their runtime complexities. It shows that while testing if a number is composite can be done in polynomial time, general number factoring is believed to require exponential time, making primality testing an important problem.
Euclid's Algorithm for Greatest Common Divisor - Time Complexity AnalysisAmrinder Arora
Euclid's algorithm for finding greatest common divisor is an elegant algorithm that can be written iteratively as well as recursively. The time complexity of this algorithm is O(log^2 n) where n is the larger of the two inputs.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignment.
I am Anthony F. I am a Math Exam Helper at liveexamhelper.com. I hold a Masters' Degree in Maths, University of Cambridge, UK. I have been helping students with their exams for the past 9 years. You can hire me to take your exam in Math.
Visit liveexamhelper.com or email info@liveexamhelper.com.
You can also call on +1 678 648 4277 for any assistance with Math Exams.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
The document discusses approximation algorithms for solving hard combinatorial optimization problems. It defines optimization problems and covers NP-hard problems like the clique, independent set, vertex cover, and traveling salesman problems. Approaches for solving NP-hard problems include exact algorithms, approximation algorithms that provide guaranteed good solutions, and heuristics without guarantees. Approximation algorithms aim to settle for good enough solutions rather than optimal ones.
This document discusses greedy algorithms and provides examples of their use. It begins by defining characteristics of greedy algorithms, such as making locally optimal choices that reduce a problem into smaller subproblems. The document then covers designing greedy algorithms, proving their optimality, and analyzing examples like the fractional knapsack problem and minimum spanning tree algorithms. Specific greedy algorithms covered in more depth include Kruskal's and Prim's minimum spanning tree algorithms and Huffman coding.
Giáo trình Phân tích và thiết kế giải thuật - CHAP 8Nguyễn Công Hoàng
The document outlines approximation algorithms for NP-hard problems including vertex cover, set cover, and traveling salesman problem (TSP). It discusses why approximation algorithms are useful for intractable but important problems. For vertex cover, it presents a 2-approximation algorithm and proves its performance ratio. For set cover, it presents the greedy algorithm and proves its bound of O(ln|X|). For TSP, it presents an algorithm that returns a tour within a factor of 2 the optimal using minimum spanning trees.
These are my note in the class of probabilistic analysis for the "average case" input. The look at:
1.- The use of the indicator function
2.- Enforcing the "Uniform Assumption"
At the end, we look at the application of the insertion sort average case.
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
The document discusses numerical methods and provides examples of how to implement them in Smalltalk. It covers frameworks for iterative processes, Newton's method for finding zeros, eigenvalue and eigenvector computation using the Jacobi method, and cluster analysis. Code examples and class diagrams are provided.
I am Anastasia S. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Masters's in Matlab from, Clemson University, USA. I have been helping students with their assignments for the past 6 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
Solving Linear Equations Over p-Adic IntegersJoseph Molina
The document discusses solving linear equations over the p-adic integers Zp. It begins by introducing p-adic integers as infinite series representations and describing how basic arithmetic operations of addition, subtraction, and multiplication can be performed on p-adic integers using techniques from modular arithmetic. It then shows that every linear equation over the p-adic integers has a unique solution, unlike equations over the usual integers Z.
This assignment discusses two algorithm design techniques - dynamic programming and decrease-and-conquer. It provides questions to design algorithms using these techniques for problems like rod cutting, shortest paths on a chessboard, insertion sort, and checking graph connectivity. Students must submit the assignment by the given deadline or face point deductions, and no presentations will be held after a certain date.
The document provides information about numerical methods topics including:
1) Lagrange's interpolation formula for finding a polynomial that passes through given data points, either equally or unequally spaced. The formula uses divided differences to find the coefficients.
2) Newton's divided difference interpolation formula for unequal intervals that also uses divided differences.
3) The nature of divided differences - for a polynomial of degree n, the nth divided difference is constant.
4) Examples of evaluating divided differences and constructing divided difference tables are given.
Here is the first set of notes for the first class in Analysis of Algorithm. I added a dedicatory for my dear Fabi... she has showed me what real idealism is....
The document discusses various optimization problems that can be solved using the greedy method. It begins by explaining that the greedy method involves making locally optimal choices at each step that combine to produce a globally optimal solution. Several examples are then provided to illustrate problems that can and cannot be solved with the greedy method. These include shortest path problems, minimum spanning trees, activity-on-edge networks, and Huffman coding. Specific greedy algorithms like Kruskal's algorithm, Prim's algorithm, and Dijkstra's algorithm are also covered. The document concludes by noting that the greedy method can only be applied to solve a small number of optimization problems.
Contents of the presentation:
- ABOUT ME
- Bisection Method using C#
- False Position Method using C#
- Gauss Seidel Method using MATLAB
- Secant Mod Method using MATLAB
- Report on Numerical Errors
- Optimization using Golden-Section Algorithm with Application on MATLAB
The document reviews concepts related to NP-completeness, including reductions between problems. It provides examples of reducing the directed Hamiltonian cycle problem to the undirected version. It also reduces 3-SAT to the clique problem by transforming a Boolean formula to a graph, then further reduces clique to vertex cover. Hundreds of problems have been shown to be NP-complete through relatively simple reductions like these that leverage previous results.
I am Stacy L. I am a Matlab Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, University of Houston. I have been helping students with their homework for the past 9 years. I solve assignments related to Data Analysis.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Data Analysis Assignments.
The document contains exercises, hints, and solutions for analyzing algorithms from a textbook. It includes problems related to brute force algorithms, sorting algorithms like selection sort and bubble sort, and evaluating polynomials. The solutions analyze the time complexity of different algorithms, such as proving that a brute force polynomial evaluation algorithm is O(n^2) while a modified version is linear time. It also discusses whether sorting algorithms like selection sort and bubble sort preserve the original order of equal elements (i.e. whether they are stable).
