arXiv subjects
Publications and source records attributed to Giacomo Micheli.
Let $K$ be a number field with ring of integers $\mathcal O$. After introducing a suitable notion of density for subsets of $\mathcal O$, generalizing that of natural density for subsets of $\mathbb Z$, we show that the density of the set of coprime $m$-tuples of algebraic integers is ${1/ζ_K(m)}$, where $ζ_K$ is the Dedekind zeta function of $K$.
In this paper we provide a complete answer to a question by Heyman and Shparlinski concerning the natural density of polynomials which are irreducible by Eisenstein's criterion after applying some shift. The main tool we use is a local to global principle for density computations over a free $\mathbb Z$-module of finite rank.
In this paper we compute the density of monic and non-monic Eisenstein polynomials of fixed degree having entries in an integrally closed subring of a function field over a finite field.
Let $\mathbb F_q$ be a finite field, $F/\mathbb F_q$ be a function field of genus $g$ having full constant field $\mathbb F_q$, $\mathcal S$ a set of places of $F$ and $H$ the holomorphy ring of $\mathcal S$. In this paper we compute the density of coprime $m$-tuples of elements of $H$. As a side result, we obtain that whenever the complement of $\mathcal S$ is finite, the computation of the density can be reduced to the computation of the $L$-polynomial of the function field. In the rational function field case, classical results for the density of coprime $m$-tuples of polynomials are obtained as corollaries.
Necessary and sufficient conditions are given on matrices $A$, $B$ and $S$, having entries in some field $\mathbb F$ and suitable dimensions, such that the linear span of the terms $A^iSB^j$ over $\mathbb F$ is equal to the whole matrix space. This result is then used to determine the cardinality of subsets of $\mathbb F[A]S\mathbb F[B]$ when $\mathbb F$ is a finite field.
Explicit monoid structure is provided for the class of canonical subfield preserving polynomials over finite fields. Some classical results and asymptotic estimates will follow as corollaries.
We present a generalized version of the knapsack protocol proposed by D. Naccache and J. Stern at the Proceedings of Eurocrypt (1997). Our new framework will allow the construction of other knapsack protocols having similar security features. We will outline a very concrete example of a new protocol using extension fields of a finite field of small characteristic instead of the prime field Z/pZ, but more efficient in terms of computational costs for asymptotically equal information rate and similar key size.
In the papers by Alvarez et al. and Pathak and Sanghi a non-commutative based public key exchange is described. A similiar version of it has also been patented (US7184551). In this paper we present a polynomial time attack that breaks the variants of the protocol presented in the two papers. Moreover we show that breaking the patented cryptosystem US7184551 can be easily reduced to factoring. We also give some examples to show how efficiently the attack works.