<h1>We are going to cover below topics in this course</h1>Block 1:<ul><li>BASICS OF AN ALGORITHM AND ITS PROPERTIES&nbsp;</li><li>ASYMPTOTIC BOUNDS&nbsp;</li><li>COMPLEXITY ANALYSIS OF SIMPLE ALGORITHMS&nbsp;</li><li>SOLVING RECURRENCES&nbsp;</li></ul>Block 2:<ul><li>GREEDY TECHNIQUE&nbsp;</li><li>Divide &amp; Conquer Technique&nbsp;</li><li>GRAPH ALGORITHMS-1&nbsp;</li></ul>Block 3:<ul><li>GRAPH ALGORITHMS- II&nbsp;</li><li>DYNAMIC PROGRAMMING TECHNIQUE&nbsp;</li><li>STRING MATCHING TESHNIQUES&nbsp;</li></ul>Block 4:<ul><li>INTRODUCTION TO COMPLEXITY CLASSES&nbsp;</li><li>NP-COMPLETENESS AND NP-HARD PROBLEMS&nbsp;</li><li>HANDLING INTRACTABILITY&nbsp;</li></ul>

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<h1>What is Algorithm?</h1> <blockquote>An algorithm is a set of well-defined instructions or rules that are used to solve a specific problem or perform a specific task. In computer science, algorithms are often used to solve computational problems, and they are expressed in programming languages as a series of steps that a computer can follow to execute a particular task.</blockquote>In general, an algorithm takes an input, performs a series of computations or manipulations on that input, and produces an output. Algorithms are used in a wide variety of fields, from mathematics and science to finance and business, and they play a fundamental role in modern computing and software development.Some examples of algorithms include sorting algorithms, which are used to arrange data in a specific order, search algorithms, which are used to find specific data within a larger set of data, and optimization algorithms, which are used to find the best possible solution to a problem.

Introduction

<h2>Characteristics of Algorithm:</h2> <ol><li>Input: There must be a finite number of inputs for the algorithm.</li><li>Output: There must be some output produced as a result of execution of the algorithm.</li><li>Definiteness: There must be a definite sequence of operations for transformation of input into output.</li><li>Effectiveness: Every step of the algorithm should be basic and essential.</li><li>Finiteness: The transformation of input to output must be achieved in finite steps, i.e. the algorithm must stop, eventually! Stopping may mean that it should produce the expected output or a response that no solution is possible.&nbsp;</li></ol>Following are desirable characteristics of an algorithm: <ul><li>The algorithm should be general and is able to solve several cases. </li><li>&nbsp;The algorithms should use resources efficiently, i.e. takes less time </li><li>and memory in producing the result. </li><li>The algorithms should be understandable so that anyone can </li><li>understand and apply it to own problem. </li><li>The algorithm should follow the uniqueness such that each </li><li>instruction of the algorithm is unambiguous and clear.&nbsp;</li></ul>

Algorithm Example

<pre class="ql-syntax" spellcheck="false">Algorithm GCD-Euclid (a, b) begin [start of Algorithm]
{
while b = 0 do {
r = a mod b;
 a = b;
b = r;
} [end of while loop] return (b)
}&nbsp;[end of Algorithm]
</pre>

Euclid’s Algorithm

The basic building blocks of an algorithm are:<ol><li>Input: The data or information that the algorithm starts with.</li><li>Output: The result or solution that the algorithm produces.</li><li>Variables: These are used to store and manipulate data within the algorithm.</li><li>Conditional statements: These are used to make decisions based on the data being processed. Common conditional statements include "if", "else", and "else if".</li><li>Loops: These are used to repeat a series of steps until a certain condition is met. Common loops include "for" and "while".</li><li>Functions: These are reusable pieces of code that perform a specific task within the algorithm. Functions can take inputs, perform operations, and return outputs.</li><li>Operations: These are the specific actions that the algorithm performs on the input data to generate the desired output. Operations can include arithmetic calculations, logical comparisons, and data manipulation.</li></ol>By using these building blocks, algorithms can be designed to solve a wide range of problems in various fields, from simple arithmetic calculations to complex machine learning models.

Basic Building Blocks

The running time of an algorithm refers to the amount of time it takes for the algorithm to complete its task. The running time can depend on several factors, including the size of the input, the specific operations performed by the algorithm, and the efficiency of the implementation.In computer science, the running time of an algorithm is often expressed in terms of its "Big O" notation. This notation provides an upper bound on the worst-case running time of the algorithm, as the size of the input grows to infinity.Common running times for algorithms include:<ol><li>O(1) - constant time, where the running time is independent of the size of the input.</li><li>O(log n) - logarithmic time, where the running time grows at a slower rate than the size of the input.</li><li>O(n) - linear time, where the running time grows linearly with the size of the input.</li><li>O(n log n) - quasilinear time, where the running time grows faster than linear but slower than quadratic.</li><li>O(n^2) - quadratic time, where the running time grows exponentially with the size of the input.</li><li>O(2^n) - exponential time, where the running time grows at an exponential rate with the size of the input.</li></ol>Efficient algorithms aim to have a lower Big O notation, as this means that they will be faster and more scalable for larger inputs. However, in practice, the running time of an algorithm can also depend on factors such as the hardware and software environment in which it is run.

Running Time Of Algorithm

Constant Time(O(1)): If the running time does not depend on theinput size (n) then it is known as constant running time. It can berepresented as&nbsp;<img 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">

Constant Time Big O one

Linear Time O(kn): If the time complexity is at most a constant factor&nbsp;times the size of the input, then it is known as linear time complexityT(n) &lt;=kn where k is a constant or T(n) =O(n). An algorithm of this type of complexity generally completes the execution in a single pass. For example to search for minimum value of a given nnumbers in an array the processing can be completed just in one pass. The following program fragment demonstrates. In this way we perform constant amount of work in processing each element of an array.&nbsp;<pre class="ql-syntax" spellcheck="false">minimum = a[1] 
for i = 2 to n
if a[i] &lt; minimum 
minimum = a[i]

end end if&nbsp;
</pre><img 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"><blockquote>If you're referring to the "linear time" complexity of an algorithm, it means that the running time of the algorithm is proportional to the size of its input. In other words, if the size of the input is doubled, the running time of the algorithm will also double.</blockquote>

Linear Constant

Logarithmic Time(log(n))

Logarithmic Time

Quadratic Time: (T(n)= O(n)2)- It occurs when the algorithm is having a pair of nested loops. The outer loop iterates O(n) time and for each iteration the inner loop takes O(n) time so we get O(n2) by multiplying these two factors of n. Practically this is useful for problem for small input size or elementary sorting algorithms. The worst case time complexity for Bubble sort, Insertion sort, Selection sort and insertion sort running time complexities are O(n2).<img 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">

Quadratic Time: (T(n)= O(n)2)

Quadratic Time

Cubic Time: (T(n)= O(n3)): It often occurs when the algorithm is having there nested loops and each loop has a maximum n iterations. Let us have one interesting example which requires cubic time complexity. Suppose we are given n sets: S1, S2,....Sn. Size of each set is n (ie each set is having n elements). The problem is to find whether some pairs of these sets are disjoint, i.e there are no common elements in these pairs and what is the time complexity ?&nbsp;<img 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">

Cubic Time: (T(n)= O(n3))

Cubic Time

Polynomial Time: (0(nk)):-This running time is obtained when the search over all subsets of a set of a size kin performed. To understand the complexity of running time we have to find how many distinct subsets of a size k of n elements of a set can be chosen. For that we have to take a combination of n elements taken k at a time .As an example let us consider a problem to find an independent set in a graph which can be defined as a set of nodes in which no pair of nodes have an edge between them. Let us formulate the independent set problem in the following way: given a constant k and a graph G having n nodes (vertices) find out an independent set of a size k.<blockquote>The brute force method to solve this problem would require searching for all subsets of k nodes and for each subset it would examine whether there is an edge connecting any two nodes for each subset s of a size k .Below is a pseudo-code for finding an independent set.&nbsp;</blockquote><img 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">

Polynomial Time: (0(nk))

Polynomial Time

Exponential Time: (O(kn)) Beyond the polynomial time complexity there are other two types of bounds :exponential time O(2N)and factorial time O(n!): let us refine the independent set problem that we are given a graph of a size n and want to find out an independent of a maximum value instead of some constant k which is less than n. The modified version of the pseudo-code is presented below.&nbsp;<blockquote>Pseudo-code : Pseudo-code for finding an Independent Set of a graph&nbsp;</blockquote><pre class="ql-syntax" spellcheck="false">Input G(V,E) {

for each subset s of n number of nodes
verify whether s is an independent set
if s is the largest among all the subsets examined so for

print “s is the largest independent set ” 
end if

end for
}end of code&nbsp;fragment
</pre><img 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">

Exponential Time: (O(kn))

Exponential Time

<blockquote>Algorithmic complexity refers to how the computational resources required to run an algorithm increase as the input size grows. In other words, it measures how quickly the runtime of an algorithm increases as the input size grows.</blockquote>The two most common measures of algorithmic complexity are time complexity and space complexity.Time ComplexityTime complexity refers to the number of operations an algorithm performs as a function of the size of its input. It is often expressed in big O notation, which provides an upper bound on the number of operations an algorithm performs as its input size approaches infinity.Space ComplexitySpace complexity, on the other hand, refers to the amount of memory an algorithm requires to execute as a function of the size of its input. It is also often expressed in big O notation, which provides an upper bound on the amount of memory an algorithm requires as its input size approaches infinity. The complexity of an algorithm M is the function f(n), which give the running time and/or storage space requirement of the algorithm in terms of the size n of the input data. Frequently, the storage space required by an algorithm is simply a multiple of the data size n. In general the term “complexity” given anywhere simply refers to the running time of the algorithm. There are 3 cases, in general, to find the complexity function f(n):&nbsp;<ol><li>Worst-case : The maximum number of steps taken on any instance of size a.</li><li>Best-case : The minimum number of steps taken on any instance of size a.</li><li>Average case : The number of steps taken on average for all instances of size n&nbsp;</li></ol>

Algorithmic Complexity

Asymptotic notation is a way of describing the growth rate of a function as its input size approaches infinity. The three most commonly used asymptotic notations are:<ol><li>Big O notation (O): It represents the upper bound of a function's growth rate. For a function f(n), O(g(n)) represents the set of functions that grow no faster than g(n), up to a constant factor. In other words, f(n) = O(g(n)) if there exists a constant c &gt; 0 and a value n0 &gt; 0 such that for all n &gt;= n0, f(n) &lt;= c * g(n).</li><li>Omega notation (?): It represents the lower bound of a function's growth rate. For a function f(n), ?(g(n)) represents the set of functions that grow at least as fast as g(n), up to a constant factor. In other words, f(n) = ?(g(n)) if there exists a constant c &gt; 0 and a value n0 &gt; 0 such that for all n =&gt; n0, f(n) &gt;= c * g(n).</li><li>Theta notation (): It represents the tight bound of a function's growth rate. For a function f(n), ?(g(n)) represents the set of functions that grow at the same rate as g(n), up to a constant factor. In other words, f(n) = ?(g(n)) if there exist constants c1 &gt; 0, c2 &gt; 0, and n0 &gt; 0 such that for all n ? n0, c1 * g(n) ? f(n) ? c2 * g(n).</li></ol>These notations are often used to describe the time complexity of algorithms. For example, if the worst-case running time of an algorithm is O(n^2), it means that the running time of the algorithm grows no faster than n^2, up to a constant factor, as the input size n increases. Similarly, if the best-case running time of an algorithm is ?(n), it means that the running time of the algorithm grows at least as fast as n, up to a constant factor, as the input size n increases. If the running time of an algorithm is ?(n), it means that the running time of the algorithm grows at the same rate as n, up to a constant factor, as the input size n increases.