Modified Procedure to Solve Fuzzy Transshipment Problem by using Trapezoidal ...inventionjournals
This paper deals with the large scale transshipment problem in Fuzzy Environment. Here we determine the efficient solutions for the large scale Fuzzy transshipment problem. Vogel’s approximation method (VAM) is a technique for finding the good initial feasible solution to allocation problem. Here Vogel’s Approximation Method (VAM) is used to find the efficient initial solution for the large scale transshipment problem.
Benginning Calculus Lecture notes 2 - limits and continuitybasyirstar
This document discusses limits and continuity in calculus. It begins by defining limits and providing examples of computing limits of functions. It then covers one-sided limits, properties of limits, and using direct substitution to evaluate limits. The document also discusses limits of trigonometric functions and infinite limits. The overall goal is to determine the existence of limits, compute limits, understand continuity of functions, and connect the ideas of limits and continuity.
This document discusses solving problems related to quantum mechanics and waves. It provides solutions to several problems involving waves on drum membranes, classical wave equations, particles in infinite and finite boxes, and the time evolution of waves. The document solves these problems through separation of variables, normal mode expansions, computing expectation values, and discussing qualitative features like dephasing and rephasing of waves. It also briefly discusses parameters for a two-slit light experiment.
I am Lawrence B. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Masters's in Matlab from, Durham University, UK. I have been helping students with their assignments for the past 5 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
This document describes solving a steady scalar transport equation using an ADI algorithm with upwind differencing of the convective terms. The problem involves convection and diffusion of a scalar quantity in a stagnation point flow. Boundary conditions are given. The algorithm is modified from Problem Set 2 to include upwind differencing. Grid independent solutions are obtained for diffusion parameters of 0.1 and 0.01. Iteration time and accuracy are analyzed for different grid sizes.
This document provides solutions to logistics management assignment problems. It addresses 6 problems related to network flows, facility location, and vehicle routing. For problem 1, it shows that the difference between the sum of outdegrees and indegrees in a network is always zero. For problem 2, it proves Goldman's majority theorem and provides an algorithm to solve the 1-median problem. The solutions utilize concepts like isthmus edges, node weights, and network separation.
Este documento resume un libro que recopila historias sobre la relación entre personas famosas y sus mascotas perros y gatos. El libro describe las experiencias de más de 70 personalidades nacionales e internacionales con sus mascotas y destaca el amor y la solidaridad que comparten. Incluye contribuciones especiales de escritores y celebridades. Algunas de las historias resaltadas son la de un periodista secuestrado y su gato, y la de una científica pionera en el campo de la clonación y su pasión por los gatos.
La creatividad humana se define por la capacidad de crear cosas de la nada, como herramientas primitivas, y mejorar continuamente el entorno. Los seres humanos son creativos por naturaleza al relacionarse activamente con el mundo, y métodos como el análisis son herramientas útiles para ordenar la información y conocer mejor la realidad.
These are my note in the class of probabilistic analysis for the "average case" input. The look at:
1.- The use of the indicator function
2.- Enforcing the "Uniform Assumption"
At the end, we look at the application of the insertion sort average case.
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
The document discusses numerical methods and provides examples of how to implement them in Smalltalk. It covers frameworks for iterative processes, Newton's method for finding zeros, eigenvalue and eigenvector computation using the Jacobi method, and cluster analysis. Code examples and class diagrams are provided.
I am Anastasia S. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Masters's in Matlab from, Clemson University, USA. I have been helping students with their assignments for the past 6 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
Solving Linear Equations Over p-Adic IntegersJoseph Molina
The document discusses solving linear equations over the p-adic integers Zp. It begins by introducing p-adic integers as infinite series representations and describing how basic arithmetic operations of addition, subtraction, and multiplication can be performed on p-adic integers using techniques from modular arithmetic. It then shows that every linear equation over the p-adic integers has a unique solution, unlike equations over the usual integers Z.
This assignment discusses two algorithm design techniques - dynamic programming and decrease-and-conquer. It provides questions to design algorithms using these techniques for problems like rod cutting, shortest paths on a chessboard, insertion sort, and checking graph connectivity. Students must submit the assignment by the given deadline or face point deductions, and no presentations will be held after a certain date.
The document provides information about numerical methods topics including:
1) Lagrange's interpolation formula for finding a polynomial that passes through given data points, either equally or unequally spaced. The formula uses divided differences to find the coefficients.
2) Newton's divided difference interpolation formula for unequal intervals that also uses divided differences.
3) The nature of divided differences - for a polynomial of degree n, the nth divided difference is constant.
4) Examples of evaluating divided differences and constructing divided difference tables are given.
Here is the first set of notes for the first class in Analysis of Algorithm. I added a dedicatory for my dear Fabi... she has showed me what real idealism is....
The document discusses various optimization problems that can be solved using the greedy method. It begins by explaining that the greedy method involves making locally optimal choices at each step that combine to produce a globally optimal solution. Several examples are then provided to illustrate problems that can and cannot be solved with the greedy method. These include shortest path problems, minimum spanning trees, activity-on-edge networks, and Huffman coding. Specific greedy algorithms like Kruskal's algorithm, Prim's algorithm, and Dijkstra's algorithm are also covered. The document concludes by noting that the greedy method can only be applied to solve a small number of optimization problems.
Contents of the presentation:
- ABOUT ME
- Bisection Method using C#
- False Position Method using C#
- Gauss Seidel Method using MATLAB
- Secant Mod Method using MATLAB
- Report on Numerical Errors
- Optimization using Golden-Section Algorithm with Application on MATLAB
The document reviews concepts related to NP-completeness, including reductions between problems. It provides examples of reducing the directed Hamiltonian cycle problem to the undirected version. It also reduces 3-SAT to the clique problem by transforming a Boolean formula to a graph, then further reduces clique to vertex cover. Hundreds of problems have been shown to be NP-complete through relatively simple reductions like these that leverage previous results.
