Lecture

Integer Factorization: Quadratic Sieve

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In course

MATH-489: Number theory II.d - Cryptography

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Description

This lecture covers the Quadratic Sieve method for integer factorization, which generalizes to allow for factoring smaller size numbers. The method involves sieving and selecting smooth numbers to efficiently factorize integers. It also discusses the complexity of the Quadratic Sieve algorithm and its application in finding B-smooth numbers. The instructor explains the process step by step, from sieving to selecting numbers to be factored, emphasizing the importance of choosing the right parameters for efficient factorization.

Instructor

Dimitar Petkov Jetchev

Official source

Ontological neighbourhood

Mathematics

Number theory: Topics in number theory

Related concepts (32)

Quadratic residue

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: Otherwise, q is called a quadratic nonresidue modulo n. Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.

Quadratic sieve

The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning that its running time depends solely on the size of the integer to be factored, and not on special structure or properties.

Integer factorization

In number theory, integer factorization is the decomposition, when possible, of a positive integer into a product of smaller integers. If the factors are further restricted to be prime numbers, the process is called prime factorization, and includes the test whether the given integer is prime (in this case, one has a "product" of a single factor). 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.

Sieve of Atkin

In mathematics, the sieve of Atkin is a modern algorithm for finding all prime numbers up to a specified integer. Compared with the ancient sieve of Eratosthenes, which marks off multiples of primes, the sieve of Atkin does some preliminary work and then marks off multiples of squares of primes, thus achieving a better theoretical asymptotic complexity. It was created in 2003 by A. O. L. Atkin and Daniel J. Bernstein. In the algorithm: All remainders are modulo-sixty remainders (divide the number by 60 and return the remainder).

Factorization

In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial factorization of x2 – 4. Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any can be trivially written as whenever is not zero.

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