A p-adic quasi-quadratic point counting algorithm

In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality $q$ with time complexity $O(n^{2+o(1)})$ and space complexity $O(n^2)$, where $n=\log(q)$. In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of theta constants. The theta null values are computed with respect to a semi-canonical theta structure of level $2^\nu p$ where $\nu >0$ is an integer and p=\mathrm{char}(\F_q)>2. The results of this paper suggest a global positive answer to the question whether there exists a quasi-quadratic time algorithm for the computation of the number of rational points on a generic ordinary abelian variety defined over a finite field.Comment: 32 page

Solving the "Isomorphism of Polynomials with Two Secrets" Problem for all Pairs of Quadratic Forms

We study the Isomorphism of Polynomial (IP2S) problem with m=2 homogeneous quadratic polynomials of n variables over a finite field of odd characteristic: given two quadratic polynomials (a, b) on n variables, we find two bijective linear maps (s,t) such that b=t . a . s. We give an algorithm computing s and t in time complexity O~(n^4) for all instances, and O~(n^3) in a dominant set of instances. The IP2S problem was introduced in cryptography by Patarin back in 1996. The special case of this problem when t is the identity is called the isomorphism with one secret (IP1S) problem. Generic algebraic equation solvers (for example using Gr\"obner bases) solve quite well random instances of the IP1S problem. For the particular cyclic instances of IP1S, a cubic-time algorithm was later given and explained in terms of pencils of quadratic forms over all finite fields; in particular, the cyclic IP1S problem in odd characteristic reduces to the computation of the square root of a matrix. We give here an algorithm solving all cases of the IP1S problem in odd characteristic using two new tools, the Kronecker form for a singular quadratic pencil, and the reduction of bilinear forms over a non-commutative algebra. Finally, we show that the second secret in the IP2S problem may be recovered in cubic time

Weak del Pezzo surfaces with irregularity

I construct normal del Pezzo surfaces, and regular weak del Pezzo surfaces as well, with positive irregularity q>0. Such things can happen only over nonperfect fields. The surfaces in question are twisted forms of nonnormal del Pezzo surfaces, which were classified by Reid. The twisting is with respect to the flat topology and infinitesimal group scheme actions. The twisted surfaces appear as generic fibers for Fano--Mori contractions on certain threefolds with only canonical singularities.Comment: 32 pages, minor changes, to appear in Tohoku Math.

Asymptotics For Primitive Roots Producing Polynomials And Primitive Points On Elliptic Curves

Let $x \geq 1$ be a large number, let $f(n) \in \mathbb{Z}[x]$ be a prime producing polynomial of degree $\deg(f)=m$, and let $u\neq \pm 1,v^2$ be a fixed integer. Assuming the Bateman-Horn conjecture, an asymptotic counting function for the number of primes $p=f(n) \leq x$ with a fixed primitive root $u$ is derived in this note. This asymptotic result has the form \pi_f(x)=\# \{ p=f(n)\leq x:\ord_p(u)=p-1 \}=\left (c(u,f)+ O\left (1/\log x )\right ) \right )x^{1/m}/\log x, where $c(u,f)$ is a constant depending on the polynomial and the fixed integer. Furthermore, new results for the asymptotic order of elliptic primes with respect to fixed elliptic curves $E:f(X,Y)=0$ and its groups of $\mathbb{F}_p$-rational points $E(\mathbb{F}_p)$, and primitive points are proved in the last chapters