I am Stacy L. I am a Matlab Assignment Expert at matlabassignmentexperts.com. I hold a Master's in Matlab, University of Houston. I have been helping students with their homework for the past 9 years. I solve assignments related to Data Analysis.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Data Analysis Assignments.
The document contains exercises, hints, and solutions for analyzing algorithms from a textbook. It includes problems related to brute force algorithms, sorting algorithms like selection sort and bubble sort, and evaluating polynomials. The solutions analyze the time complexity of different algorithms, such as proving that a brute force polynomial evaluation algorithm is O(n^2) while a modified version is linear time. It also discusses whether sorting algorithms like selection sort and bubble sort preserve the original order of equal elements (i.e. whether they are stable).
Modified Procedure to Solve Fuzzy Transshipment Problem by using Trapezoidal ...inventionjournals
This paper deals with the large scale transshipment problem in Fuzzy Environment. Here we determine the efficient solutions for the large scale Fuzzy transshipment problem. Vogel’s approximation method (VAM) is a technique for finding the good initial feasible solution to allocation problem. Here Vogel’s Approximation Method (VAM) is used to find the efficient initial solution for the large scale transshipment problem.
Benginning Calculus Lecture notes 2 - limits and continuitybasyirstar
This document discusses limits and continuity in calculus. It begins by defining limits and providing examples of computing limits of functions. It then covers one-sided limits, properties of limits, and using direct substitution to evaluate limits. The document also discusses limits of trigonometric functions and infinite limits. The overall goal is to determine the existence of limits, compute limits, understand continuity of functions, and connect the ideas of limits and continuity.
This document discusses solving problems related to quantum mechanics and waves. It provides solutions to several problems involving waves on drum membranes, classical wave equations, particles in infinite and finite boxes, and the time evolution of waves. The document solves these problems through separation of variables, normal mode expansions, computing expectation values, and discussing qualitative features like dephasing and rephasing of waves. It also briefly discusses parameters for a two-slit light experiment.
I am Lawrence B. I am a Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Masters's in Matlab from, Durham University, UK. I have been helping students with their assignments for the past 5 years. I solve assignments related to Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com. You can also call on +1 678 648 4277 for any assistance with Signal Processing Assignments.
This document describes solving a steady scalar transport equation using an ADI algorithm with upwind differencing of the convective terms. The problem involves convection and diffusion of a scalar quantity in a stagnation point flow. Boundary conditions are given. The algorithm is modified from Problem Set 2 to include upwind differencing. Grid independent solutions are obtained for diffusion parameters of 0.1 and 0.01. Iteration time and accuracy are analyzed for different grid sizes.
This document provides solutions to logistics management assignment problems. It addresses 6 problems related to network flows, facility location, and vehicle routing. For problem 1, it shows that the difference between the sum of outdegrees and indegrees in a network is always zero. For problem 2, it proves Goldman's majority theorem and provides an algorithm to solve the 1-median problem. The solutions utilize concepts like isthmus edges, node weights, and network separation.
Este documento resume un libro que recopila historias sobre la relación entre personas famosas y sus mascotas perros y gatos. El libro describe las experiencias de más de 70 personalidades nacionales e internacionales con sus mascotas y destaca el amor y la solidaridad que comparten. Incluye contribuciones especiales de escritores y celebridades. Algunas de las historias resaltadas son la de un periodista secuestrado y su gato, y la de una científica pionera en el campo de la clonación y su pasión por los gatos.
La creatividad humana se define por la capacidad de crear cosas de la nada, como herramientas primitivas, y mejorar continuamente el entorno. Los seres humanos son creativos por naturaleza al relacionarse activamente con el mundo, y métodos como el análisis son herramientas útiles para ordenar la información y conocer mejor la realidad.
Live cardiff blues vs scarlets on tv coverageartanartene9556
The document provides a link to watch an online live stream of a rugby match between the Cardiff Blues and Scarlets teams. Viewers can watch the full game stream by visiting the website www.superrugbyonline.net. The stream allows fans to follow the match-up between the two Welsh regional rugby union teams in real-time.
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Este documento presenta una sopa de letras que contiene las letras de seis ciudades estadounidenses: Chicago, Tampa, Phoenix, Orlando, Las Vegas y San Diego.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
Proyecto de vida mateo wenceslao urrea guzman 2IE Simona Duque
Mateo Wenceslao Urrea Guzmán tiene 13 años, nació en San Rafael, Antioquia el 7 de marzo de 1997 y es el menor de 4 hermanos. Actualmente está terminando el bachillerato y le gusta pasar tiempo con amigos escuchando rock y metal, además sueña con estudiar aviación y convertirse en piloto.
Sebastián nació en 1996 en Rio Negro, Antioquia. Asistió a la escuela primaria allí y actualmente vive en Marinilla con su madre, padre, y tres hermanos. Planea estudiar Administración de Empresas en la universidad para administrar el negocio familiar de banquetes y establecer su propia empresa. Su meta es convertirse en un profesional exitoso y ayudar a sus hermanos a alcanzar metas similares.
This document discusses opening a restaurant that targets tourists, noting there are over 10,000 tourists per day and 2938 existing restaurants as the target market. It estimates break-even would be reached with 585 picture sales per year at £4.99 each, and outlines next steps to acquire customers and share on social media.
Los animales son una parte importante de la vida. Cada animal tiene su propia personalidad y nos brindan compañía. Debemos cuidar y proteger a todos los animales.
Este documento describe los diferentes tipos de mutaciones a nivel celular, cromosómico y genético. Explica las mutaciones somáticas y germinales, los efectos de las mutaciones y los agentes mutagénicos. También resume los cambios cromosómicos numéricos y estructurales, así como las mutaciones génicas y su papel en la evolución y el cáncer.
