It is well known that duality theorems are of utmost importance for the arithmetic of local and global fields and that Brauer groups appear in this context unavoidably. The key word here is class field theory.In this paper we want to make evident that these topics play an important role in public key cryptopgraphy, too. Here the key words are Discrete Logarithm systems with bilinear structures.Almost all public key crypto systems used today based on discrete logarithms use the ideal class groups...

As a subproduct of the Schoof-Elkies-Atkin algorithm to count points on elliptic curves defined over finite fields of characteristic $p$, there exists an algorithm that computes, for $\ell$ an Elkies prime, $\ell$-torsion points in an extension of degree $\ell -1$ at cost $\tilde{O}(\ell max{(\ell ,logq)}^2)$ bit operations in the favorable case where $\ell \le p/2$.We combine in this work a fast algorithm for computing isogenies due to Bostan, Morain, Salvy and Schost with the $p$-adic approach followed by Joux and Lercier to get an algorithm valid without any limitation...

Solutions of the equations y² = xⁿ+k (n = 3,4) in a finite field are given almost explicitly in terms of k.

We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let $e_E\left(p\right)$ be the exponent of the group of rational points of the reduction modulo p of E over the finite field ${}_p$. Let be the family of elliptic curves $E_{a,b}:y^2=x^3+ax+b$, where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, $1/\left|\right|{\sum }_{E\in }{\sum }_{p\le x}{e}_E^k\left(p\right)=C_{k}li\left(x^{k+1}\right)+O(\left(x^{k+1}\right)/{\left(logx\right)}^c$)$$as x → ∞, as long as $A,B>exp\left(c_1{\left(logx\right)}^{1/2}\right)$ and $AB>x{\left(logx\right)}^{4+2c}$,...

In this article the discrete logarithm problem in degree 0 class groups of curves over finite fields given by plane models is studied. It is proven that the discrete logarithm problem for non-hyperelliptic curves of genus 3 (given by plane models of degree 4) can be solved in an expected time of $\tilde{O}\left(q\right)$, where $q$ is the cardinality of the ground field. Moreover, it is proven that for every fixed natural number $d\ge 4$ the following holds: We consider the discrete logarithm problem for curves given by plane models...

We give upper and lower bounds for the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.