Asymtotic Notations

In computer science, a computing problem is a task or a question that can be solved or answered by a computer program. Computing problems can be classified into different categories based on their characteristics and requirements. Here are some common categories of computing problems:<ol><li>Computational problems: These are problems that require some form of computation or mathematical operations to solve, such as finding the shortest path between two points in a graph or sorting a list of numbers.</li><li>Decision problems: These are problems that require a binary decision, such as whether a number is prime or not, or whether a graph is connected or not.</li><li>Optimization problems: These are problems that require finding the best solution among a set of possible solutions, such as finding the shortest path that visits all cities in a traveling salesman problem.</li><li>Search problems: These are problems that require finding a solution that satisfies certain constraints, such as finding a set of moves in a chess game that leads to a checkmate.</li><li>Simulation problems: These are problems that require simulating a system or a process, such as simulating a weather model or a traffic flow model.</li></ol>In general, computing problems can be solved using algorithms, which are step-by-step procedures for solving a specific problem. The efficiency of an algorithm in solving a computing problem is typically measured by its time and space complexity, as well as its accuracy and scalability. The types of problems in computing are limitless, and are categorized into a few areas to make it easy for researchers to address types of problems while addressing the algorithm field. Following are the some commonly known problem types: <ul><li>Sorting </li><li>Searching </li><li>&nbsp;Graph problems </li><li>&nbsp;Combinatorial problems </li><li>&nbsp;Geometric problems </li><li>&nbsp;Numerical problems&nbsp;</li></ul>

Computing Problems

The divide and conquer approach is a problem-solving technique in computer science and mathematics. It involves breaking down a complex problem into smaller, more manageable subproblems, solving each subproblem recursively, and then combining the solutions to the subproblems to form a solution to the original problem.The divide and conquer approach consists of three steps:<ol><li>Divide: The first step is to break down the problem into smaller subproblems. This can be done in a variety of ways, depending on the nature of the problem. For example, if the problem involves sorting a list of numbers, the list can be divided into two smaller lists, each of which can be sorted independently.</li><li>Conquer: The second step is to solve each subproblem recursively. This can involve applying the divide and conquer approach again to the subproblems, or using a different algorithm or approach.</li><li>Combine: The final step is to combine the solutions to the subproblems to form a solution to the original problem. This typically involves merging or aggregating the results of the subproblems in some way.</li></ol>The divide and conquer approach is particularly useful for solving problems that can be broken down into smaller, independent subproblems, and where the solutions to the subproblems can be combined easily.Following are the examples of the problems that can efficiently be solved using divide and conquer approach.&nbsp;<ul><li>&nbsp;Binary Search. </li><li>&nbsp;Quick Sort. </li><li>&nbsp;Merge Sort. </li><li>&nbsp;Strassen's Matrix Multiplication. </li><li>&nbsp;Closest Pair of Points.&nbsp;</li></ul>

Divide and Conquer Approach

The greedy technique is a problem-solving approach in computer science and mathematics. It involves making locally optimal choices at each step of a problem in the hope of finding a globally optimal solution. In other words, it involves making the best choice at each step without considering the consequences of that choice on future steps.The greedy technique consists of the following steps:<ol><li>Define the problem: The first step is to define the problem and identify the objective function or metric that needs to be optimized.</li><li>Identify the choices: The next step is to identify the choices that can be made at each step of the problem.</li><li>Define the decision rule: The decision rule is the algorithm or rule that is used to choose the locally optimal solution at each step. The decision rule is usually based on the objective function or metric identified in step 1.</li><li>Iterate: The final step is to iterate the decision rule until a globally optimal solution is found or a stopping criterion is met.</li></ol>The greedy technique is particularly useful for solving problems that have optimal substructure and exhibit the "greedy choice property". The greedy choice property states that a globally optimal solution can be obtained by making locally optimal choices at each step. However, the greedy technique does not always produce a globally optimal solution, and it is important to analyze the problem and the decision rule to ensure that the solution is correct.Consequently, Greedy algorithms are often very easy to design for the optimisation problems. Following are some of the examples of the greedy approach.&nbsp;<ul><li>Kruskal’s Minimum Spanning Tree </li><li>&nbsp;Prim's Minimal Spanning Tree </li><li>&nbsp;Dijkstra's shortest path </li><li>&nbsp;Knapsack Problem&nbsp;</li></ul>

Greedy Technique

<blockquote>Dynamic programming is a technique used to solve complex optimization problems that can be broken down into smaller sub-problems. It involves solving each sub-problem only once and storing the solution for future reference, which can significantly reduce the computational time required to solve the overall problem.</blockquote>The basic idea behind dynamic programming is to build up a solution to the problem by solving smaller sub-problems and using the solutions to those sub-problems to solve larger sub-problems. This process continues until the final solution to the overall problem is obtained.Dynamic programming is often used in problems that involve a sequence of decisions that must be made in order to reach a final goal. For example, it can be used to solve problems related to scheduling, routing, and resource allocation.The working style of dynamic programming is similar to divide and conquer approach. Both solve a problem by breaking it down into a several sub problems that can be solved recursively. The drawback of divide and conquer method is the calling of a recursive function with same output/result repeatedly which is overcome in dynamic programming by maintaining a table to store the results. It is dynamically decided whether to call a function or retrieve values from the table, that’s why the word dynamic is used in it.One of the most famous examples of dynamic programming is the Knapsack problem, where you have a bag with a limited capacity and a set of items with different weights and values. The goal is to find the combination of items that maximize the total value while staying within the capacity of the bag.Dynamic programming is a powerful technique that can be applied to a wide range of problems, but it requires careful analysis and can be difficult to implement in some cases.

Dynamic Programming

Branch and bound algorithm is a general problem-solving method for optimization problems that involve searching through a large space of possible solutions. It is commonly used to solve combinatorial optimization problems, such as the traveling salesman problem or the knapsack problem.The branch and bound algorithm works by dividing the problem into smaller sub-problems, which can be solved independently. The algorithm then explores each sub-problem and eliminates those that are not promising, using a "bounding" technique to determine the maximum or minimum possible value of the objective function for each sub-problem.The algorithm continues to recursively divide the sub-problems into smaller sub-problems until a solution is found or all possibilities have been explored. At each level of recursion, the algorithm "branches" into two or more sub-problems, each of which is a subset of the original problem.The branching process involves selecting a variable or constraint and assigning it a value or a range of possible values. The algorithm then proceeds to explore each sub-problem, starting with the one that is most promising based on the bounding technique.The bounding technique is used to determine the minimum or maximum possible value of the objective function for each sub-problem. This allows the algorithm to eliminate sub-problems that cannot possibly yield a better solution than the ones that have already been found.The branch and bound algorithm can be very effective for solving combinatorial optimization problems, especially when the number of possible solutions is very large. However, it can be computationally intensive and requires careful design to ensure that the branching process is efficient and effective.

Branch And Bound Algorithm

Randomized algorithms are algorithms that use randomization to solve computational problems. They often provide probabilistic solutions, meaning that the output of the algorithm may not always be correct, but it is likely to be correct with a high probability.Randomized algorithms are useful when the problem space is very large or when a deterministic algorithm would be too slow. By introducing randomness into the algorithm, it is possible to reduce the amount of time or resources required to find a solution.There are several types of randomized algorithms, including:<ol><li>Monte Carlo algorithms: These algorithms use random sampling to approximate a solution with a certain level of confidence. The output of the algorithm may not be correct, but the probability of it being correct can be controlled.</li><li>Las Vegas algorithms: These algorithms always produce the correct output, but the running time may vary. The algorithm runs until it produces the correct output, which may take a random amount of time.</li><li>Randomized rounding algorithms: These algorithms use randomization to round a fractional solution to an integer solution. They are often used to approximate the solution to optimization problems.</li></ol>Randomized algorithms have a wide range of applications, including cryptography, machine learning, and network analysis. They can be particularly useful when dealing with large datasets or when the problem is inherently stochastic in nature.

Branch And BoRandomized Algorithmsund Algorithm

Randomized Algorithms

Backtracking is a general algorithmic technique that involves exploring all possible solutions to a problem by systematically trying out different options and backtracking from those that fail to reach a valid solution. It is particularly useful when there are many possible solutions to a problem and it is impractical to evaluate all of them in advance.The basic idea behind the backtracking algorithm is to start with an empty solution, and then incrementally build up the solution one step at a time. At each step, the algorithm makes a choice between several possible options, and then recursively explores each of these options until a valid solution is found, or all possibilities have been exhausted.&#x1D6C9;rc

Backtracking Algorithm

<h1>We are going to cover below topics in this course</h1>Block 1:<ul><li>SOFTWARE ENGINEERING AND ITS MODELS&nbsp;&nbsp;</li><li>PRINCIPLES OF SOFTWARE REQUIREMENTS ANALYSIS&nbsp;</li><li>SOFTWARE DESIGN&nbsp;</li><li>SOFTWARE QUALITY AND SECURITY&nbsp;&nbsp;</li></ul>Block 2:<ul><li>SOFTWARE PROJECT PLANNING&nbsp;</li><li>RISK MANAGEMENT AND PROJECT SCHEDULING&nbsp;</li><li>SOFTWARE TESTING&nbsp;</li><li>SOFTWARE CHANGE MANAGEMENT&nbsp;</li></ul>Block 3:<ul><li>WEB SOFTWARE ENGINEERING&nbsp;</li><li>MOBILE SOFTWARE ENGINEERING&nbsp;</li><li>CASE TOOLS&nbsp;</li><li>ADVANCED TOPICS IN SOFTWARE ENGINEERING&nbsp;</li></ul>Block 4:<ul><li>SOFTWARE PROCESS IMPROVEMENT&nbsp;</li><li>EMERGING TRENDS IN SOFTWARE ENGINEERING&nbsp;</li><li>INTRODUCTION TO UML&nbsp;</li><li>DATA SCIENCE FOR SOFTWARE ENGINEERS&nbsp;</li></ul>

Software Engineering Home page

Software engineering is the process of designing, developing, testing, and maintaining software in a systematic and efficient manner. It involves a combination of engineering principles, programming languages, and tools to create high-quality software that meets the needs of users.The software engineering process typically begins with requirements gathering, where the software team works with stakeholders to identify the needs and goals of the software project. This is followed by design, where the team creates a blueprint of the software architecture and user interface. Then, development begins, where the code is written and tested to ensure it functions as intended. Finally, the software is deployed and maintained, with updates and bug fixes as needed.Software engineering is a multi-disciplinary field that encompasses many areas of computer science, including algorithms, data structures, databases, user interface design, and software testing. It also involves working with a variety of programming languages and development tools, such as Java, Python, C++, Git, and JIRA.One important aspect of software engineering is ensuring that the software is secure, reliable, and scalable. This involves following best practices for coding, testing, and deployment, as well as using tools and frameworks that are specifically designed for these purposes.Another key aspect of software engineering is working collaboratively as a team. This often involves using agile development methodologies, such as Scrum or Kanban, to manage the software development process and ensure that all team members are working together effectively.Overall, software engineering is an important field that plays a critical role in the development of modern software applications. By following best practices and leveraging the latest tools and technologies, software engineers can create software that is secure, reliable, and user-friendly, while also meeting the needs and goals of stakeholders.

Software Engineering Introduction

Software engineering has come a long way since its inception in the 1960s. Here is an overview of its evolution:<ol><li>Traditional Software Development (1960s-1980s): In the early days, software development was mostly done by individual programmers who worked in isolation. The focus was on developing software that was functional, reliable, and efficient. Software development methodologies such as Waterfall were popular during this time.</li><li>Object-Oriented Programming (1980s-1990s): Object-oriented programming (OOP) became popular in the 1980s. This approach to software development emphasizes the use of objects, which are instances of classes, to represent and manipulate data. OOP led to the development of new software development methodologies such as Rational Unified Process (RUP) and Agile.</li><li>Agile Methodologies (2000s-2010s): Agile methodologies, which prioritize iterative and incremental development, became popular in the 2000s. The Agile Manifesto, published in 2001, outlines the values and principles of Agile. Popular Agile methodologies include Scrum, Kanban, and Extreme Programming (XP).</li><li>DevOps (2010s-Present): DevOps is a set of practices that combines software development (Dev) and IT operations (Ops) to shorten the systems development life cycle. It emphasizes collaboration and communication between development and operations teams. DevOps has become increasingly popular in recent years, with many organizations adopting it to improve their software delivery processes.</li><li>Continuous Integration and Continuous Delivery (CI/CD) (2010s-Present): CI/CD is a set of practices that automate the software delivery process, making it faster and more reliable. CI/CD involves continuously building, testing, and deploying software changes. Many organizations use CI/CD tools such as Jenkins, Travis CI, and CircleCI.</li></ol>Overall, software engineering has evolved from a solo practice to a highly collaborative one that emphasizes iterative development and automation. As technology continues to evolve, software engineering is likely to continue to evolve as well.