Inteligencia emocional-daniel goleman By MAYTA VASQUEZ Y MÁSMayVaskz
Este documento presenta información sobre las emociones y la inteligencia emocional. Se discuten varios temas como las funciones de las emociones, el desarrollo del cerebro, las respuestas de lucha o huida, la memoria emocional y la importancia de regular las emociones. También se mencionan conceptos como la inteligencia interpersonal, las diferentes formas en que las personas experimentan sus emociones, y la relación entre las emociones y el rendimiento académico. El documento cita al psicólogo Daniel Goleman y
This document compares and contrasts a student's preliminary video project to their final video project. It analyzes improvements made in several areas, including camera shots, editing, costume/mise-en-scene choices, sound design, and purposeful close-ups. The student demonstrates growth in their skills by eliminating wasted space, creating continuity through editing, applying meaningful costumes, incorporating diegetic and non-diegetic sounds appropriately, and using close-ups strategically to further the story.
Es hora de decir adiós al presupuesto tradicionalbicongroup
El documento critica el proceso presupuestario tradicional por consumir demasiados recursos de tiempo y energía de las empresas. Señala que la cultura del presupuesto es una barrera para el cambio en las organizaciones y propone un enfoque de planeación continua como alternativa, el cual actualiza periódicamente las proyecciones para incorporar cambios en el mercado. Finalmente, sugiere que aunque la transición no será fácil, las empresas aprenderán a preferir este nuevo método sobre el presupuesto tradicional.
The document describes a seminar report on using a divide and conquer algorithm to find the closest pair of points from a set of points in two dimensions. It discusses implementing both a brute force algorithm that compares all pairs, taking O(n^2) time, and a divide and conquer algorithm that recursively divides the point set into halves and finds the closest pairs in each subset and near the dividing line, taking O(nlogn) time. It provides pseudocode for both algorithms and discusses the history and improvements made to the closest pair problem over time, reducing the number of distance computations needed.
The document discusses several topics in cryptography including prime numbers, primality testing algorithms, factorization algorithms, the Chinese Remainder Theorem, and modular exponentiation. It defines prime numbers and describes algorithms for determining if a number is prime like the trial division method and Miller-Rabin primality test. Factorization algorithms are used to break encryption. The Chinese Remainder Theorem can be used to solve simultaneous congruences and speed up computations performed modulo composite numbers. Euler's theorem and its generalization are also covered.
This document summarizes a research paper that proposes using H-infinity optimization to derive a causal approximation for spline interpolation. Spline interpolation is commonly used in image processing but requires filtering past and future data, making it non-causal. The paper formulates designing a causal approximation as an H-infinity optimization problem to minimize the worst-case error over all possible input signals. For cubic splines, a closed-form optimal causal filter is derived. Numerical methods can solve for optimal filters for higher-order splines or when constraining the filter to be finite impulse response. An example is provided to demonstrate the effectiveness of the proposed causal approximation using H-infinity optimization.
Mean Absolute Percentage Error for regression models, presentation of the paper published in Neurocomputing, 2016.
http://www.sciencedirect.com/science/article/pii/S0925231216003325
We approach the screening problem - i.e. detecting which inputs of a computer model significantly impact the output - from a formal Bayesian model selection point of view. That is, we place a Gaussian process prior on the computer model and consider the $2^p$ models that result from assuming that each of the subsets of the $p$ inputs affect the response. The goal is to obtain the posterior probabilities of each of these models. In this talk, we focus on the specification of objective priors on the model-specific parameters and on convenient ways to compute the associated marginal likelihoods. These two problems that normally are seen as unrelated, have challenging connections since the priors proposed in the literature are specifically designed to have posterior modes in the boundary of the parameter space, hence precluding the application of approximate integration techniques based on e.g. Laplace approximations. We explore several ways of circumventing this difficulty, comparing different methodologies with synthetic examples taken from the literature.
Authors: Gonzalo Garcia-Donato (Universidad de Castilla-La Mancha) and Rui Paulo (Universidade de Lisboa)
This document presents and compares three approximation methods for thin plate spline mappings that reduce the computational complexity from O(p3) to O(m3), where m is a small subset of points p. Method 1 uses only the subset of points to estimate the mapping. Method 2 uses the subset of basis functions with all target values. Method 3 approximates the full matrix using the Nyström method. Experiments on synthetic grids show Method 3 has the lowest error, followed by Method 2, with Method 1 having the highest error. The three methods trade off accuracy, computation time, and the ability to do principal warp analysis.
Problem Solving by Computer Finite Element MethodPeter Herbert
This document discusses using finite element methods and the cotangent Laplacian to solve partial differential equations numerically. It begins by explaining how to generate simplicial meshes by dividing a region into basic pieces. It then introduces the cotangent Laplacian, which approximates the Laplacian operator, and how it is calculated based on angles in triangles. Finally, it demonstrates applying the cotangent Laplacian to solve sample Dirichlet and Neumann boundary value problems and compares the approximate solutions to exact solutions, showing convergence as the mesh is refined.
This document discusses PageRank, an algorithm used by Google Search to rank websites in their search results. It describes how PageRank works by modeling the web as a directed graph and calculating an importance score for each page based on the page's inlinks. It discusses how PageRank can be formulated as the principal eigenvector of the stochastic link matrix or as the stationary distribution of a random walk on the web graph. It also covers techniques like random teleportation to address issues like spider traps and dead ends.
Linear regression is an approach for modeling the relationship between one dependent variable and one or more independent variables.
Algorithms to minimize the error are
OLS (Ordinary Least Square)
Gradient Descent and much more.
Let me know if anything is required. Ping me at google #bobrupakroy
This document discusses algorithms for solving the point in polygon problem for arbitrary polygons. It presents two main concepts: the even-odd rule and the winding number rule. It shows that both concepts are closely related and can be based on determining the winding number. The document derives an incremental angle algorithm for computing the winding number and modifies it to accelerate the computation and handle special cases. It compares the resulting winding number algorithm to those found in literature.
