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  2. Integer factorization - Wikipedia

    en.wikipedia.org/wiki/Integer_factorization

    A general-purpose factoring algorithm, also known as a Category 2, Second Category, or Kraitchik family algorithm, [10] has a running time which depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method.

  3. Factorization - Wikipedia

    en.wikipedia.org/wiki/Factorization

    The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. 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.

  4. Integer factorization records - Wikipedia

    en.wikipedia.org/wiki/Integer_factorization_records

    Integer factorization is the process of determining which prime numbers divide a given positive integer.Doing this quickly has applications in cryptography.The difficulty depends on both the size and form of the number and its prime factors; it is currently very difficult to factorize large semiprimes (and, indeed, most numbers that have no small factors).

  5. Fermat's factorization method - Wikipedia

    en.wikipedia.org/wiki/Fermat's_factorization_method

    If the approximate ratio of two factors (/) is known, then a rational number / can be picked near that value. N u v = c v ⋅ d u {\displaystyle Nuv=cv\cdot du} , and Fermat's method, applied to Nuv , will find the factors c v {\displaystyle cv} and d u {\displaystyle du} quickly.

  6. Special number field sieve - Wikipedia

    en.wikipedia.org/wiki/Special_number_field_sieve

    In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integers of the form r e ± s , where r and s are small (for instance Mersenne numbers ).

  7. Factorization of polynomials - Wikipedia

    en.wikipedia.org/wiki/Factorization_of_polynomials

    Modern algorithms and computers can quickly factor univariate polynomials of degree more than 1000 having coefficients with thousands of digits. [3] For this purpose, even for factoring over the rational numbers and number fields, a fundamental step is a factorization of a polynomial over a finite field.

  8. Quadratic sieve - Wikipedia

    en.wikipedia.org/wiki/Quadratic_sieve

    The number π(B), denoting the number of prime numbers less than B, will control both the length of the vectors and the number of vectors needed. Use sieving to locate π(B) + 1 numbers a i such that b i = (a i 2 mod n) is B-smooth. Factor the b i and generate exponent vectors mod 2 for each one.

  9. General number field sieve - Wikipedia

    en.wikipedia.org/wiki/General_number_field_sieve

    When using such algorithms to factor a large number n, it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order n 1/2. The size of these values is exponential in the size of n (see below). The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the ...