Evolution Of Software Enginnering

Software products differ from other engineering products in several ways. Here are some of the key differences:<ol><li>Intangible vs Tangible: One of the most significant differences is that software products are intangible, while other engineering products are tangible. Software products are essentially lines of code that run on a computer or other electronic device, while tangible products can be physically touched and manipulated.</li><li>Complexity: Software products tend to be more complex than other engineering products because they are designed to perform complex tasks and handle large amounts of data. This complexity requires a high degree of precision and attention to detail in the design and implementation process.</li><li>Reproducibility: Unlike other engineering products, software products can be reproduced infinitely without any additional cost or effort. This means that once a software product is developed, it can be distributed to an unlimited number of users without requiring any additional resources.</li><li>Evolution: Software products are often subject to frequent updates and upgrades, whereas other engineering products typically remain static once they are built. This means that software products must be designed to be flexible and adaptable, and must be able to evolve over time as new features and functionality are added.</li><li>Testing: Testing software products can be more difficult than testing other engineering products, because software can exhibit unexpected behavior in a variety of circumstances. This requires thorough testing and quality assurance processes to ensure that the software functions as expected under all conditions.</li><li>Engineering products can be perfectly designed, but in the case of software, however good the design, it can never be 100% error free. Even the best quality software is not completely error free. A software is called good quality software if it performs the required operation, even if it has a few errors.&nbsp;</li><li>The Software is not vulnerable to external factors like environmental effects. But the same external factors may harm hardware. The hardware component may be replaced with spare parts in the case of failure, whereas the failure of a software component may indicate the errors in design.&nbsp;</li></ol> Overall, software products differ from other engineering products in significant ways, requiring a unique set of skills and approaches to design, develop, test, and maintain them effectively.

Software vs Other Enginnering Products 

Software engineering plays a critical role in the development, maintenance, and evolution of software products. Here are some of the key reasons why software engineering is important:<ol><li>Efficient Software Development: Software engineering provides a structured approach to software development, which helps ensure that software is developed efficiently and effectively. By following a defined process, software engineers can minimize errors and ensure that the software is delivered on time and within budget.</li><li>Quality Assurance: Software engineering includes a range of quality assurance practices, such as testing and code reviews, that help ensure that software is reliable, secure, and meets the needs of users. This can help minimize the risk of software failure or security breaches, which can have serious consequences for users and organizations.</li><li>Maintainability: Software engineering helps ensure that software is maintainable over the long term. By following best practices for software design and development, engineers can create software that is easier to update and modify as new requirements emerge or bugs are discovered.</li><li>Collaboration: Software engineering emphasizes collaboration and communication among team members, which is essential for successful software development. By working together effectively, software engineers can share ideas, identify problems early on, and find creative solutions to complex challenges.</li><li>Innovation: Software engineering plays a critical role in driving innovation in the technology industry. By applying new techniques and approaches to software development, engineers can create software that is faster, more efficient, and more capable than ever before.</li></ol>Overall, software engineering is essential for developing high-quality software that meets the needs of users and organizations. Without software engineering, software development would be less efficient, less reliable, and less able to adapt to changing requirements over time.

Importance Of Software Enginnering

Software development models are frameworks that provide a structure for planning, designing, building, and testing software. Here are some of the most commonly used software development models:<ol><li>Build &amp; Fix Model: It is a simple two phase model. In one phase, code is developed and in another, code is fixed.</li><li>Waterfall Model: The waterfall model is a linear sequential approach to software development. It follows a series of phases in a sequential order, such as requirements gathering, design, implementation, testing, and maintenance.</li><li>Agile Model: The agile model is an iterative and incremental approach to software development. It involves collaboration between cross-functional teams and emphasizes customer satisfaction through continuous delivery of working software.</li><li>Spiral Model: The spiral model is a risk-driven model that emphasizes identifying and mitigating risks throughout the software development process. It involves a series of iterations that include planning, risk analysis, engineering, and evaluation.</li><li>Prototype Model: The prototype model is a model that involves building a working prototype of the software before the final version. This model allows users to provide feedback and make changes before the final software is delivered.</li><li>V-Model: The V-model is a software development model that emphasizes testing at every stage of the software development life cycle. It involves creating test cases early in the process and executing them as the software is built.</li><li>Incremental Model: The incremental model involves breaking the software development process into small, incremental pieces. Each increment is built and tested, and then integrated into the larger system.</li><li>DevOps Model: The DevOps model is an approach that combines software development and IT operations. It involves continuous integration, testing, and deployment, and emphasizes collaboration and communication between developers and operations teams.</li></ol>Each software development model has its own strengths and weaknesses, and the choice of model will depend on the specific needs and goals of the project.

Software Development Models

Water Fall Model

The Spiral Model is a software development model that combines elements of both the Waterfall Model and the iterative development approach. It was first introduced by Barry Boehm in 1986 and is based on the idea of repeatedly cycling through a series of development phases in a spiral pattern.The Spiral Model consists of four main phases:<ol><li>Planning: In this phase, the project objectives, requirements, and constraints are identified and analyzed. The project team also identifies the risks associated with the project and develops a plan to address them.</li><li>Risk Analysis: In this phase, the project team conducts a detailed analysis of the risks identified in the previous phase. This involves evaluating the probability of each risk occurring, its potential impact on the project, and the cost of addressing it.</li><li>Engineering: In this phase, the project team develops and tests the software based on the requirements identified in the first phase. The development process is iterative, with each cycle of development focused on a specific set of requirements.</li><li>Evaluation: In this phase, the software is evaluated to determine whether it meets the requirements and objectives of the project. This involves testing the software and gathering feedback from stakeholders.</li></ol>After each cycle of development, the project team reviews the results and decides whether to continue with the next cycle, adjust the plan, or terminate the project. The cycle then repeats, with each iteration incorporating the feedback and lessons learned from the previous iteration.The Spiral Model is particularly well-suited for projects with a high degree of complexity or uncertainty, where the requirements may be difficult to define or subject to change. It emphasizes risk management and allows for flexibility in the development process, making it a popular approach for software development in industries such as aerospace, defense, and telecommunications. However, it can also be more time-consuming and resource-intensive than other development models, and may not be suitable for projects with tight deadlines or limited resources.

Spiral Model

Prototype Model

Capability Maturity Models (CMMs) are a set of frameworks that provide organizations with a roadmap to improve their processes and increase their capability to deliver high-quality products or services. CMMs are used to evaluate an organization's maturity level in a particular area and provide guidelines for continuous improvement.There are several CMMs that have been developed for various industries and areas of focus, including software engineering, project management, and cybersecurity. The most widely used CMM is the Capability Maturity Model Integration (CMMI), which was developed by the Software Engineering Institute (SEI) at Carnegie Mellon University.The CMMI framework consists of five maturity levels, ranging from level 1 (initial) to level 5 (optimized). Each level represents a higher degree of process maturity and organizational capability. The five levels are:<ol><li>Initial: Processes are ad hoc, and success depends on individual effort.</li><li>Managed: Processes are defined and standardized, and outcomes are predictable.</li><li>Defined: Processes are well-established and documented, and there is a focus on continuous improvement.</li><li>Quantitatively Managed: Processes are measured and controlled, and there is a focus on quantitative analysis and data-driven decision making.</li><li>Optimizing: Processes are continuously improved, and there is a focus on innovation and learning.</li></ol>The CMMI framework is widely used in the software industry to evaluate the maturity of an organization's software development processes. It provides a structured approach to process improvement and helps organizations to identify areas for improvement and implement best practices.Other CMMs, such as the Project Management Maturity Model (PMMM) and the Cybersecurity Capability Maturity Model (C2M2), provide similar frameworks for process improvement in specific areas.

CAPABILITY MATURITY MODELS

In Capability Maturity Models (CMMs), maturity levels refer to a set of well-defined stages of process improvement that organizations go through as they strive to improve their processes and increase their capability to deliver high-quality products or services.In the Capability Maturity Model Integration (CMMI), the most widely used CMM, there are five maturity levels, ranging from level 1 (initial) to level 5 (optimized):<ol><li>Level 1 - Initial: At this level, an organization's processes are ad hoc and unstructured. The success of the organization depends on the individual efforts of the people involved in the process.</li><li>Level 2 - Managed: At this level, an organization has established basic project management processes, and the outcomes are predictable. The focus is on managing project requirements, schedules, and budgets.</li><li>Level 3 - Defined: At this level, an organization has a well-established and documented set of processes for project management, and there is a focus on continuous process improvement. The focus is on defining and standardizing processes across the organization.</li><li>Level 4 - Quantitatively Managed: At this level, an organization has a well-defined set of processes that are measured and controlled. The focus is on quantitative analysis and data-driven decision making.</li><li>Level 5 - Optimizing: At this level, an organization has a culture of continuous improvement and innovation. The focus is on improving processes and incorporating new technologies and best practices.</li></ol>Each maturity level builds upon the previous level, and organizations can use the CMMI framework to evaluate their maturity level in a particular area and implement best practices to improve their processes and increase their capability.

Maturity Levels In CMM

Key Process Areas (KPAs) are critical activities that an organization must perform to achieve a particular maturity level in Capability Maturity Models (CMMs). KPAs provide a roadmap for organizations to identify and prioritize the processes that need improvement to achieve higher levels of maturity.In the Capability Maturity Model Integration (CMMI), there are a set of KPAs for each maturity level, and they are categorized into four different areas:<ol><li>Process Management: KPAs in this area focus on establishing and maintaining an effective process for managing projects and ensuring that the processes are followed consistently. Some examples of KPAs in this area include process planning, process measurement, and process control.</li><li>Project Management: KPAs in this area focus on the management of individual projects within the organization. These KPAs include project planning, project monitoring and control, and risk management.</li><li>Engineering: KPAs in this area focus on the technical aspects of the product or service being developed. Examples of KPAs in this area include requirements development, design, and verification and validation.</li><li>Support: KPAs in this area focus on the support activities required to enable the other three areas to function effectively. Examples of KPAs in this area include configuration management, quality assurance, and training.</li></ol>Each KPA has a set of specific goals and practices that an organization must achieve to demonstrate the maturity level required for that area. By focusing on the critical activities outlined in the KPAs, organizations can improve their processes and increase their capability to deliver high-quality products or services.

Key Process Areas

Software Requirements Analysis is the process of identifying, eliciting, documenting, analyzing, validating, and managing the requirements for a software system. Effective requirements analysis is critical for the success of any software project, and it requires a structured approach and adherence to a set of principles.Here are some principles of software requirements analysis:<ol><li>Stakeholder Involvement: Requirements analysis should involve all stakeholders, including end-users, developers, project managers, and other relevant parties. It is essential to understand the needs and expectations of each stakeholder to develop a comprehensive set of requirements.</li><li>Requirements Elicitation: Requirements analysis should involve a variety of techniques to elicit requirements from stakeholders. Techniques could include interviews, surveys, workshops, and observation.</li><li>Requirements Documentation: Requirements should be documented in a clear and concise manner, and the documentation should be easily understandable by all stakeholders. The documentation should include all relevant information, such as functional and non-functional requirements, constraints, assumptions, and dependencies.</li><li>Requirements Analysis: The requirements should be analyzed to ensure that they are complete, consistent, and verifiable. The analysis should also identify any conflicts or ambiguities in the requirements.</li><li>Requirements Validation: The requirements should be validated to ensure that they meet the stakeholders' needs and expectations. This can be achieved through reviews, walkthroughs, and inspections.</li><li>Requirements Management: The requirements should be managed throughout the software development life cycle. Changes to the requirements should be documented, analyzed, and validated before being implemented.</li><li>Continuous Improvement: The requirements analysis process should be continually improved to ensure that the software system's requirements are met effectively and efficiently.</li></ol>By adhering to these principles, organizations can ensure that the software system's requirements are effectively analyzed, validated, and managed, resulting in a higher likelihood of success for the software project.

Principle Of Software Requirement Analysis

Software requirements analysis involves various techniques and approaches to identify, analyze, document, and validate the requirements for a software system. Here are some of the common types of requirements analysis:<ol><li>Business Requirements Analysis: Business requirements analysis is focused on understanding the overall business goals and objectives and how the software system can support them. This involves identifying the business processes and workflows, the system's functionality, and the business rules that must be incorporated into the system.</li><li>Functional Requirements Analysis: Functional requirements analysis is focused on understanding the specific features and functionality that the software system must provide. This includes identifying the inputs, outputs, and processing requirements of the system.</li><li>Non-functional Requirements Analysis: Non-functional requirements analysis is focused on understanding the performance, security, and usability requirements of the software system. This includes identifying the system's response time, availability, reliability, scalability, and other characteristics that are critical to the system's success.</li><li>User Requirements Analysis: User requirements analysis is focused on understanding the needs and expectations of the system's end-users. This involves identifying the user's tasks, goals, and requirements for the system's interface and usability.</li><li>System Requirements Analysis: System requirements analysis is focused on understanding the technical requirements of the software system, including the hardware and software components, network infrastructure, and integration with other systems.</li><li>Domain Requirements Analysis: Domain requirements analysis is focused on understanding the specific domain or industry in which the software system will be used. This involves identifying the industry-specific standards, regulations, and best practices that must be incorporated into the system.</li></ol>By using a combination of these types of requirements analysis, software development teams can ensure that they identify and document all the necessary requirements for a software system and deliver a system that meets the stakeholders' needs and expectations.

