Deterministic polynomial factoring over finite fields: A uniform approach viaP-schemes

𝒫-schemes and Deterministic Polynomial Factoring over Finite Fields

$\mathcal{P}$-schemes and Deterministic Polynomial Factoring over Finite Fields

We introduce a family of mathematical objects called $\mathcal{P}$-schemes, where $\mathcal{P}$ is a poset of subgroups of a finite group $G$. A $\mathcal{P}$-scheme is a collection of partitions of the right coset spaces $H\backslash G$, indexed by $H\in\mathcal{P}$, that satisfies a list of axioms. These objects generalize the classical notion of association schemes as well as the notion of $...

P-schemes: a unifying framework for deterministic polynomial factoring over finite fields

We introduce a family of mathematical objects called P-schemes, generalizing the notions of association schemes andm-schemes [IKS09]. Based on these objects, we develop a unifying framework for deterministic polynomial factoring over finite fields under the generalized Riemann hypothesis (GRH). It allows us to not only recover most of the known results but also discover new ones. In particular,...

Deterministic Polynomial Factoring and Association Schemes

The problem of finding a nontrivial factor of a polynomial f(x) over a finite field Fq has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the art by focusing on prime degree polynomials; let n be the degree. If (n − 1) has a ‘larg...

On the Deterministic Complexity of Factoring Polynomials over Finite Fields

We present a new deterministic algorithm for factoring polynomials over Z p of degree n. We show that the worst-case running time of our algorithm is O(p 1=2 (log p) 2 n 2+), which is faster than the running times of previous deterministic algorithms with respect to both n and p. We also show that our algorithm runs in polynomial time for all but at most an exponentially small fraction of the p...