Discrete Mathematics - Vol 3 | 7. Lecture - 55: Modular Arithmetic by Abraham | Learn Smarter
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7. Lecture - 55: Modular Arithmetic

The chapter discusses modular arithmetic, key algorithms related to it, and their relevance to computer science, particularly in cryptography. It outlines congruence relations, arithmetic rules in modular systems, and emphasizes the inefficiencies of naive algorithms for modular exponentiation in favor of a more efficient square and multiply method. The chapter wraps up with insights into the complexity of modular arithmetic operations.

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Sections

  • 7.1

    Discrete Mathematics

    This section introduces modular arithmetic and its relevance in computer science, particularly in cryptography.

    Learning Practice

  • 7.2

    Lecture - 55: Modular Arithmetic

    Modular arithmetic is a fundamental concept in number theory that is crucial for applications in computer science, particularly in cryptography.

    Learning Practice

  • 7.3

    Introduction To Number Theory

    This section introduces the fundamentals of number theory, specifically focusing on modular arithmetic, properties of prime numbers, and algorithms relevant to cryptography.

    Learning Practice

  • 7.4

    Modular Arithmetic

    This section covers the fundamentals of modular arithmetic, including its definition, properties, and practical applications in cryptography.

    Learning Practice

  • 7.4.1

    Definition And Concept Of Modulus

    Modulus is a fundamental concept in modular arithmetic, defining the remainder of division of one integer by another.

    Learning Practice

  • 7.4.2

    Congruence With Respect To Modulo

    This section introduces modular arithmetic and the concept of congruence, specifically focusing on how two integers can be considered equal under a given modulus.

    Learning Practice

  • 7.4.2.1

    Definition Of Congruence

    This section introduces the concept of congruence in modular arithmetic, explaining how it relates integers and their remainders when divided by a modulus.

    Learning Practice

  • 7.4.2.2

    Arithmetic Rules In Modular Arithmetic

    This section discusses the fundamental arithmetic rules governing modular arithmetic and their applications in number theory and computer science.

    Learning Practice

  • 7.4.3

    Arithmetic Rules Of Modular Arithmetic

    This section explores the arithmetic rules of modular arithmetic, emphasizing addition, subtraction, multiplication, and modular exponentiation, particularly in the context of cryptographic applications.

    Learning Practice

  • 7.4.3.1

    Addition And Subtraction Rules

    This section explores the rules of addition and subtraction under modular arithmetic, illustrating how to handle congruence relations.

    Learning Practice

  • 7.4.3.2

    Multiplication Rules

    This section discusses modular arithmetic, particularly focusing on the rules of multiplication and their role in cryptography.

    Learning Practice

  • 7.4.3.3

    Division In Modular Arithmetic

    This section introduces modular arithmetic in number theory, focusing on division and its distinctions from other operations such as addition and multiplication.

    Learning Practice

  • 7.5

    Algorithms For Modular Arithmetic

    This section discusses modular arithmetic algorithms, focusing on their properties and significance in cryptography, especially modular exponentiation.

    Learning Practice

  • 7.5.1

    Modular Addition, Subtraction And Multiplication

    This section introduces modular arithmetic, focusing on addition, subtraction, and multiplication within a defined modulus, and discusses properties and algorithms crucial for computational applications, especially in cryptography.

    Learning Practice

  • 7.5.2

    Complexity Measurement

    This section focuses on the complexity measurement in modular arithmetic operations critical to number theory and computer science.

    Learning Practice

  • 7.5.3

    Modular Exponentiation

    This section covers the concept of modular exponentiation, its significance in cryptography, and efficient algorithms for its computation.

    Learning Practice

  • 7.5.3.1

    Naive Approach For Modular Exponentiation

    This section discusses the naive approach for modular exponentiation, highlighting its inefficiency and introducing a more optimal method for computation.

    Learning Practice

  • 7.5.3.2

    Square And Multiply Approach

    The Square and Multiply Approach is an efficient method for modular exponentiation, which significantly reduces the number of multiplications needed by leveraging the binary representation of the exponent.

    Learning Practice

  • 7.5.3.3

    Pseudocode For Square And Multiply

    This section introduces the Square and Multiply algorithm for efficient modular exponentiation, emphasizing its significance in cryptography.

    Learning Practice

  • 7.6

    Summary And References

    This section introduces modular arithmetic, its properties, and algorithms relevant to number theory and cryptography.

    Learning Practice

References

ch56.pdf

Class Notes

Memorization

What we have learnt

  • Modular arithmetic involves...
  • Congruence relations are es...
  • The square and multiply met...

Final Test

Revision Tests