Types Of Requirement Analysis

SRS stands for "Software Requirements Specification." It is a document that describes in detail the requirements for a software system. The SRS typically includes a description of the functional and non-functional requirements of the software, as well as any constraints, assumptions, and acceptance criteria that must be met.The purpose of an SRS is to provide a clear and unambiguous description of what the software system is supposed to do, so that developers, testers, and stakeholders can all have a common understanding of the system's requirements. The SRS is usually created early in the software development lifecycle, and serves as a reference throughout the development process to ensure that the software is meeting its requirements. The requirement analysis involves obtaining a clear and thorough understanding of the product to be developed. Thus, SRS should be consistent, correct, unambiguous &amp; complete, document. The developer of the system can prepare SRS after detailed communication with the customer. An SRS clearly defines the following:&nbsp;<ul><li>External Interfaces of the system: They identify the information which is to flow „from and to’ to the system.</li><li>Functional and non-functional requirements of the system. They stand for the finding of run time requirements.</li><li>Design constraints:&nbsp;</li></ul>Outlines of SRSIntroduction:<ul><li>Purpose</li><li>Scope</li><li>Definitions, acronyms, and abbreviations</li><li>References</li><li>Overview</li></ul>Overall description:<ul><li>Product perspective</li><li>Product functions</li><li>User characteristics</li><li>Constraints</li><li>Assumptions and dependencies</li></ul>Specific requirements:<ul><li>External Interfaces</li><li>Functional requirements</li><li>Performance requirements</li><li>Logical Database requirements</li><li>Design Constraints</li><li>Software system attributes</li><li>Organising the specific requirements</li><li>Additional Comments</li></ul>Supporting information:<ul><li>Table of contents and index</li><li>Appendixes&nbsp;</li></ul>

Software Requirements Specification(SRS)

Software Requirements Specification

<h1>We are going to cover below topics in this course</h1>Block 1:<ul><li>The Process Of Communication&nbsp;</li><li>Telephone Techniques</li><li>Job Application and Interviews</li><li>Group Discussions</li><li>Managing Organisational&nbsp;Structure</li></ul>Block 2:<ul><li>Meetings</li><li>Presentation skills- I</li><li>Presentation skills- II</li><li>Developing Interpersonal skill for successful life at the work place</li><li>Work Ethics and Social Media Etiquette</li><li>Copyright and plagiarism</li></ul>

Home Page MCS 214. Professional Skills and Ethics

Communication is the act of conveying information from one person or group to another through various channels such as speech, writing, body language, or technology. It involves the exchange of messages or ideas between two or more individuals with the intent of sharing information, expressing thoughts, and conveying emotions.Effective communication requires both a sender and a receiver to understand the message being communicated. Communication can take many forms, such as verbal communication, nonverbal communication, written communication, visual communication, and digital communication.Good communication skills are important in almost every aspect of life, including personal relationships, business, education, and healthcare. Effective communication can help build strong relationships, increase understanding, and facilitate collaboration and problem-solving. It is a key component of successful teamwork and effective leadership.

What is Communication

Process Of Communication

There are several common barriers to communication that can prevent effective communication from taking place. These include:<ol><li>Language barriers: When people speak different languages, or even different dialects of the same language, it can be difficult for them to communicate effectively.</li><li>Cultural barriers: People from different cultures may have different communication styles, values, and beliefs, which can cause misunderstandings and misinterpretations.</li><li>Physical barriers: Environmental factors such as noise, distance, or poor lighting can make communication difficult.</li><li>Emotional barriers: Emotional states such as anxiety, anger, or fear can interfere with effective communication.</li><li>Perception barriers: Different people may perceive the same message differently, based on their own experiences, biases, and expectations.</li><li>Information overload: When there is too much information to process, it can be difficult to focus on and understand the most important points.</li><li>Technology barriers: Technical issues such as poor audio quality, slow internet connections, or incompatible software can disrupt communication.</li></ol>To overcome these barriers, it is important to use clear and concise language, actively listen to the other person, ask questions to clarify understanding, and be aware of cultural differences and other potential barriers. Additionally, choosing the appropriate communication channel (such as face-to-face, email, or phone) can also help to improve communication effectiveness.

Communication Barriers

There are several types of communication, including:<ol><li>Verbal communication: This is communication that involves the use of spoken or written words, such as face-to-face conversations, phone calls, emails, and memos.</li><li>Nonverbal communication: This includes communication that does not involve the use of words, such as body language, facial expressions, gestures, and tone of voice.</li><li>Written communication: This is communication that is written, such as reports, memos, emails, and letters.</li><li>Visual communication: This involves communication that uses visual aids such as diagrams, graphs, charts, and videos to convey information.</li><li>Interpersonal communication: This is communication that takes place between two or more people, such as face-to-face conversations or phone calls.</li><li>Group communication: This involves communication that takes place within a group, such as team meetings, group presentations, and group discussions.</li><li>Mass communication: This is communication that reaches a large audience, such as through television, radio, newspapers, and social media.</li></ol>Each type of communication has its own strengths and weaknesses, and the appropriate type of communication to use will depend on the specific situation and goals of the communication.

Types Of Communications

Expression is an important aspect of communication as it helps to convey emotions, attitudes, and intentions. There are several ways in which expression can be used in communication:<ol><li>Facial expressions: Facial expressions, such as smiles, frowns, and eye contact, can convey a range of emotions, including happiness, sadness, anger, and interest.</li><li>Body language: Body language, such as posture, gestures, and movements, can also convey emotions and attitudes, such as confidence, nervousness, and openness.</li><li>Tone of voice: The tone of voice used in communication can convey emotions, attitudes, and intentions. For example, a sarcastic tone of voice can convey humor or criticism, while a soft tone of voice can convey empathy or concern.</li><li>Written expression: In written communication, punctuation, font, and formatting can also be used to convey emotions and attitudes. For example, using all caps can convey yelling or excitement, while using a smaller font can convey shyness or hesitation.</li><li>Artistic expression: Artistic forms of expression, such as painting, music, and dance, can also be used in communication to convey emotions and attitudes.</li></ol>Using expression effectively in communication can help to create a more engaging and effective communication experience, as it allows for a deeper level of understanding and connection between the individuals involved. However, it is important to be aware of cultural differences in expression, as some forms of expression may be more or less appropriate in certain cultures.

Communication Expressions

A conference call is a telephone call in which three or more people participate at the same time. The call can be set up so that everyone can hear each other, or it can be arranged so that one person speaks at a time and others listen. Conference calls are commonly used in business settings for meetings, team collaborations, and interviews where multiple participants from different locations need to communicate and share information at the same time. The participants can be located in different cities, countries, or even continents, and connect to the call using a telephone or video conferencing software.Benefits Of Conference CallThere are several benefits of using conference calls, including:<ol><li>Cost-effective: Conference calls are typically less expensive than traveling to a meeting location, which can save time and money.</li><li>Increased efficiency: Conference calls can be set up quickly and easily, allowing for faster decision-making and problem-solving.</li><li>Convenience: Conference calls can be conducted from anywhere, which means that participants can join from their office, home, or even while traveling.</li><li>Enhanced collaboration: Conference calls allow participants to share information, ask questions, and provide feedback in real-time, which can lead to more effective collaboration.</li><li>Improved communication: Conference calls can help reduce miscommunications that can occur with emails or other forms of written communication.</li><li>Increased productivity: By eliminating the need for travel and allowing for faster decision-making, conference calls can increase productivity and save time for all participants.</li></ol>Overall, conference calls are a cost-effective and efficient way to communicate and collaborate with colleagues, clients, and partners regardless of their location.

Conference Call, Group Call, Calls in group

Conference Call

There are many types of questions that can be asked during an interview, but here are a few common ones:<ol><li>Behavioral Questions: These are questions designed to learn more about how you have behaved in specific situations in the past. Examples of behavioral questions might include: "Can you tell me about a time when you had to handle a difficult situation with a co-worker?" or "Can you give me an example of a time when you had to make a tough decision?"</li><li>Technical Questions: These are questions that assess your knowledge and understanding of the technical aspects of the job you are applying for. For example, if you were applying for a software engineering position, you might be asked a technical question like: "How would you go about optimizing the performance of a web application?"</li><li>Situational Questions: These are questions designed to gauge how you would handle hypothetical situations that you might encounter on the job. Examples of situational questions might include: "If a customer complained about a product, how would you handle the situation?" or "If a team member was not meeting their deadlines, how would you address the issue?"</li><li>Personal Questions: These are questions that seek to learn more about you as a person, your interests, hobbies, and goals. For example, you might be asked a personal question like: "What do you like to do in your free time?" or "What motivated you to pursue a career in this field?"</li><li>Open-Ended Questions: These are questions that allow you to provide more detailed and nuanced responses. Examples of open-ended questions might include: "Can you tell me about yourself?" or "Why do you want to work for this company?"</li></ol>

What Types of question asked during interview, MCS-214 Notes Material

Types Of Question During Interview

Here are some tips on how to succeed in a group discussion:<ol><li>Listen actively: Pay attention to what others are saying and show that you are engaged in the discussion. This means avoiding distractions such as your phone, and making eye contact with the person speaking.</li><li>Respect others' opinions: You may not always agree with what others are saying, but it is important to show respect for their opinions. Avoid interrupting, and wait for your turn to speak.</li><li>Contribute meaningfully: Speak clearly and confidently, and make sure your contributions are relevant to the topic at hand. Avoid speaking just to hear yourself talk, and make sure your comments add value to the discussion.</li><li>Stay on topic: Make sure your comments are focused on the topic being discussed. If you have a new idea or suggestion, try to tie it back to the main topic.</li><li>Build on others' ideas: If someone else makes a good point, try to build on it or expand on it. This shows that you are engaged in the discussion and can help move the conversation forward.</li><li>Be open-minded: Be willing to consider new ideas and perspectives, even if they differ from your own. This can lead to a more productive discussion and help everyone learn from each other.</li><li>Manage conflicts constructively: If there is a disagreement or conflict, try to resolve it constructively by listening to both sides and finding common ground. Avoid personal attacks and focus on the issue at hand.</li></ol>

Best Way to Scceede in Group Discussion, MCS-214 Notes Material

Best Way to Scceede in Group Discussion

Presentation skills refer to the abilities and techniques used by an individual to deliver a message or information effectively to an audience. It includes various elements, such as verbal communication, body language, tone of voice, visual aids, and overall presentation style.Good presentation skills involve planning and organizing the content, understanding the audience, establishing a clear purpose and message, using effective communication techniques, and engaging the audience throughout the presentation. It also involves being confident, composed, and delivering the message in a clear and concise manner.Developing strong presentation skills is important in many professional and personal settings, as it helps individuals to communicate effectively, persuade and influence others, and achieve their desired outcomes.

Presenttaion Skills, MSC-214 Notes

Presenttaion Skills

<h1>We are going to cover below topics in this course</h1>Block 1:<ul><li>Cyber Security Issues and Challenges</li><li>Cryptography Mechanism</li><li>Data Security and Management</li></ul> Block 2:<ul><li> </li></ul>

MCS-215 Topic to be covered, MSC-215 Notes

<blockquote>Cybersecurity is the practice of protecting computer systems, networks, devices, and data from theft, damage, or unauthorized access. It involves a range of technologies, processes, and practices designed to safeguard information and prevent cyber threats, such as hacking, phishing, malware, and other types of cyber attacks.</blockquote>The field of cybersecurity is constantly evolving as new threats and vulnerabilities emerge. Some of the key cybersecurity practices and technologies include firewalls, antivirus software, encryption, access control, security assessments, incident response planning, and employee training and awareness.Ensuring cybersecurity is essential for organizations of all sizes, as cyber attacks can lead to significant financial losses, reputational damage, and legal liability. It is also important for individuals to protect their personal information and devices from cyber threats, by using strong passwords, keeping software up-to-date, and being cautious when sharing sensitive information online.