CHAPTER TENObjectiveA brief introduction of the basic conceptTawnaDelatorrejs
CHAPTER TEN
Objective:
A brief introduction of the basic concepts of Forecasting Tools like Moving average, Weighted Moving Average, Exponential Smoothing will be used to develop projection models.
Chapter Content:
Forecasting techniques:
Into our class we will use a simple product to manufacturer. A plush eraser will be our product (Note: Don’t blame my drawing, only look at and enjoy it.).
The first technique will be Moving Average (MA). This forecasting technique consists in the estimate of a average value from historical data that move as the new present value it’s know. This average is determinated by a series of terms or established periods (n). The quantity of periods (n) will be based in the variation that it exists between the historical data. If there is large variation, the value (n) must be greater to reflect the variation. If there is small variation, the value (n) can be smaller.
Let us suppose that the following table shown the eraser’s demand for first six months of production.
Period (Month)
Demand
1
1250
2
1590
3
1340
4
1510
5
1486
6
1440
Using the Moving Average equation:
(
)
n
i
t
A
t
MA
å
-
=
)
(
Where: MA(t) is the forecasting for period t
A(t-i) is the present for period t-i
(n) is the number of periods to average
If we looking for the forecasting for the fifth period, using n=2 and n=3, which would be the answer?
N=2
N=3
A(4) = 1510
A(4) = 1510
A(3) = 1340
A(3) = 1340
---------------
A(2) = 1590
Σ = 2850
Σ = 4440
n = 2
n = 3
MA(5) = 1425
MA(5) = 1480
The average changes of period when calculating the next forecasting. When forecast the sixth period, the terms to be used for the average change according to the following example:
N=2
N=3
A(5) = 1486
A(5) = 1486
A(4) = 1510
A(4) = 1510
---------------
A(3) = 1340
Σ = 2996
Σ = 4336
n = 2
n = 3
MA(5) = 1498
MA(5) = 1445.3
The next technique known like Weighted Moving Average, this technique to difference of regular moving average, each period have a weight assigned as output probability. The Moving average to divide the periods sum between the value (n), indirectly,
it’s giving the same probability o weight to each period to determine the forecasting.
(
)
(
)
å
-
-
=
i
t
xW
i
t
A
t
WMA
)
(
Where t is the hoped period and i value run from 1 to n.
Example, determining the sixth forecasting, with (n) = 2
Mov. Average Reg
Weighted MA
A(5) = 1486 x (50%)
A(5) = 1486 x (75%)
A(4) = 1510 x (50%)
A(4) = 1510 x (25%)
MA(6) = 1498
WMA(6) = 1492
This technique allow to assign a weight or probability according to expect behavior from marketing influences. I.e. to assign greater weight to the period value most recent a cause of a promotion. The quantity of periods or term to be used for estimate the forecast depends of the variation that exists between the historical data. That means, follow the same concept of moving average.
The third technique is the Exponential Smoothing. This forecasting technique allows a ...
This document summarizes key topics from a lecture on linear regression analysis, including: initial data analysis, defining the linear model, testing hypotheses about parameters, and methods for obtaining confidence intervals and regions with or without assuming normality, such as permutation tests and bootstrapping. Key analysis steps like checking assumptions, fitting models, and comparing models are demonstrated in R code.
This document discusses probability density functions (PDFs) and their properties. It defines PDFs and provides examples of common PDFs like the normal, exponential, chi-squared, and Student's t distributions. It also covers topics like cumulative distribution functions, expected values, variances, joint and multivariate PDFs, and the central limit theorem. Statistical techniques for modeling data with PDFs are summarized, including least squares fitting, the singular value decomposition, covariance, and model resolution.
A New Deterministic RSA-Factoring AlgorithmJim Jimenez
This document proposes a new deterministic algorithm for factoring RSA numbers (n = p * q) and describes how it works. The algorithm uses schoolboy multiplication and counting/probability concepts to sequentially produce possible values for the prime factors p and q in a way that their product equals the original RSA number n. It has two main procedures: 1) A Producer procedure that sequentially generates values for the digits of p and q to match the first half of the digits in n. 2) An Eliminator procedure that eliminates combinations of p and q that do not match the second half of digits in n, leaving the correct factors. Pseudocode is provided to demonstrate how it works on a sample number. The document concludes by analyzing the running
This document discusses optimal receivers for additive white Gaussian noise (AWGN) channels. It begins by modeling the digital communication system and channel as a vector channel with additive noise. It defines optimal receivers as those that minimize the error probability. The document then derives the maximum likelihood (ML) and maximum a posteriori probability (MAP) decision rules, and shows that the ML rule is to choose the message with highest probability density given the received vector. It also discusses estimating bits individually and relates bit and symbol error probabilities. Preprocessing is discussed, showing it cannot reduce the error rate of an optimal receiver.
This document describes an algorithm to connect randomly ordered 2D points into a minimal nearest-neighbor closed contour. It begins by explaining that the algorithm takes in x and y coordinate points and connects them in either a clockwise or counterclockwise direction starting from a specified point. It then provides examples of applying the algorithm to continuous points in both normal and pathological cases, as well as to a square grid. The algorithm also allows filtering out points farther than a given distance limit.
This document discusses dynamic programming and algorithms for solving all-pair shortest path problems. It begins by defining dynamic programming as avoiding recalculating solutions by storing results in a table. It then describes Floyd's algorithm for finding shortest paths between all pairs of nodes in a graph. The algorithm iterates through nodes, calculating shortest paths that pass through each intermediate node. It takes O(n3) time for a graph with n nodes. Finally, it discusses the multistage graph problem and provides forward and backward algorithms to find the minimum cost path from source to destination in a multistage graph in O(V+E) time, where V and E are the numbers of vertices and edges.
