Factoring polynomials over large finite fields

Abstract

Abstract.This paper reviews some of the known algorithms for factoring polynomials over finite fields and presents a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GF(p"!) to the problem of finding the roots in GF(p) of certain other polynomials over GF(p). The amount of computation and the storage space required by these algorithms are algebraic in both the degree of the polynomial to be factored and the logarithm of the order of the finite field.Certain observations on the application of these methods to the factorization of polynomials over the rational integers are also included.

Algorithmic Results in List Decoding

“…Otherwise, using a univariate polynomial factorization algorithm (such as Berlekamp's randomized factoring algorithm [6]), compute the list E 1 (X), E 2 (X), . .…”

“…For univariate polynomial factorization (and hence also root finding) of a polynomial f (X) ∈ F p t [X], where p is the characteristic of the field, Berlekamp [6] (see also [76,Exercise 14.40]) gave a deterministic algorithm with running time polynomial in degree(f ), t, and p (this was achieved via a deterministic polynomial-time reduction from factoring in F p t [X] to root finding in F p [X]). Combined with our reduction, we get a deterministic algorithm for the above bivariate root finding problem in time polynomial in q, k, as desired.…”

“…The characteristic ofF is at most q, and its degree over the underlying prime field is at most q log q. Therefore, we can find all roots of R(Y 1 ) by a deterministic algorithm running in time polynomial in d, q [6] (see discussion in Section 4.5.2). Each of the roots will be a polynomial in F[X] of degree less than q − 1.…”

The Distribution of Lucas and Elliptic Pseudoprimes

“…However, this is the only indeterminacy in (3), since primes p which split in K have a 2 2 unique representation up to units as ß +ry . 1 1 The representation of p as ß + ry can be found in random polynomial time by factoring the polynomial x + r in F , using Berlekamp's algorithm [3]. Once a number c is found such that c + r = 0 (mod p), one may use the method of Cornacchia [4] to determine ß and y .…”

A review of cryptographic properties of S-boxes with generation and analysis of crypto secure S-boxes.

“…Later appropriate coding techniques of Polynomials over Galois Field GF(p q ) had been illustrated with example [TK68]. The previous idea of factorizing Polynomials over Galois Field GF(p q ) [EB67] had also been extended to Large value of P or Large Finite fields [EB70]. Later Few Probabilistic Algorithms to find IPs over Galois Field GF(p q ) for degree q had been elaborated with example [MR80].…”