Cyber Security Itroduction, MSC-215 Notes

Cyber Security Itroduction

Digital security is a broader term which encompasses within itself protection of online identity data assets Technology with the use of various tools like software, Web Services, biometrics, firewalls, proxies, vulnerability scanner, instant message or telephone encryption tools etc. Digital security provides protection against cyber-attacks unauthorised access, online malicious activities etc.<blockquote>The three pillars of digital security are confidentiality, integrity, and availability, commonly referred to as the CIA triad.</blockquote><ol><li>Confidentiality: Confidentiality refers to the protection of sensitive information from unauthorized access or disclosure. This involves implementing measures such as encryption, access controls, and data classification to ensure that only authorized individuals or systems have access to sensitive information.</li><li>Integrity: Integrity refers to the protection of the accuracy and completeness of information, ensuring that it has not been tampered with or modified in any unauthorized way. Measures such as data validation, checksums, and digital signatures can be implemented to ensure the integrity of information.</li><li>Availability: Availability refers to the ability to access information and resources when needed. This involves implementing measures such as backups, redundancy, and disaster recovery planning to ensure that systems and data are available and accessible to authorized users when needed.</li></ol>These three pillars are essential for ensuring the security of digital systems and protecting sensitive information from unauthorized access, modification, or disclosure.Digital Security: Pros :<ul><li>It helps in protecting personal information stored in devices. </li><li>Suspicious or unauthorized access to devices can be blocked through digital security and thus preventing possible harm.</li><li> Security based on biometrics is capable of providing a higher degree of protection against attacks as it's difficult to steal biometric information. ? Digital security enables oneself to fearlessly communicate, transact, work etc. in online mode. </li><li> Protects the computer from crashing or slowing down and thus protects business, transactions, communication etc. happening over computer, network or system 3 </li><li> Digital security thus may help in fostering the economy of the State as it cuts down on many costs.&nbsp;</li></ul>Digital Security Cons:<ul><li> Availing services or procuring tools for digital security can be a costly affair. Web services or tools may or may not be compatible with the device of the user.</li><li> Digital security services or tools may be difficult to configure at times and needs to be updated regularly.</li><li>Services or tools may slow down functioning of user’s device or at times may intervene even normal functioning of another programme&nbsp;</li></ul>

Digital Security Pros and Cons | MCS-215 Online Notes

Digital Security Pros and Cons

There have been numerous security issues and breaches in cyberspace in recent years. Here are some examples:<ol><li>Data breaches: Large-scale data breaches have become increasingly common, with cybercriminals targeting organizations to steal personal and financial information. For example, in 2020, the software company SolarWinds experienced a major data breach that affected multiple federal agencies and businesses.</li><li>Ransomware attacks: Ransomware is a type of malware that encrypts an organization's data, making it inaccessible until a ransom is paid. In 2021, the Colonial Pipeline in the United States was hit by a ransomware attack, causing widespread disruption and fuel shortages.</li><li>Phishing attacks: Phishing is a social engineering technique that involves tricking individuals into giving away sensitive information, such as login credentials. In 2021, the social media platform Twitter experienced a high-profile phishing attack, which resulted in several high-profile accounts being compromised.</li><li>Insider threats: Insider threats occur when employees or contractors misuse their access to an organization's systems or data. In 2020, a former employee of Twitter was charged with hacking into the accounts of high-profile individuals.</li><li>Internet of Things (IoT) security: IoT devices, such as smart home devices and industrial control systems, are vulnerable to cyber attacks. In 2020, the healthcare industry was hit by a series of attacks on connected medical devices.</li><li>Malware attacks: Malware refers to any software designed to harm or exploit computer systems. Some of the most famous malware attacks include the WannaCry ransomware attack in 2017 and the NotPetya malware attack in 2016, both of which caused significant damage to systems and networks worldwide.</li><li>Distributed Denial of Service (DDoS) attacks: DDoS attacks involve overwhelming a system or network with traffic from multiple sources, causing it to become unavailable. These attacks can be used to disrupt services or to extort money from organizations by threatening to launch a DDoS attack.</li></ol> These are just a few examples of the many security issues and breaches that have occurred in cyberspace in recent years. As technology continues to evolve, it is likely that new types of security threats will emerge.

Security Breaches in Cyberspace | MCS-215 Online Notes

Security Breaches in Cyberspace

Below are the technological solution for Cyber Security Breaches1. Unauthorised access - Strong passwords, endpoint security, two factor authentications, physical security practices, monitoring user activity are few of the common practices employed for protection against unauthorized access. 2. Distributed Denial of Service Attack – Various infocom security tools are used to protect against DOS Attack like anti malware, firewall, spam filtering, switches and routers, Intrusion prevention systems, DDOS defence systems, content delivery network etc.(GOsafeonline,2014) .3.Malwares -<ul><li>Botnets - VPN, secured network architecture, traffic management tools etc. can help in detecting and preventing botnet attack.</li><li>Ransomwares - Backing up data, disabling unnecessary ports or services such as RDP can help in preventing ransomware attack</li><li>Trojan - robust firewall, anti-spywares, maintaining cyber hygiene are helpful against trojans.</li><li>Virus - Malware removal software, automatic scans can reduce the risk of virus attacks.</li><li>Worms - Internet connection firewall, network intrusion detection software, use of Domain Message Authentication Reporting (DMARC), Domain Name System Security Extension (DNSSEC) can be helpful.</li></ul>4. Spywares - Anti spywares, sandbox protection, ad pop up blockers can play significant role in averting spywares.5. Social Engineering attacks – network traffic analysis, firewalls, updated software, spam protection guards, updated blacklists(Chizari, Hassan &amp;Zulkurnain, et al (2015) can aid in preventing social engineering attacks. However, it is considered that the most efficient way to tackle social engineering attacks are awareness about such attacks. Social engineering attacks thrive on people’s ignorance and hence awareness is the key to fight against such attacks.&nbsp;&nbsp;6. Phishing - SPAM filters, web filters, patching, use of SSL certificate to secure traffic are a few technologies which may assist in combating Phishing attacks.7. Cryptojacking- anti crypto mining extensions, endpoint protection, web filtering tools, network monitoring solutions etc are helpful in preventing crypto jacking.(Micheal Nadeau,2021).8. Exploiting vulnerability – Patching operating systems, enabling SMB signing, network segmentation which limits access to systems etc. may reduce the chances of exploitation of vulnerability.&nbsp;9. Cyber physical attacks- Adequate control on physical access, cyber nodes, enabling remote access only when necessary, two factor authentications etc. are few of the practices which have turned out beneficial in combating cyber physical attacks.10. IOT attacks - Firmware updates with cryptographic signatures, proper identity management, hardened toolchains, libraries and framework, etc may be used for protection against IOT attacks.11. Web Jacking - Web Server firewalls, X frames options etc may be used to prevent web jacking.&nbsp;12. Drive by download - Updating website components, web security software etc. can provide protection against drive-by download.&nbsp;

Technological Solution For Cyber Security | MCS-215 Free Notes Online

Technological Solution For Cyber Security

In India, there are several cybersecurity laws and regulations aimed at protecting the country's critical infrastructure, data, and systems. Here are some of the most notable ones:<ol><li>Information Technology Act, 2000: This is the primary law governing cybersecurity in India. It provides legal recognition for electronic documents and digital signatures, and it also outlines offenses related to hacking, cyberterrorism, and unauthorized access to computer systems.</li><li>The Indian Cyber Crime Coordination Centre (I4C): I4C is a government agency tasked with combating cybercrime and promoting cybersecurity in India. It serves as a central coordination point for cybercrime investigations and provides technical assistance to law enforcement agencies.</li><li>National Cyber Security Policy: This policy was introduced in 2013 and outlines India's strategy for securing its cyberspace. It focuses on developing cybersecurity capabilities, enhancing awareness and education, and collaborating with other countries and international organizations to combat cyber threats.</li><li>Personal Data Protection Bill, 2019: This is a proposed law that would regulate the collection, storage, and processing of personal data in India. The bill imposes obligations on businesses to obtain consent from individuals before collecting their data and to protect that data from unauthorized access or disclosure.</li><li>Reserve Bank of India (RBI) Guidelines on Cyber Security: RBI has issued guidelines to banks and financial institutions for ensuring cybersecurity and protecting customer data. The guidelines outline specific security measures that banks must implement, including access controls, encryption, and incident response planning.</li></ol>These are some of the key cybersecurity laws and regulations in India. However, it is important to note that the cybersecurity landscape is constantly evolving, and new laws and regulations may be introduced in the future to address emerging threats and challenges.

Cyber Security Laws in India | Free MCS-215 Notes online 

Cyber Security Laws in India

The message or data to be encrypted, also known as the plaintext, is transformed by a function that is parameterized by a KEY. The output of the encryption process, known as the cipher text, is then transmitted through the insecure communication channel. The art of breaking ciphers is called cryptanalysis. The art of devising ciphers (cryptography) and breaking them (cryptanalysis) is collectively known as cryptology. Mathematically, C = Ek (P) meaning that the encryption of the plaintext P using key K gives the cipher text C. Similarly, P = Dk(C) implies the decryption of C to get the plaintext again. It then follows that Dk (Ek (P)) = P.CyphersCiphers are cryptographic algorithms used to convert plaintext into ciphertext. There are various types of ciphers, including:<ol><li>Transposition cipher: In this type of cipher, the letters in the plaintext are rearranged, but the letters themselves are not changed. One popular example of transposition cipher is the Rail Fence cipher, where the plaintext is written diagonally across a grid, and then the letters are read row by row to get the ciphertext.</li><li>Block cipher: This type of cipher encrypts fixed-size blocks of plaintext at once, and each block is encrypted independently of the others. Examples of block ciphers include AES, DES, and Blowfish.</li><li>Stream cipher: In this type of cipher, the plaintext is combined with a random stream of bits to create the ciphertext. Examples of stream ciphers include RC4 and ChaCha.</li><li>Polyalphabetic cipher: This type of cipher uses multiple substitution alphabets to encrypt the plaintext, making it harder to crack. One popular example of polyalphabetic cipher is the Vigenère cipher.</li><li>Polyalphabetic substitution cipher: In this cipher, a blended alphabet is utilized to encrypt the plaintext, but at random points it would convert to a special uncommon mixed alphabet which denotes the alteration with an uppercase letter in the Ciphertext. </li><li> Permutation Cipher: The locations assumed by plaintext are moved to a uniform system in this cipher so that the ciphertext amounts to a permutation of the plaintext. </li><li> Private-key Cryptography: In this cipher, even the attacker is mindful of the plaintext and corresponding ciphertext. The sender and receiver must have a pre-shared key. The shared key is kept undisclosed from all other parties and is used for encryption as well as decryption. DES and AES algorithms are examples of this type of cipher. This cryptography is also known as "symmetric key algorithm". </li><li> Public-key Cryptography: In this cipher, two different keys - public key and private key - are used for encryption and decryption. The sender uses the public key to carry out encryption, whereas the receiver is kept in the dark about the private key. This is also known as asymmetric key algorithm.</li></ol>These are some of the most commonly used types of ciphers, and each type has its own strengths and weaknesses. It is important to choose the right cipher for the specific application to ensure the security of the data being encrypted.

Cryptography | Free MCS-215 Notes online 

Cryptography

Cryptography has several important functions in information security, including:<ol><li>Confidentiality: Cryptography can be used to ensure that information remains confidential and can only be accessed by authorized individuals. This is done by encrypting the data in such a way that it is unreadable without the correct key or password.</li><li>Integrity: Cryptography can also be used to ensure that information has not been tampered with or modified in transit. This is done by using cryptographic hash functions that generate a unique fingerprint of the data that can be used to verify its authenticity.</li><li>Authentication: Cryptography can be used to authenticate the identity of individuals or systems. This is done by using digital signatures or other authentication mechanisms that use cryptographic techniques to verify the authenticity of the sender.</li><li>Non-repudiation: Cryptography can also be used to provide non-repudiation, which means that a sender cannot deny sending a message or document. This is done by using digital signatures that are tied to the sender's identity and cannot be repudiated.</li></ol>Overall, cryptography is a crucial component of modern information security, providing a way to secure and protect sensitive information from unauthorized access and manipulation.

Functions Of Cryptography | Free MCS-215 Notes Online

Functions Of Cryptography

Steganography is the practice of hiding secret information within a non-secret message or file, such as an image, audio, or video file. Unlike cryptography, which encrypts the message to keep it secret, steganography hides the existence of the message in the first place.It can be divided into following five types: <ul><li>Text Steganography</li><li> Image Steganography </li><li>Video Steganography </li><li>Audio Steganography </li><li>Network Steganography</li></ul>Steganography techniques involve modifying the original message or file in a way that is imperceptible to human senses but can be detected using specialized software or techniques. For example, one common steganography technique is to embed secret data within the least significant bits of a digital image or audio file. These bits are typically ignored by humans but can be extracted using specialized software.Steganography can be used for both legitimate and malicious purposes. For instance, it can be used by journalists to hide sensitive information from oppressive regimes, or by law enforcement agencies to hide covert communication during investigations. However, it can also be used by cybercriminals to hide malware or other malicious content within seemingly harmless files, making it harder to detect and block.<blockquote>In summary, steganography is a technique used to hide secret information within a non-secret message or file. It can be used for both legitimate and malicious purposes, and it requires specialized software or techniques to detect the hidden message.</blockquote>

What is Steganography | Free MCS-215 Notes Online

Steganography

Steganography techniques involve hiding secret data within a cover media, such as images, audio, video, or text files, in a way that is imperceptible to the human senses. Here are some common steganography techniques:<ol><li>LSB (Least Significant Bit) Technique: This is a popular steganography technique used for images and audio files. It involves replacing the least significant bit of each pixel or sample with a bit of the secret data. This change is usually imperceptible to the human eye or ear, but can be detected by specialized software.</li><li>Spread Spectrum Technique: This technique involves spreading the secret data across a wide range of frequencies in an audio or video signal. The secret data is then extracted at the receiving end by filtering out the carrier frequencies.</li><li>Text Steganography: This technique involves hiding secret data within a text file by replacing or modifying certain characters, words, or spaces. For example, a secret message could be hidden within a longer document by using a code to indicate which characters or words to read.</li><li>Image Steganography: In this technique, secret data is hidden within an image file by modifying the color or pixel values of certain areas in the image. For example, a small area in the image could be slightly darker or lighter to represent a 1 or 0 in the secret message.</li><li>Audio Steganography: This technique involves hiding secret data within an audio file by manipulating the audio waveform or frequency spectrum. For example, a low volume tone or noise could be added to represent a 1 or 0 in the secret message.</li></ol>These are just a few examples of steganography techniques. There are many other techniques and variations that can be used to hide secret data within different types of media.

