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Distinct-degree factorization algorithm tests every d not greater than half the degree of the input polynomial. Rabin's algorithm takes advantage that the factors are not needed for considering fewer d. Otherwise, it is similar to distinct-degree factorization algorithm. It is based on the following fact.
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 .
Since f is of degree d with integer coefficients, if a and b are integers, then so will be b d ·f(a/b), which we call r. Similarly, s = b e · g ( a / b ) is an integer. The goal is to find integer values of a and b that simultaneously make r and s smooth relative to the chosen basis of primes.
In mathematics, particularly computational algebra, Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields). The algorithm consists mainly of matrix reduction and polynomial GCD computations.
The first algorithm for polynomial decomposition was published in 1985, [6] though it had been discovered in 1976, [7] and implemented in the Macsyma/Maxima computer algebra system. [8] That algorithm takes exponential time in worst case, but works independently of the characteristic of the underlying field .
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 ).
The algorithm [4] in Python code is: def chen_fox_lyndon_factorization ( s : list [ int ]) -> list [ int ]: """Monoid factorisation using the Chen–Fox–Lyndon theorem. Args: s: a list of integers Returns: a list of integers """ n = len ( s ) factorization = [] i = 0 while i < n : j , k = i + 1 , i while j < n and s [ k ] <= s [ j ]: if s [ k ...
The mechanism of Sudan's Algorithm is almost the same as the algorithm of Berlekamp–Welch Algorithm, except in the step 1, one wants to compute a bivariate polynomial of bounded (,) degree. Sudan's list decoding algorithm for Reed–Solomon code which is an improvement on Berlekamp and Welch algorithm, can solve the problem with t = ( 2 n d ...