The document discusses various topics related to algorithms including introduction to algorithms, algorithm design, complexity analysis, asymptotic notations, and data structures. It provides definitions and examples of algorithms, their properties and categories. It also covers algorithm design methods and approaches. Complexity analysis covers time and space complexity. Asymptotic notations like Big-O, Omega, and Theta notations are introduced to analyze algorithms. Examples are provided to find the upper and lower bounds of algorithms.
The document discusses Euler's generalization of Fermat's Little Theorem to composite moduli called the Theorem of Euler-Fermat. It explains that for any integer a coprime to a composite number m, a raised to the totient function of m (φ(m)) is congruent to 1 modulo m. It also provides formulas for calculating the totient function for prime powers and products of coprime integers. The Chinese Remainder Theorem, which states that a system of congruences with coprime moduli always has a solution, is introduced as well.
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Vinay Chandra, IJRIT-148
International Journal of Research in Information
Technology (IJRIT)
www.ijrit.com ISSN 2001-5569
FPGAImplementation of Encryption Primitives
Vinay Chandra1
, Kiran Kumar V G 2
1
IV Semester, M.Tech, Electronics and Communication Engineering, Sahyadri College of Engineering and
Management, Mangaluru, Karnataka, India
vinaychandra2009@gmail.com
2
Associate professor, Department of Electronics and Communication Engineering, Sahyadri College of
Engineering and Management, Mangaluru, Karnataka, India
kiran.ec@sahyadri.edu.in
Abstract
In my project, circuit design of an arithmetic module applied to cryptography i.e. Modulo Multiplicative
Inverse used in Montgomery algorithm is presented and results are simulated using Xilinx. This algorithm is useful in
doing encryption algorithms in binary arithmetic because all computers only deal with binary numbers. Encryption
algorithms rely on modulo arithmetic in which a modulus p is not a power of two unlike in binary arithmetic. This
makes encryption algorithms more complex as we would need to carry out modulus p operations especially residue
computation in binary arithmetic. Also encryption using this multiplicative inverse is presented with simulation results.
This multiplicative inverse function has iterative computations of multiplication, division and subtraction with variable
loop times.
Keywords: cryptography, simulated, Xilinx, residue, iterative, multiplicative inverse.
1. Introduction
With the increasing importance of information security, research works on cryptography and cipher
design become more and more significant. As the cryptographer’s mathematics, modular arithmetic, which
is also called clock arithmetic, is the central mathematical concept in cryptography and used in almost any
cipher from Caesar Cipher to the RSA Cipher. Different from some basic mod calculations, modulo
multiplicative inverse is a relatively complex iterative procedure and time-consuming calculation with
unfixed loop times.
Most used public key cryptosystems are based on modular arithmetic. For example RSA requires
exponentiation mod n where n is a product of two primes; Diffie-Helman, ElGamal and DSA are based on
exponentiation modulus a prime; ECC, etc. P. Montgomery proposed a method for computing modular
multiplication efficiently. He proposed to move the representation of numbers from the ring Zn to a
different domain, called Montgomery Residual representation or Montgomery Domain. It is well suited to
hardware implementations. Montgomery multiplication is a method for computing a × b mod p for positive
integers a, b, and p. It reduces execution time on a computer when there are a large number of
multiplications to be done with the same modulus p.
2. System description
Montgomery algorithm is useful in doing encryption algorithms in binary arithmetic because all
computers only deal with binary numbers. Encryption algorithms rely on modulo arithmetic in which a
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modulus p is not a power of two unlike in binary arithmetic. This makes encryption algorithms more
complex as we would need to carry out modulus p operations especially residue computation in binary
arithmetic.
2.1 Modulo Arithmetic
Basic binary arithmetic operations
Let u and v be the two operands
1. u + v % p = u + v – p [u + v / p ] = u + v – p ,if u + v ≥ p (1)
u + v if u + v < p
2. u – v % p = u – v – p [u – v / p ] = u – v – p ,if –p < u – v < 0 (2)
u – v if 0 ≤ u – v < p
3. u × v % p = u × v – p [u × v / p ] (3)
4. u / v % p = u × v-1
– p [u × v-1
/ p ] (4)
v × v-1
- p × z = 1
v-1
is called multiplicative inverse of v.
In 3 and 4 since u and v can be as large as p – 1, in worst case we may have log p iteration of
division with repeated subtraction. v-1
is that number which when multiplied with v gives remainder of 1.
This makes the two numbers congruent. These four operations may have to be done repeatedly in
encryption algorithms where mod p is a prime or a prime power.
2.2 Congruence (Equivalence) mod p
Two integers are said to be congruent mod p and we write, u ≡ v (% p) or just u ≡ v % p if and only if
u – v = p × z, for some integer z.
Example: 11 ≡ 5 % 6, 2 ≡ 4 ≡ 6 % 2. Here u and v need not be less than p.
When we look at any number mod p, the result is any number between 0 and p – 1. And other numbers
will be mapped between 0 and p – 1. For example p will be mapped to 0, p + 1 will be mapped to 1 and so
on. All numbers outside 0 and p – 1 will be mapped to any number between 0 and p – 1. We can write u – v
or v – u, both are the same. Congruence is an equivalence relation, i.e. all numbers are congruent to itself.
Congruence is also symmetric, i.e. if a is congruent to b then b is congruent to a, whatever mod we pick.
Congruence is also transitive, i.e. if a is congruent to b and b is congruent to c, then a is congruent to c.
So what we do here is take all integers (plus and minus) and divide them into p classes (equivalence
classes). And these classes are divided into 0, 1, 2…. P – 1. These are called equivalence classes under
congruence relationship.
Note 1: The congruence relation over a modulus p is an equivalence relation and partitions all integers
into p congruence classes. Class 0, class 1, ...class p – 1.
Note 2: An inverse u-1
of an integer u mod p exists if and only if u × u-1
≡ 1 % p. i.e. the multiplication
(u × u-1
) must be congruent to 1.
With this relationship we will be able to do multiplication operation under congruence. And will be
helpful in understanding the Montgomery algorithm.