Steganography Techniques | Free MCS-215 Notes Online

Steganography Techniques

Encryption and decryption are two important processes in cryptography that are used to secure and protect information.<blockquote>Encryption is the process of converting plain text or data into a coded message that cannot be easily read or understood without a secret key or password. This is done using an encryption algorithm that transforms the original data into a ciphertext that is unreadable to anyone who does not have the key. Encryption is used to protect sensitive information during transmission or storage, such as credit card numbers, passwords, and other personal data.</blockquote><blockquote>Decryption is the process of converting the encrypted message or ciphertext back into its original form using a secret key or password. This is done using a decryption algorithm that reverses the encryption process and converts the ciphertext back into the original plaintext. Decryption is necessary to retrieve and read the original message or data that was encrypted.</blockquote>Encryption and decryption are critical to many applications of modern cryptography, including secure communication, data protection, and privacy. For example, when you visit a secure website (HTTPS), your browser encrypts the data being sent to the website using a cryptographic protocol, and the website decrypts it using a secret key to ensure that your data is protected from eavesdropping or tampering.Overall, encryption and decryption are essential techniques used in cryptography to ensure the confidentiality, integrity, and authenticity of information.

Encription and Decription | Free MCS-215 Notes Online

Encription and Decription

Public key encryption, also known as asymmetric encryption, is a type of encryption scheme that uses two different keys - a public key and a private key - to encrypt and decrypt information.In a public key encryption scheme, the public key is freely distributed and used by anyone to encrypt messages intended for the owner of the corresponding private key. The private key, on the other hand, is kept secret by the owner and used to decrypt the encrypted messages. This allows for secure communication over insecure channels, such as the internet.The process of distributing public and private keys in a public key encryption scheme is known as key distribution. There are several ways to distribute public keys, including:<ol><li>Public Key Infrastructure (PKI): A PKI is a system of digital certificates, certificate authorities, and other registration authorities that verify and authenticate the identities of parties involved in a communication. PKI is commonly used in web browsing, email, and other internet applications.</li><li>Key exchange protocols: These are algorithms and protocols that allow two parties to securely exchange keys over an insecure channel. One example of a key exchange protocol is the Diffie-Hellman key exchange.</li><li>Self-signed certificates: In this method, an individual or organization creates and signs their own digital certificate using a public key, rather than relying on a trusted third-party authority. This is commonly used for testing or internal communication within an organization.</li></ol>Overall, public key encryption provides a powerful and flexible method for secure communication and data protection. However, key distribution is an important issue that must be carefully managed to ensure the security and integrity of the communication.

Public key encryption | Free MCS-215 Notes Online

Public key encryption

Advantages of symmetric-key cryptography&nbsp;<ol><li> Symmetric-key ciphers can be designed to have high rates of data throughout. Some hardware implementations achieve of hundreds of megabytes per second, while software implementations may attain throughput rates in the megabytes per second range. Keys for symmetric-key ciphers are relatively short.</li><li> Symmetric-key ciphers can be employed as primitives to construct various cryptographic mechanisms including pseudorandom number generators, hash functions and computationally efficient digital signature schemes, to name just a few. </li><li> Symmetric-key ciphers can be composed to produce stronger ciphers. Simple transformations which are easy to analyse, but are weak on their own weak, can be used to construct strong product ciphers. </li><li> Symmetric-key encryption is perceived to have an extensive history, although it must be acknowledged that, notwithstanding the invention of rotor machines earlier, much of the knowledge in this area has been acquired subsequent to the invention of the digital computer, and in particular, the design of the Data Encryption Standard in the early 1970s.</li></ol>Disadvantages of symmetric-key cryptography<ol><li>In a two-party communication, the key must remain secret at both ends. </li><li>In a large network, there are many key pairs to be managed. Consequently, effective key management requires the use of an unconditionally trusted TTP. </li><li>In a two-party communication between entities µ and ¶, sound cryptographic practice dictates that the key be changed frequently and perhaps for each communication session. </li><li>Digital signature mechanisms arising from symmetric-key encryption typically require either large keys for the public verification function or the use of a TTP.</li></ol>Advantages of public-key cryptography<ol><li>Only the private key must be kept secret (authenticity of public keys must, however, be guaranteed). </li><li>The administration of keys on a network requires the presence of only a functionally trusted TTP as opposed to an unconditionally trusted TTP. Depending on the mode of usage, the TTP might only be required in an “off-line” manner, as opposed to in real time.</li><li>Depending on the mode of usage, a private key/public key pair may remain unchanged for considerable periods of time, e.g., many sessions (even several years). </li><li>Many public-key schemes yield relatively efficient digital signature mechanisms. The key used to describe the public verification function is typically much smaller than for the symmetric-key counterpart. </li><li>In a large network, the number of keys necessary may be considerably smaller than in the symmetric-key scenario.</li></ol>Disadvantages of public-key encryption<ol><li>Throughput rates for the most popular public-key encryption methods are several orders of magnitude slower than the best-known symmetric-key schemes. </li><li>Key sizes are typically much larger than those required for symmetric-key encryption and the size of public-key signatures is larger than that of tags providing data origin authentication from symmetric-key techniques. </li><li>No public-key scheme has been proven to be secure (the same can be said for block ciphers). The most effective public-key encryption schemes found to date have their security based on the presumed difficulty of a small set of number-theoretic problems. </li><li>Public-key cryptography does not have as extensive a history as symmetric-key encryption, being discovered only in the mid 1970s.&nbsp;</li></ol>

Encription Schemes Comparison  | Free MCS-215 Notes Online

Encription Schemes Comparison

The Data Encryption Standard (DES) is a symmetric-key block cipher algorithm used for encryption of electronic data. It was developed by IBM in the 1970s and was adopted by the National Institute of Standards and Technology (NIST) as a federal standard for encryption in 1977.DES operates on 64-bit blocks of plaintext and uses a 56-bit key for encryption. Here's how it works:<ol><li>Key Generation: A 56-bit key is generated using a method that involves permuting and transforming an input key.</li><li>Initial Permutation: The 64-bit block of plaintext is first subjected to an initial permutation.</li><li>Round Function: DES employs 16 rounds of the encryption process. Each round uses a round function that takes a 32-bit half-block of the data as input, along with a 48-bit subkey derived from the original key, and produces a 32-bit half-block as output.</li><li>Subkey Generation: Each subkey is generated from the original key using a method that involves shifting and permuting the key bits.</li><li>Expansion: The 32-bit half-block is expanded to 48 bits using a permutation.</li><li>XOR: The expanded half-block is XORed with the subkey.</li><li>S-Box: The result is divided into eight 6-bit pieces, each of which is substituted using a different S-Box, which maps 6 bits to 4 bits.</li><li>Permutation: The resulting 32 bits are subjected to a permutation.</li><li>Swap: After the 16 rounds are completed, the two 32-bit half-blocks are swapped.</li><li>Final Permutation: The resulting 64-bit block is subjected to a final permutation, which is the inverse of the initial permutation.</li><li>The resulting ciphertext is 64 bits long and can only be decrypted using the same 56-bit key that was used for encryption. Despite its widespread use, DES has been shown to be vulnerable to certain types of attacks, and it has been largely replaced by the more secure Advanced Encryption Standard (AES) in most applications.</li></ol>

Data Encryption Standard(DES)  | Free MCS-215 Notes Online

Data Encryption Standard(DES)

The RSA algorithm is a widely used public-key cryptosystem for secure data transmission. It was invented by Ron Rivest, Adi Shamir, and Leonard Adleman at MIT in 1977 and is named after their surnames.The RSA algorithm uses a pair of keys, one for encryption and the other for decryption. The encryption key is public and can be shared with anyone, while the decryption key is kept secret and known only to the intended recipient.Here's how RSA works:<ol><li>Key Generation: The first step in RSA is to generate a pair of keys. To do this, we select two large prime numbers, p and q. We then multiply these primes together to obtain n = pq, which is used as the modulus for the public and private keys. We also calculate the totient of n, which is (p-1)(q-1).</li><li>Public Key Generation: We then select a public exponent e, which is a positive integer that is relatively prime to the totient of n. The public key consists of the modulus n and the public exponent e.</li><li>Private Key Generation: We then calculate the private exponent d, which is the inverse of e modulo the totient of n. The private key consists of the modulus n and the private exponent d.</li><li>Encryption: To encrypt a message M, we first represent it as a number between 0 and n-1. We then compute the ciphertext C as C = M^e (mod n).</li><li>Decryption: To decrypt the ciphertext C, the recipient uses their private key to compute M = C^d (mod n).</li></ol>RSA is based on the fact that it is very difficult to factor the product of two large prime numbers. The security of the RSA algorithm depends on the length of the keys used. Keys of 1024 bits or more are considered secure, while keys of 2048 bits or more are recommended for sensitive applications.While RSA is widely used for secure data transmission, it is important to note that it is computationally intensive and can be slow for large messages. Therefore, hybrid encryption schemes that combine RSA with symmetric-key encryption algorithms are often used in practice.

RSA Algorithm  | Free MCS-215 Notes Online

RSA Algorithm

An electronic signature is a digital signature used to authenticate the identity of a person signing a document or electronic communication. It serves the same purpose as a traditional handwritten signature, but it is created using electronic means and is often considered more secure and efficient.Electronic signatures can take many different forms, but they generally fall into three categories:<ol><li>Simple electronic signatures: These are basic electronic signatures that are created using a stylus or a finger on a touch screen, or by typing in a name or initials. They can be as simple as a scanned image of a handwritten signature.</li><li>Advanced electronic signatures: These are more sophisticated electronic signatures that require the use of a secure digital certificate issued by a trusted authority. They provide additional security features, such as encryption and tamper detection, and can be used for legal purposes.</li><li>Qualified electronic signatures: These are the most advanced electronic signatures and are used in highly regulated industries, such as finance and healthcare. They are created using a secure digital certificate that has been issued by a qualified trust service provider, and they provide the highest level of security and legal validity.</li></ol>Electronic signatures are often used to sign contracts, agreements, and other legal documents, and they can be used in a variety of industries, including healthcare, finance, and real estate. They offer several benefits over traditional signatures, including increased security, efficiency, and convenience. However, it is important to ensure that electronic signatures are legally valid in your jurisdiction and that they meet the requirements of any relevant laws or regulations.

Electronic Signature  | Free MCS-215 Notes Online

Electronic Signature

A digital signature is a type of electronic signature that uses a cryptographic technique to provide authenticity and integrity of a digital document or message. It is created using a combination of a private key and a public key and is used to verify the identity of the signer and ensure that the document has not been tampered with.Here's how digital signatures work:<ol><li>Key Generation: The signer generates a pair of keys, one private and one public, using a cryptographic algorithm.</li><li>Signature Creation: To create a digital signature, the signer uses their private key to apply a mathematical function to the document or message being signed. This produces a unique value, known as a hash, that is specific to the document and the private key.</li><li>Signature Verification: To verify the digital signature, the recipient uses the signer's public key to apply the same mathematical function to the document or message. This produces a hash value that should match the one generated by the signer. If the two hash values match, the signature is verified, and the recipient can be sure that the document or message has not been tampered with and that it was indeed signed by the claimed signer.</li></ol>Digital signatures provide several advantages over traditional signatures. They are more secure, as it is virtually impossible to forge or alter a digital signature without access to the signer's private key. They also provide a high level of confidence that the document or message has not been tampered with, as any change in the document or message would result in a different hash value and the signature would fail verification.Digital signatures are widely used in industries such as finance, healthcare, and government, where the integrity and authenticity of documents and messages are critical. They are also recognized as legally binding in many jurisdictions and are governed by laws and regulations that govern their use.

