Computational hardness assumptions | Integer factorization algorithms | Unsolved problems in computer science | Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit RSA modulus would take about 500 times as long. Not all numbers of a given length are equally hard to factor. The hardest instances of these problems (for currently known techniques) are semiprimes, the product of two prime numbers. When they are both large, for instance more than two thousand bits long, randomly chosen, and about the same size (but not too close, for example, to avoid efficient factorization by Fermat's factorization method), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical; that is, as the number of digits of the primes being factored increases, the number of operations required to perform the factorization on any computer increases drastically. Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem—for example, the RSA problem. An algorithm that efficiently factors an arbitrary integer would render RSA-based public-key cryptography insecure. (Wikipedia).
Factoring by using a sum of cubes - Online tutor
👉 Learn how to factor polynomials using the sum or difference of two cubes. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. To factor an algebraic expression means to break it up into expression
From playlist How to factor a polynomial to a higher power
Factor a polynomial expression completely over real numbers
Learn how to factor higher order trinomials. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. To factor an algebraic expression means to break it up into expressions that can be multiplied togeth
From playlist How to Factor Higher Order #Polynomial
Using the sum of two cubes with a fraction
👉 Learn how to factor polynomials using the sum or difference of two cubes. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. To factor an algebraic expression means to break it up into expression
From playlist How to factor a polynomial to a higher power
How to factor a polynomial using the difference of two cubes
👉 Learn how to factor polynomials using the sum or difference of two cubes. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. To factor an algebraic expression means to break it up into expression
From playlist How to factor a polynomial to a higher power
Factoring a binomial to the fourth power by the difference of two squares
👉 Learn how to factor polynomials using the difference of two squares for polynomials raised to higher powers. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. To factor an algebraic expression m
From playlist How to factor a polynomial by difference of two squares
Factoring a binomial using the difference of two cubes
👉 Learn how to factor polynomials using the sum or difference of two cubes. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. To factor an algebraic expression means to break it up into expression
From playlist How to factor a polynomial to a higher power
Factoring a polynomial raised to the 4th power
Learn how to factor higher order trinomials. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. To factor an algebraic expression means to break it up into expressions that can be multiplied togeth
From playlist How to Factor Higher Order #Polynomial
Factoring a polynomial using the difference of two cubes
👉 Learn how to factor polynomials using the sum or difference of two cubes. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. To factor an algebraic expression means to break it up into expression
From playlist How to factor a polynomial to a higher power
How to factor using the sum of two cubes
👉 Learn how to factor polynomials using the sum or difference of two cubes. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. To factor an algebraic expression means to break it up into expression
From playlist How to factor a polynomial to a higher power
Ring Theory: As an application of all previous ideas on rings, we determine the primes in the Euclidean domain of Gaussian integers Z[i]. Not only is the answer somewhat elegant, but it contains a beautiful theorem on prime integers due to Fermat. We finish with examples of factorization
From playlist Abstract Algebra
Rings 16 Factorization of polynomials
This lecture is part of an online course on rings and modules. We discuss the problem of factorising polynomials with integer coefficients, and in particular give some tests to see whether they are irreducible. For the other lectures in the course see https://www.youtube.com/playlist?lis
From playlist Rings and modules
Keith Conrad - Prime Factorization From Euclid to Noether
This talk was part of Number Theory Day 2023, at UConn. More information about the event can be found here: https://alozano.clas.uconn.edu/number-theory-day/
From playlist Number Theory Day
2016 Putnam Exam A1 | Putnam Polynomials Puzzle
Today we solve problem A1 from the 2016 Putnam competition, usually just called the Putnam exam! We're asked to find the smallest positive integer j, so if we take the jth derivative of any polynomial with integer coefficients, this will be divisible by 2016 when evaluated at any integer k
From playlist Coffee Time Math with Wrath of Math
[ANT13] Dedekind domains, integral closure, discriminants... and some other loose ends
In this video, we see an example of how badly this theory can fail in a non-Dedekind domain, and so - regrettably - we finally break our vow of not learning what a Dedekind domain is.
From playlist [ANT] An unorthodox introduction to algebraic number theory
Numbers & Sets: Lecture 10/33 - Bézout's Identity, Euclid's Algorithm
This video series is not endorsed by the University of Cambridge. These videos are primarily inspired from Dexter Chua's lecture notes and Hammack's Book of Proof. Dexter Chua's lecture notes can be found here: https://dec41.user.srcf.net/notes/IA_M/numbers_and_sets.pdf Additionally, prob
From playlist Summer of Math Exposition 2 videos
Diophantine Equations: Polynomials With 1 Unknown ← number theory ← axioms
Learn how to solve a Diophantine Equation that's a polynomial with one variable. We'll cover the algorithm you can use to find any & all integer solutions to these types of equations. written, presented, & produced by Michael Harrison #math #maths #mathematics you can support axioms on
From playlist Number Theory
R Programming: Introduction: Factors (R Intro-04)
[my R script is here https://github.com/bionicturtle/youtube/tree/master/r-intro] Factors are categorical vectors. Specifically, they are (integer) vectors that store categorical values, or ordinal values. Ordinal values are *ranked* categories (but they are not intervals).Factors can only
From playlist R Programming: Intro
[ANT01] Algebraic number theory: an introduction, via Fermat's last theorem
The existence of the Euclidean algorithm is what makes multiplication in Z so nice. But some other rings have Euclidean algorithms too. Here's how we can exploit this for profit.
From playlist [ANT] An unorthodox introduction to algebraic number theory
Direct Proofs: Beginner Examples (Even/Odd)
Let's prove some basic statements using the direct proof method! This is intended for beginners to direct proof or proof in general. Timestamps: 0:00 Introduction 00:50 Definitions 3:31 Example 1 8:14 Example 2 12:18 Example 3 15:53 Example 4 Thanks for watching! Comment below with quest
From playlist Proofs
What are the formulas for the sum and difference of two cubes
👉 Learn how to factor polynomials using the sum or difference of two cubes. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. To factor an algebraic expression means to break it up into expression
From playlist How to factor a polynomial to a higher power