Note 3: For any prime number p and a positive integer r, there exists two integers v and z such that
r × v – p × z = 1 or v ≡ r -1
% p. (5)
Proof:
Given that p is prime there exists two integers v, t < p such that
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Vinay Chandra, IJRIT-150
(r % p) × v – p × t = 1. (6)
Or
(r – p [r / p]) × v – p × t = 1 (7)
r × v – p ([r / p] × v + t) = 1 ↔ r × v – p × z =1 where (8)
z = [r / p] × v + t ↔ r × v % p = 1 or v ≡ r -1
% p (9)
Where r is relatively prime to p or when p is a prime, then it happens by itself then we get an inverse
for that r mod p. So if we pick any r and a prime, then r is always invertible with respect to mod p and that
gives us r inverse (r -1
) which we call v because that’s what the formula indicates.
2.3 Idea
Idea is to transform mod p arithmetic to mod r arithmetic where p is a prime number, r = 2k
, k is a
positive integer and 2k-1
< p < 2k
.r picked in note 3 had no restriction on it. We can pick any r as long as p
is prime. The inverse relation applies. I.e. there is an inverse of r mod p. Inverting r is very critical in
computation of mod p arithmetic in binary.
2.4 Montgomery pair
Montgomery pair of a positive integer u < p with respect to a prime modulus p is defined as:
v = g (u) = u × r % p (10)
We take u < p and map it to another number u × r % p which is still less than p. This relation is one to
one and onto.
The inverse Montgomery pair of v is defined as:
u = g -1
(v) = v × r -1
% p (11)
It is obvious that
u = g -1
(v) = v × r -1
% p = (u × r % p) r -1
% p = u (12)
Since (u × r % p) r -1
% p = (u × r × r-1
% p) % p = (u % p) % p = u (13)
2.5 Montgomery Transformation
We start with the direct product of two numbers. We begin with a pair and the pair of numbers we are
interested in is (u, v).
Fig. 1 Montgomery transform
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Multiplication in transformed domain is defined as.
z = g (u) × g (v) × r -1
% p (14)
To transform back to original domain, we have to multiply by r -1
.
u × r × v × r × r -1
× r -1
% p = u × v % p (15)
The curved path gives same result as the vertical path on the left.
The diagram is commutative.
Vertical edge on left gives:
(u, v) → u × v % p (16)
Curved path gives:
(u, v) → (g (u), g (v)) → (u × r % p, v × r % p) → u × r × v × r × r-1
% p = u
× r × v %p → u × r × v × r -1
% p = u × v % p (17)
2.6 Montgomery algorithm comes in here
Let x be an integer, 0 ≤ x ≤ p × r. The inverse Montgomery transform is defined by a pair.
m = x % r × z % r (18)
t = (x + m × p) / r (19)
x could be the product of u and v (u × v). We have r > p, therefore u × v < p × r (since u, v < p). Also r
is a power of 2 (2k
)
r = 2k
(20)
2k-1
< p < 2k
Mod r means truncating the rightmost k bits because r = 2k
. And divide by r means shift k bits to the
right. Montgomery showed that t is x × r -1
mod p. And also t is an integer. So after doing divide by r, we
don’t get a fraction which later requires flooring.
2.7 Forward and backward transforms
Here we see how to compute u × r % p. conversion of an integer u mod p to its Montgomery pair g (u)
is done.
u × r % p = (u × r -1
× r2
) % p
= (u × (r2
% p)) × r -1
% p (21)
Conversion of an integer u mod p to its inverse Montgomery pair g -1
(u) is directly computed by
inverse Montgomery transform. We have complicated the process only a little bit by introducing one extra
multiplication (u × (r2
% p)). And r2
% p is a constant because r and p are fixed. Computation of the
remaining part (i.e. r -1
% p) is seen previously. Modulo addition and subtraction mod p can also be
transformed to Montgomery domain and transformed back using commutative diagram.
This can be done as follows:
u → u × r % p
v → v × r % p
In transformed domain
u × r % p ± v × r % p = (u ± v) × r % p (22)
(u ± v) × r × r -1
% p = (u ± v) % p (23)
3. Hardware Implementation
Described below are hardware architectures used in VHDL.
3.1 multiplicative inverse using extended Euclid algorithm
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Vinay Chandra, IJRIT-152
Given: b mod a
Compute: inverse of b (i.e. multiplicative inverse of b) given by b-1
.
Procedure:
Initialize: r1 ← a,
r2 ← b,
t1 ← 0 and
t2 ← 1
While (r2 > 0)
{
q = r1 / r2;
r = r1 – q × r2;
r1 = r2;
r2 = r;
t = t1 – q × t2;
t1 = t2;
t2 = t;
}
If (r1 = 1) then b-1
= t1
Note: if t1 is negative then b-1
= t1 + a
Example:
p = 65437
r = 216
= 65536
65536 mod 65437
Compute inverse of 65536
Table 1 Multiplicative inverse computation by tabular method
q r1 r2 r t1 t2 t
0 65437 65536 65437 0 1 0
1 65536 65437 99 1 0 1
660 65437 99 97 0 1 -660
1 99 97 2 1 -660 661
48 97 2 1 -660 661 -32388
2 2 1 0 661 -32388 65437
1 -32388
Inverse of 65536 mod 65437 is: – 32388 + 65437 = 33049
Check step:
r × r -1
– p × z = 1
65536 × 33049 – 65437 × z = 1
z = 33099
3.2 Divider
Long division in decimal (135 divided by 13) is shown below.
