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Register a new userFactoring polynomials over global fields
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Let K be a global field and f in K[X] be a polynomial. We present an efficient algorithm which factors f in polynomial time.
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In this paper we introduce the additive analogue of the index of a polynomial over finite fields. We study several problems in the theory of polynomials over finite fields in terms of their additive indices, such as value set sizes, bounds on multiplicative character sums, and characterizations of permutation polynomials.
Let $D$ be a negative integer congruent to $0$ or $1bmod{4}$ and $mathcal{O}=mathcal{O}_D$ be the corresponding order of $ K=mathbb{Q}(sqrt{D})$. The Hilbert class polynomial $H_D(x)$ is the minimal polynomial of the $j$-invariant $ j_D=j(mathbb{C}/mathcal{O})$ of $mathcal{O}$ over $K$. Let $n_D=(mathcal{O}_{mathbb{Q}( j_D)}:mathbb{Z}[ j_D])$ denote the index of $mathbb{Z}[ j_D]$ in the ring of integers of $mathbb{Q}(j_D)$. Suppose $p$ is any prime. We completely determine the factorization of $H_D(x)$ in $mathbb{F}_p[x]$ if either $p mid n_D$ or $p mid D$ is inert in $K$ and the $p$-adic valuation $v_p(n_D)leq 3$. As an application, we analyze the key space of Oriented Supersingular Isogeny Diffie-Hellman (OSIDH) protocol proposed by Col`o and Kohel in 2019 which is the roots set of the Hilbert class polynomial in $mathbb{F}_{p^2}$.
In this paper, we present three classes of complete permutation monomials over finite fields of odd characteristic. Meanwhile, the compositional inverses of these complete permutation polynomials are also proposed.
In 2005, Kayal suggested that Schoofs algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then explain details of a generalization involving Pilas algorithm for abelian varieties.
We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $mathbb{F}_p(t)$, when $p$ is prime and $rge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, we compute the $L$-function of $J$ over $mathbb{F}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $mathbb{F}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^ u +1$, and $K_d := mathbb{F}_p(mu_d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the Tate-Shafarevich group of $J$ over $K_d$ is $[J(K_d):V]^2$. When $r>2$, we prove that the new part of $J$ is isogenous over $overline{mathbb{F}_p(t)}$ to the square of a simple abelian variety of dimension $phi(r)/2$ with endomorphism algebra $mathbb{Z}[mu_r]^+$. For a prime $ell$ with $ell mid pr$, we prove that $J[ell](L)={0}$ for any abelian extension $L$ of $overline{mathbb{F}}_p(t)$.
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