Digital Signature  | Free MCS-215 Notes Online

Digital Signature

Digital signature and electronic signature are two types of signatures used in digital transactions, but they are not the same thing.<blockquote>An electronic signature is a broad term that refers to any electronic symbol or process used to indicate that a person has agreed to the contents of a document or message. It can include a scanned image of a handwritten signature, a typed name, a checkbox, a PIN, or even a biometric identifier like a fingerprint. Electronic signatures are often used to speed up transactions and to reduce the amount of paper used in business processes.</blockquote><blockquote>On the other hand, a digital signature is a specific type of electronic signature that uses cryptographic techniques to provide additional security and integrity to a digital document or message. It is created using a combination of a private key and a public key, and it is used to verify the identity of the signer and ensure that the document or message has not been tampered with. Digital signatures are often used in more sensitive transactions where legal requirements for authenticity and security are high.</blockquote>While both electronic signatures and digital signatures are used to signify agreement to a digital document or message, digital signatures offer a higher level of security and can be used in situations where legal validity is required. They are typically more complex to set up and require additional infrastructure and key management, while electronic signatures are often simpler and more convenient to use.

Electronic Vs Digital Signature  | Free MCS-215 Notes Online

Electronic Vs Digital Signature

A hash function is a mathematical function that takes an input (often a message or data file) and produces a fixed-size output called a hash value or message digest. The output of a hash function is a unique representation of the input, and even a small change in the input will result in a completely different hash value.Hash functions are widely used in cryptography and computer security to provide data integrity and authenticity. Here are some of the key characteristics of a good hash function:<ol><li>Deterministic: A given input should always produce the same output.</li><li>One-way: It should be computationally infeasible to determine the input from the hash value.</li><li>Collision-resistant: It should be extremely difficult to find two inputs that produce the same hash value.</li><li>Fixed-size output: The output should be of a fixed size, regardless of the size of the input.</li></ol>Hash functions are used in various cryptographic applications, including digital signatures, message authentication codes, and password storage. When a user creates a password, it is passed through a hash function and the resulting hash value is stored. When the user enters the password, the hash value of the input is compared to the stored hash value, and if they match, the user is granted access.In summary, a hash function is a crucial component of modern cryptography and computer security. It is used to provide data integrity, authenticity, and confidentiality, and is an essential tool for protecting sensitive information in digital systems.Benefits:Hash functions provide several benefits in various areas of computer science, cryptography, and security. Here are some of the key benefits of hash functions:<ol><li>Data Integrity: Hash functions are widely used to ensure data integrity in digital systems. By generating a unique hash value for each data file or message, hash functions can detect any unauthorized changes to the data.</li><li>Data Authenticity: Hash functions can also be used to verify the authenticity of digital messages or files. By generating a hash value using a secret key, the recipient of the message can verify that the sender is authentic and that the message has not been tampered with.</li><li>Password Security: Hash functions are commonly used to store passwords securely in digital systems. By hashing the password and storing only the hash value, the system can verify passwords without storing the actual password, which reduces the risk of password theft or unauthorized access.</li><li>Data Privacy: Hash functions are used in many privacy-preserving applications, such as pseudonymization, where sensitive information is replaced with a unique identifier (a hash value), while preserving the overall structure and meaning of the data.</li><li>Efficient Data Retrieval: Hash functions are often used to index data in databases or data structures, enabling fast and efficient retrieval of data items. Hash tables are a common example of such data structures.</li></ol>In summary, hash functions provide a wide range of benefits in various areas of computer science and security, including data integrity, data authenticity, password security, data privacy, and efficient data retrieval.

Hash Function | Free MCS-215 Notes Online

Hash Function  | Free MCS-215 Notes Online

Hash Function

A hash function is a function that maps data of arbitrary size to a fixed-size value. The most common algorithm used in hash functions are:<ol><li>MD5 (Message Digest 5): A widely used algorithm that generates a 128-bit hash value. It has been widely used in the past, but it is now considered insecure for cryptographic purposes due to vulnerabilities.</li><li>SHA-1 (Secure Hash Algorithm 1): Another widely used algorithm that generates a 160-bit hash value. It is also considered insecure for cryptographic purposes and has been replaced by stronger hash functions such as SHA-256 and SHA-512.</li><li>SHA-2 (Secure Hash Algorithm 2): A family of algorithms that includes SHA-256, SHA-384, and SHA-512. These algorithms are widely used for secure cryptographic applications and generate hash values of 256, 384, and 512 bits, respectively.</li><li>SHA-3 (Secure Hash Algorithm 3): A newer family of hash algorithms that was chosen in a public competition organized by NIST. It includes SHA3-256, SHA3-384, and SHA3-512 and is designed to provide better security than SHA-2.</li><li>BLAKE2: A family of hash functions that includes BLAKE2b and BLAKE2s, designed to provide better performance than SHA-3 on modern processors.</li><li>The RIPEMD is an acronym for RACE Integrity Primitives Evaluation Message Digest. This set of hash functions was devised by open research community and generally known as a family of European hash functions</li><li>Whirlpool is a 512-bit hash function which has stemmed from the revised edition of Advanced Encryption Standard (AES). One of the designer was Vincent Rijmen, a co-creator of the AES.</li></ol>These algorithms are widely used in various applications such as digital signatures, message authentication codes, password storage, and privacy-preserving techniques. It's important to note that the security of a hash function depends on the algorithm's strength, the size of the hash value, and the implementation of the algorithm.

Hash Function Algorithm | Free MCS-215 Notes Online

Hash Function Algorithm  | Free MCS-215 Notes Online

Hash Function Algorithm

<h1>We are going to cover below topics in this course</h1>Block 1:<ul><li>INTRODUCTION TO INTERNET&nbsp;</li><li>DATA TRASNMISSION BASICS &amp; TRANSMISSION MEDIA&nbsp;</li><li>DATA ENCODING AND MULTIPLEXING&nbsp;</li><li>MULTIPLEXING AND SWITCHING&nbsp;</li></ul>Block 2:<ul><li> </li></ul>

MCS-218 Home Page | MCS-218 Free Notes

The internet is a global network of interconnected computers and other electronic devices that communicate with each other through a standardized communication protocol known as the Internet Protocol (IP). It was created in the 1960s by the United States Department of Defense as a means of connecting their computer systems and has since grown to become one of the most important technological innovations in human history.The internet allows individuals, businesses, and governments to communicate, share information, conduct commerce, and access a vast array of resources from anywhere in the world. It has revolutionized the way we live, work, and interact with each other, making it possible to instantly connect with people and information on a global scale.The internet is made up of various components, including web pages, email, social media, online videos, and other digital content. These components are accessed through a web browser or other internet-enabled device, such as a smartphone or tablet.While the internet has brought many benefits, it has also raised concerns about privacy, security, and the spread of misinformation. As technology continues to evolve, it is likely that the internet will continue to play a significant role in shaping the future of society.

Introduction of Internet | MCS-218 Free Notes

The history of the internet dates back to the 1960s, when the U.S. Department of Defense created the Advanced Research Projects Agency Network (ARPANET) to connect research institutions and defense contractors. ARPANET was a packet-switching network that used the Transmission Control Protocol (TCP) and Internet Protocol (IP) to transmit data between different nodes.In the 1970s, the development of email and other networking technologies led to the expansion of ARPANET beyond military and academic communities. The term "internet" was first used in 1974 to describe a network of networks that used TCP/IP protocols.In the 1980s, the National Science Foundation (NSF) created the NSFNET, which became the backbone of the internet. The introduction of domain names and the World Wide Web (WWW) in the early 1990s brought the internet to the masses, and its use exploded in the following years.The 2000s saw the emergence of social media, mobile devices, and the cloud, which transformed the internet into a ubiquitous platform for communication, commerce, entertainment, and education. Today, the internet connects billions of people and devices around the world, enabling global collaboration, innovation, and knowledge sharing.

History Of Internet | MCS-218 Free Notes

History Of Internet

ISP:An Internet Service Provider (ISP) is a company that provides internet connectivity services to individuals, businesses, and other organizations. ISPs offer a range of internet connection options, including dial-up, DSL, cable, fiber optic, and satellite, depending on the location and infrastructure availability.ISPs provide the necessary equipment, such as modems, routers, and other networking devices, to enable their customers to connect to the internet. They also offer a variety of service plans, with different speeds and data allowances, to meet the needs of different types of users.ISPs are responsible for managing their networks and ensuring that their customers have reliable and secure access to the internet. They also provide technical support to their customers to help them troubleshoot any issues with their internet connection.There are three types of ISP<ol><li>Tier-1 ISPs</li><li>Tier-2 ISPs</li><li>Tier-3 ISPs</li></ol>Internet Backbone:Internet backbone is named for core network, can handle traffic at very high speed, interconnecting various networks to allow transmission of data over the worldwide network. Internet backbone mainly consists of tier-1 ISP’s networks altogether. It connects networks of all tier -1 ISP and carries the majority of Internet traffic. Internet backbone mainly contains a large number of routers, switches and other intermediate devices interconnected with very high speed links (optical fiber links). Many companies may build their own backbone infrastructure for interconnectivity among offices at different geographical locations. Similar is the case with Internet backbones, where the computer connects to an access network of a tier -3 ISP, which further connects to the main Internet backbone (tier -2 or tier -1 ISP’s network)to access the worldwide network.&nbsp;

ISP and Internet Backbone | MCS-218 Free Notes

ISP and Internet Backbonet | MCS-218 Free Notes

ISP and Internet Backbone

A taxonomy of networks refers to the classification or categorization of different types of computer networks based on their characteristics, topologies, architectures, and protocols. Here are some common types of networks based on their characteristics:<ol><li>Local Area Network (LAN): A LAN is a network that is confined to a small geographic area, such as a building, office, or campus. It is usually owned and managed by a single organization and is used for sharing resources such as printers, files, and applications.</li><li>Wide Area Network (WAN): A WAN is a network that covers a large geographic area, such as a country or multiple countries. It connects multiple LANs and enables users to share resources and data over long distances.</li><li>Metropolitan Area Network (MAN): A MAN is a network that covers a metropolitan area, such as a city or a town. It is usually owned and managed by a single organization or a group of organizations and is used for sharing resources and data within the metropolitan area.</li><li>Wireless Local Area Network (WLAN): A WLAN is a type of LAN that uses wireless technology to connect devices within a small geographic area, such as a home or office.</li><li>Storage Area Network (SAN): A SAN is a specialized network that provides high-speed access to storage devices, such as hard drives and tape libraries. It is used for data backup, disaster recovery, and other storage-related tasks.</li><li>Virtual Private Network (VPN): A VPN is a network that provides secure and encrypted communication between remote users and a private network, such as a corporate network or a university network.</li><li>Personal Area Network (PAN): PAN is for a person. It is a network established for a person for a small area of the range of few metere. It can be wired (using USB data cable) or wireless (using infrared, ZigBee, Bluetooth). To establish PAN no third party device is required, the devices connect to each other autonomously detects and connect with each other. Examples of wireless PAN, or WPAN are: controlling TV with remote using infrared, connecting mobile headsets with mobile using Bluetooth, wireless keyboards and mice, wireless printers , wireless CCTV cameras etcusing Wi-Fi, connecting printer, scanner, bar code scanners and game consoles using USB connection.&nbsp;</li></ol>

Taxonomy of Network | MCS-218 Free Notes

Taxonomy of Network

A Wide Area Network (WAN) is a computer network that covers a large geographic area, typically spanning multiple cities, states, or even countries. WANs can be used to connect multiple local area networks (LANs) together to allow users to share resources, communicate with each other, and access the Internet.Here are some common types of WANs:<ol><li>Circuit-Switched WAN: Circuit-switched WANs use dedicated physical circuits, such as leased lines or ISDN lines, to connect remote locations. This type of WAN is typically used for high-speed, reliable connections, but can be expensive.</li><li>Packet-Switched WAN: Packet-switched WANs use virtual circuits to transmit data over shared public networks, such as the Internet or a private IP-based network. This type of WAN is more cost-effective than circuit-switched WANs, but may not provide the same level of reliability and security.</li><li>MPLS WAN: MPLS (Multiprotocol Label Switching) WANs use a combination of circuit-switched and packet-switched technologies to provide high-speed, reliable connections between remote locations. MPLS WANs are commonly used by businesses and organizations that require secure, high-speed data transmission between multiple locations.</li><li>Satellite WAN: Satellite WANs use satellite communications technology to connect remote locations that are not easily accessible by traditional WAN technologies, such as wired or cellular networks. This type of WAN can provide reliable connectivity in areas with limited or no terrestrial infrastructure, but may be affected by weather conditions and can be relatively expensive.</li><li>Cellular WAN: Cellular WANs use cellular data networks, such as 3G, 4G, or 5G, to provide connectivity between remote locations. This type of WAN is often used for mobile applications, such as remote workers or vehicles, and can provide reliable connectivity in areas with good cellular coverage. However, cellular WANs can be affected by network congestion and signal strength.</li></ol>There are three categories of Wide Are Network (WAN)<ul><li>Enterprise network: A network owned and managed by a single organization is named as enterprise network. It is the interconnected version of all the LANs of an organization/institution.&nbsp;</li><li>Global network: Networks of several organizations over a wide area collectively known as a global network. </li><li>Internet: A network of networks of broad area category. It is the largest network worldwide. It includes many LANs, WANs, MANs, and millions of computers. No single authority controls it; rather the local, national authority controls every segment of the Internet.&nbsp;</li></ul>

