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Del curso dictado por University of Colorado System

Mathematical Foundations for Cryptography

50 calificaciones

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University of Colorado System

Mathematical Foundations for Cryptography

50 calificaciones

Curso 2 de 4 en SpecializationIntroduction to Applied Cryptography

Welcome to Course 2 of Introduction to Applied Cryptography. In this course, you will be introduced to basic mathematical principles and functions that form the foundation for cryptographic and cryptanalysis methods. These principles and functions will be helpful in understanding symmetric and asymmetric cryptographic methods examined in Course 3 and Course 4. These topics should prove especially useful to you if you are new to cybersecurity. It is recommended that you have a basic knowledge of computer science and basic math skills such as algebra and probability.

De la lección

Integer Foundations

Building upon the foundation of cryptography, this module focuses on the mathematical foundation including the use of prime numbers, modular arithmetic, understanding multiplicative inverses, and extending the Euclidean Algorithm. After completing this module you will be able to understand some of the fundamental math requirement used in cryptographic algorithms. You will also have a working knowledge of some of their applications.

Course Introduction3:39

Conoce a los instructores

William Bahn
Lecturer
Computer Science
Richard White
Assistant Research Professor
Computer Science
Sang-Yoon Chang
Assistant Professor
Computer Science

Welcome to Mathematical Foundations for Cryptology.

This is the second course in the undergraduate introduction to applied

cryptology specialization.

Modern cryptographic systems,

particularly public key systems based on asymmetric cryptographic algorithms,

rely heavily on certain topics from the field of discrete mathematics.

That allow us to design problems that are efficiently solvable if you have a small

amount of critical information, but

astronomically difficult to solve if you don't.

The purpose of this course is to provide you with sufficient grounding in integer

and modular arithmetic so that you can understand and work with these algorithms.

The main topics we will be covering include the basic notion of divisibility

in prime numbers, as well as what the greatest common divisor is and

how to find it using the Euclidean Algorithm.

We'll see that while division is not defined in modular arithmetic,

multiplicative inverses are.

But they have some non-intuitive properties and don't always exist.

When they do exist,

we'll learn how to use the Extended Euclidean Algorithm to find them.

We'll explore the extremely important topic of Modular Exponentiation and

see how the totient of the modulus is critical being able to work

efficiently with exponents, which live in a different modular world.

Will also learn about discrete logarithms and the Chinese Remainder Theorem.

Finally, we'll consider the difficulty of factoring large integers

in determining whether a large number is prime.

So let's look at how these are broken up into the four modules that make up

this course.

In the first module, we'll cover the core concepts of discrete math.

We'll first look at key integer concepts,

such as divisibility and prime numbers, and learn how to

use the Euclidean Algorithm to find the greatest common divisor of two integers.

We'll then learn about the principles of modular arithmetic, and with that in hand,

turn to the notion of multiplicative inverses in modular arithmetic world.

And see that we can extend the Euclidean Algorithm to find these inverses.

In the second module, we'll explore modular exponentiation, and

discover that exponential expressions that might take thousands of years to

calculate, if done directly, can often times be done very efficiently,

even by hand, in a matter of minutes.

Along the way, we'll also discover that exponents do not live in the same modular

world and the rest of the expression.

A fact that actually forms the basis from many of the cryptographic algorithms

that protect our most sensitive secrets.

Well then learn how the modulus of exponent is related to

the totient of the expression modulus, and how to calculate that torsion.

We'll finish this module by looking at discrete logarithms and see how they

behave very differently from the ordinary logarithms we're familiar with.

The third module will let us explore a very powerful tool known as

the Chinese Remainder Theorem or CRT, which lets us work with

unimaginably large numbers quite efficiently, even with just pen and paper.

We'll learn how to represent integers via the Chinese Remainder Theorem and

how to convert back.

We'll then look at what things we can do easily with CRT represented numbers

as well as some of the things that we can't.

The final module will explore prime numbers in more depth,

including the difficulty of factoring large numbers.

And of testing numbers to see if they are prime in efficient ways.

We'll look at a couple of methods that can give us a high degree of confidence, but

not a certainty, that a number is prime.

And also touch on methods that can absolutely prove whether a number is prime

and consider why they are seldom used in practice.

Once you complete this course, you should be in a reasonable position to appreciate

and handle most of the math in the remaining courses in this specialization.

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