Table 2 Decimal long division
1 0
13 1 3 5
1 3 ↓
0 5
0
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5
Quotient: 10
Remainder: 5
Long division in binary (135 divided by 13)
Table 3 Long division in binary arithmetic
1 0 1 0
1101 1 0 0 0 0 1 1 1
1 1 0 1 ↓ ↓ ↓
0 0 1 1 1 ↓ ↓
0 0 0 0 ↓ ↓
1 1 1 1 ↓
1 1 0 1 ↓
0 0 1 0 1
0 0 0 0
1 0 1
Algorithm used to do division in VHDL
Step1: store numerator in combination n1: n2
Step2: store denominator d
Step3: repeat 4 times
Shift n1: n1 left one bit
If n1 ≥ d
n1 = n1 – d
n2 (0) = 1
Step4: quotient = n2
remainder = n1 (3:0)
Table 4 Division architecture in VHDL
n1 10000 1110 n2 shl n1: n2
d 01101
n1 00011 1111 n2 n1 = n1 – d, n2(0) = 1
n1 00111 1110 n2 shl n1: n2
d 01101
n1 01111 1100 n2 shl n1: n2
d 01101
n1 00010 1101 n2 n1 = n1 – d, n2(0) = 1
n1 00101 1010 n2
Remainder Quotient
Note: if quotient is more than 4 bits, this technique doesn’t work.
E.g. 10110101 / 10000
Quotient = 00010110
Remainder = 101
So to overcome this problem we do the 4 bit division twice.
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Fig. 2 Eight bit by four bit divider
00001011 / 1000 = q (00000001) r (011)
00110101 / 1000 = q (00000110) r (101)
Therefore the quotient is 00010110 and the remainder is 101 after the division is complete.
3.3 Multiplier
Below is an example for four bit by four bit multiplier. Append four zeros to the multiplicand i.e.
make it 8 bits and store it in bp. Initially set eight bit product to zero. Multiplier ‘a’ is the input control line
to the mux. And because it is a four bit by four bit multiplier, we have four copies of the same circuit i.e. a
adder, a mux and a shift left. Depending on the value of the control line of the mux, the sum of bp and p is
passed through or only p is passed. If control input to the mux is ‘1’, then sum of bp and p is passed
through. And if the control input to the mux is ‘0’, then only p is passed through. Also note that on every
interation, bp is shifted left by one bit position.
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Vinay Chandra, IJRIT-155
Fig. 3 Four bit by four bit multiplier
3.4 r computation
Given a prime p
Compute r = 2k
Such that 2k-1
< p < 2k
Say for example the prime number is stored in 8 bits. Create another variable temp with one extra
bit i.e. 9 bits in this case, set its most significant bit to one and set remaining all bits to zero. Shift left the
contents of pime till its most significant bit becomes one. Simultaniously shift the contents of temp to the
right. And all this operation is put inside a loop. At the end of the loop when most significant bit of the
prime becomes, temp holds the value or r = 2k
.
Fig. 4 r computation
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3.5 Random number generation
Random numbers can be generated by simply shuffling the bits randomly. Initially to start with any
random number is taken. Here bit positions 4 to 7 is exored with bit positions 9 to 12 and result is stored in
bit position 12 to 15 on every clock cycle. Also bit position 4 to 15 is shifted to bit position 2 to 13 on every
clock cycle. Thus on every clock cycle a new number is generated.
Fig. 5 Random number generator
4. Simulation Results
4.1 Encryption
Encrypting the word “sahyadrian” by entering the ascii value of each letter.
Fig. 6 ASCII table
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Fig. 7 Simulation result for encryption
4.2 Random number inverse computation
Result below shows multiplicative inverse generated for the random numbers.
Fig. 8 Simulation result for random number generation
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4.3 Decryption
Result below shows multiplicative decrypting the encrypted word “sahyadrian”.
Fig. 9 Simulation result for decryption
4.3 Decryption
Figure below shows the synthesis report of Montgomery transform.
Fig. 10 Synthesis report
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5. Conclusion
As most encryption algorithms rely on modulus operation, proposed method describes the design
of a circuit module applied to cryptography i.e. Modulo Multiplicative Inverse used in Montgomery
algorithm and results are simulated using Xilinx. Montgomery multiplication computes a × b mod p for
positive integers a, b, and p. Execution time on a computer is reduced when there are a large number of
multiplications to be done with the same modulus p. The multiplicative inverse is computed using extended
Euclid algorithm. Technique used for division is shift and subtract. And the multiplier block is
implemented using shift and add operation. Also a simple technique used to generate random numbers is
presented. And finally the word “sahyadrian” is encrypted and decrypted.
This technique can be used for image encryption. Random numbers generated using the technique
described can be used for encrypting each pixel of the image. This technique of pixel by pixel encryption
makes the encryption process very fast as it eliminates long and tedious task of modulus operation by
simple shift and truncate operations.
The shift and add multiplier can be replaced by a more efficient multiplier which could save
hardware resources. Various Vedic multipliers like the Karatsuba multiplier can also be used for fast
multiplication. The technique can also be extended to encrypt text files by encrypting each character.
6. References
[1] Xiaoying, “Modulo Multiplicative Inverse Circuit Design”, 1Department of Computer and Information
Science, FST, University of Macau, Macao, China
[2] A. Daly, W. Marnane, “Efficient architectures for implementing montgomery modular multiplication
and RSA modular exponentiation on reconfigurable logic”, Proceedings of the 2002 ACM/SIGDA tenth
international symposium on Field-programmable gate arrays, Monterey, California, USA, pp.40 – 49,
2002.
[3] D. W. Matula, A. Fit-Florea, M. A. Thornton, “Table Lookup Structures for Multiplicative Inverses
Modulo 2^k“, 17th IEEE Symposium on Computer Arithmetic (ARITH'05) pp. 156-163, 2005
[4] Xilinx, “Xilinx System Generator”, Version 6.2, Xilinx Inc., USA,
http://www.xilinx.com/ise/optional_prod/system_generator.htm.
[5] Thomas Blum, “High Radix Montgomery Modular Exponentiation on Reconfigurable Hardware”, ECE
Department Worcester Polytechnic Institute 100 Institute Road, Worcester, MA 01609, USA