Types of WAN | MCS-218 Free Notes

Types of WAN

Some rules are necessary to make any communication successful, this set of rules is known as the protol. A protocol allows computers to talk with each other. A protocol defines the data type may be exchanged, set of commands used to exchange (send/ receive) data, and the way data to be transfered are how it is confirmed that data is received.&nbsp;In the Internet various protocols are designed for networking applications. Like Ethernet protocol is used in wired network, IEEE 802.11 protocol is defined for wireless communication. An Internet protocol suite is a set of standard protocols used for transmitting data over the Internet.&nbsp;Standard Internet protocols may be classified into four major catagories: <ol><li>Application layer – Hyper Text Transfer Protocol (HTTP), Simple Mail Transfer Protocol (SMTP). </li><li>Transport layer – Transmission Control Protocol (TCP), User Datagram Protocol (UDP) </li><li>Network layer – Internet Protocol (IP) version 4 and version 6 </li><li>Data link layer – Ethernet, IEEE 802.11 (Wi-Fi)&nbsp;</li></ol>

Standard Internet Protocols | MCS-218 Free Notes

Standard Internet Protocols

Sorting algorithms are essential algorithms in computer science and are used to arrange elements in a specific order, typically ascending or descending. There are various sorting algorithms, each with its advantages and disadvantages in terms of efficiency and ease of implementation. Here are some of the most common sorting algorithms:<ol><li>Bubble Sort:</li></ol><ul><li class="ql-indent-1">Simple but inefficient.</li><li class="ql-indent-1">Repeatedly steps through the list, compares adjacent elements, and swaps them if they are in the wrong order.</li><li class="ql-indent-1">Time complexity: O(n^2) in the worst and average cases.</li></ul><ol><li>Selection Sort:</li></ol><ul><li class="ql-indent-1">Divides the input list into two parts: the sorted and the unsorted portion.</li><li class="ql-indent-1">Repeatedly selects the smallest (or largest) element from the unsorted portion and moves it to the sorted portion.</li><li class="ql-indent-1">Time complexity: O(n^2) in the worst and average cases.</li></ul><ol><li>Insertion Sort:</li></ol><ul><li class="ql-indent-1">Builds a sorted array one element at a time by repeatedly taking an element from the unsorted part and inserting it into its correct position in the sorted part.</li><li class="ql-indent-1">Efficient for small datasets.</li><li class="ql-indent-1">Time complexity: O(n^2) in the worst and average cases.</li></ul><ol><li>Merge Sort:</li></ol><ul><li class="ql-indent-1">Divides the list into smaller sublists, sorts those sublists, and then merges them back together.</li><li class="ql-indent-1">Known for its stability and consistent O(n log n) time complexity in the worst, average, and best cases.</li></ul><ol><li>Quick Sort:</li></ol><ul><li class="ql-indent-1">Chooses a 'pivot' element and partitions the array into two subarrays - elements less than the pivot and elements greater than the pivot.</li><li class="ql-indent-1">Recursively sorts the subarrays.</li><li class="ql-indent-1">Average time complexity is O(n log n), but worst-case is O(n^2) if not implemented carefully.</li></ul><ol><li>Heap Sort:</li></ol><ul><li class="ql-indent-1">Builds a max (or min) heap from the elements and repeatedly extracts the maximum (or minimum) element from it.</li><li class="ql-indent-1">Time complexity: O(n log n) in the worst, average, and best cases.</li></ul><ol><li>Radix Sort:</li></ol><ul><li class="ql-indent-1">A non-comparative integer sorting algorithm that works by distributing elements into buckets based on their digits.</li><li class="ql-indent-1">Time complexity: O(nk) where n is the number of elements and k is the number of digits in the largest number.</li></ul><ol><li>Bucket Sort:</li></ol><ul><li class="ql-indent-1">Divides the input into a finite number of buckets, then sorts each bucket using another sorting algorithm or recursively using bucket sort.</li><li class="ql-indent-1">Works well for uniformly distributed data.</li><li class="ql-indent-1">Time complexity depends on the chosen sorting algorithm for buckets.</li></ul><ol><li>Counting Sort:</li></ol><ul><li class="ql-indent-1">A non-comparative integer sorting algorithm that counts the number of occurrences of each element and uses this information to place the elements in the correct order.</li><li class="ql-indent-1">Time complexity: O(n + k) where n is the number of elements and k is the range of input values.</li></ul>The choice of sorting algorithm depends on the specific requirements of your application, including the size of the dataset, memory constraints, and desired time complexity. Different sorting algorithms have different trade-offs, so it's important to choose the one that best suits your needs.

Sorting, Bubble Sort, Quick Sort, Selection Sort, Merge Sort

MCS 211 | Sorting Algorithm and their functions

Sorting

Bubble Sort is a simple sorting algorithm that repeatedly steps through a list of elements, compares adjacent elements, and swaps them if they are in the wrong order. The pass through the list is repeated until no swaps are needed, which means the list is sorted. While it's not efficient for large datasets, it's easy to understand and implement. Here's an explanation of Bubble Sort with Java Code:public class BubbleSort {&nbsp;&nbsp;public static void main(String[] args) {&nbsp;&nbsp;&nbsp;&nbsp;int[] arr = {64, 34, 25, 12, 22, 11, 90};&nbsp;&nbsp;&nbsp;&nbsp;bubbleSort(arr);&nbsp;&nbsp;&nbsp;&nbsp;System.out.println("Sorted array: ");&nbsp;&nbsp;&nbsp;&nbsp;for (int num : arr) {&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;System.out.print(num + " ");&nbsp;&nbsp;&nbsp;&nbsp;}&nbsp;&nbsp;} &nbsp;&nbsp;public static void bubbleSort(int[] arr) {&nbsp;&nbsp;&nbsp;&nbsp;int n = arr.length;&nbsp;&nbsp;&nbsp;&nbsp;boolean swapped; &nbsp;&nbsp;&nbsp;&nbsp;for (int i = 0; i &lt; n - 1; i++) {&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;swapped = false; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;for (int j = 0; j &lt; n - i - 1; j++) {&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if (arr[j] &gt; arr[j + 1]) {&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;// Swap arr[j] and arr[j+1]&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;int temp = arr[j];&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;arr[j] = arr[j + 1];&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;arr[j + 1] = temp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;swapped = true;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;} &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;// If no two elements were swapped in inner loop, the array is already sorted&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;if (!swapped) {&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;break;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}&nbsp;&nbsp;&nbsp;&nbsp;}&nbsp;&nbsp;}In this code:<ul><li>arr is the input array to be sorted.</li><li>We use two nested loops. The outer loop iterates through the array from 0 to n - 1, where n is the number of elements in the array.</li><li>The inner loop compares adjacent elements arr[j] and arr[j + 1]. If arr[j] is greater than arr[j + 1], they are swapped.</li><li>The swapped variable is used to optimize the algorithm. If no swaps are made in a pass through the inner loop, it means the array is already sorted, and we can break out of the outer loop early.</li><li>Bubble Sort has a time complexity of O(n^2) in the worst case, making it inefficient for large arrays.</li></ul>

MCS 211 | Bubble Sort Explanation with Java

Bubble Sort

<h1>We are going to cover below topics in this course</h1>Block 1:<ul><li>INTRODUCTION TO ARTIFICIAL INTELLIGENCE&nbsp;</li><li>PROBLEM SOLVING USING SEARCH</li><li>UNINFORMED &amp; INFORMED SEARCH</li><li>PREDICATE ANDPROPOSITIONAL LOGIC&nbsp;</li></ul>Block 2:<ul><li>FIRST ORDER PREDICATE LOGIC&nbsp;&nbsp;</li><li>RULE BASED SYSTEMS AND OTHER FORMALISM</li><li>PROBABILISTIC REASONING&nbsp;</li><li>FUZZY AND ROUGH SETS</li></ul>Block 3:<ul><li>INTRODUCTION TO MACHINE LEARNING METHODS&nbsp;</li><li>CLASSIFICATION&nbsp;</li><li>REGRESSION</li><li>NEURAL NETWORKS AND DEEP LEARNING&nbsp;</li></ul>Block 4:<ul><li>FEATURE SELECTION AND EXTRACTION&nbsp;</li><li>ASSOCIATION RULES</li><li>CLUSTERING</li><li>MACHINE LEARNING PROGRAMMING USING PYTHON</li></ul>

MCS 224 | IGNOU Study Adda

Artificial Intelligence (AI) is the simulation of human intelligence in machines that are programmed to think and learn like humans. It involves the development of computer algorithms that can perform tasks that usually require human intelligence, such as visual perception, speech recognition, decision-making, natural language processing, and problem-solving.AI is achieved through a combination of machine learning, deep learning, natural language processing, robotics, and other technologies. These techniques enable machines to learn from data, improve their accuracy and performance over time, and make predictions or decisions based on the information they have been trained on.AI is being used in various industries, including healthcare, finance, transportation, manufacturing, and retail, among others, to automate processes, increase efficiency, and improve outcomes. However, the rapid advancement of AI also poses ethical and societal challenges, such as job displacement, bias, privacy, and accountability, which require careful consideration and regulation.

AI Introduction | MCS 224 | IGNOU Study Adda

AI Introduction

Artificial Intelligence (AI) is a broad field that encompasses a range of techniques and approaches to create intelligent machines that can perform tasks that typically require human-like intelligence. Here are some basics of AI:<ol><li>Machine learning: It is a technique used in AI where machines learn from data and improve their performance over time without being explicitly programmed. It involves training algorithms on data sets to identify patterns and make predictions or decisions based on that data.</li><li>Deep learning: It is a subset of machine learning that uses artificial neural networks to process and analyze data. It is inspired by the structure and function of the human brain and is used for tasks such as image recognition, natural language processing, and speech recognition.</li><li>Natural language processing (NLP): It is the branch of AI that deals with the interaction between computers and human language. It enables machines to understand, interpret, and generate human language, including text and speech.</li><li>Robotics: It involves designing, building, and programming robots that can perform tasks autonomously or with human interaction. Robots are used in various industries, including manufacturing, healthcare, and transportation.</li><li>Computer vision: It is a field of AI that focuses on enabling machines to interpret and understand visual data from the world around them, including images and videos. It is used in applications such as object recognition, facial recognition, and autonomous driving.</li></ol>AI has numerous applications across various industries and is rapidly advancing with the development of new techniques and technologies. However, there are also ethical and societal implications of AI that need to be carefully considered and regulated

Basics Of AI | MCS 224 | IGNOU Study Adda

Basics Of AI

Types Of AI | MCS 224 | IGNOU Study Adda

Types Of AI

Artificial Narrow Intelligence (ANI) : Artificial Narrow Intelligence (ANI), also called Weak AI or Narrow AI: Weak AI is a term for thinking that is "simulated." Such systems seem to act intelligently, but they don't have any awareness of what they are doing. For example, a chatbot might talk to you in a way that seems natural, but it doesn't know who it is or why it's talking to you. Artificial intelligence is a system that was built to do a certain job.&nbsp;Artificial General Intelligence (AGI): Strong or General Artificial Intelligence, also called "actual" thinking. That is, acting like a smart human and thinking like one with a conscious, subjective mind. For instance, when two humans talk, they probably both know who they are, what they're doing, and why. Systems with strong or general artificial intelligence can do things that people can do. These systems tend to be harder to understand and more complicated. They are set up to handle situations where they might need to solve problems on their own without help from a person. Uses for these kinds of systems include self-driving cars and operating rooms in hospitals.&nbsp;Artificial Super Intelligence (ASI): Super intelligence: The term "super intelligence" usually refers to a level of general and strong AI that is smarter than humans, if that's even possible. The ASI is seen as the logical next step after the AGI because it can do more than humans can. This includes making decisions, making rational decisions, and even things like building emotional relationships. Their is a marginal difference between AGI and ASI.

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Master Of Computer Application (MCA NEW)
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Master Of Computer Application (MCA NEW) NOTES

Master Of Computer Application (MCA NEW)
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