arXiv subjects
Publications and source records attributed to Dimitrios Poulakis.
In this paper we propose a signature scheme based on two intractable problems, namely the integer factorization problem and the discrete logarithm problem for elliptic curves. It is suitable for applications requiring long-term security and provides a more efficient solution than the existing ones.
The polynomials with quaternion coefficients have two kind of roots: isolated and spherical. A spherical root generates a class of roots which contains only one complex number $z$ and its conjugate $\bar{z}$, and this class can be determined by $z$. In this paper, we deal with the complex roots of quaternion polynomials. More precisely, using Bézout matrices, we give necessary and sufficient conditions, for a quaternion polynomial to have a complex root, a spherical root, and a complex isolated root. Moreover, we compute a bound for the size of the roots of a quaternion polynomial.
We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations $F(X,Y) = b$ defined over a totally real number field $K$, assuming that $F(X,1)$ has a root $α$ such that $K(α)$ is a CM-field. Furthermore, we give an algorithm for the computation of the integral solutions of such an equation.
We obtain a quantitative version of the classical Chevalley-Weil theorem for curves. Let $ϕ: \tilde{C} \to C$ be an unramified morphism of non-singular plane projective curves defined over a number field $K$. We calculate an effective upper bound for the norm of the relative discriminant of the number field $K(Q)$ over $K$ for any point $P\in C(K)$ and $Q\inϕ^{-1}(P)$
Let $G$ be a finite abelian group $G$ with $N$ elements. In this paper we give a O(N) time algorithm for computing a basis of $G$. Furthermore, we obtain an algorithm for computing a basis from a generating system of $G$ with $M$ elements having time complexity $O(M\sum_{p|N} e(p)\lceil p^{1/2}\rceil^{μ(p)})$, where $p$ runs over all the prime divisors of $N$, and $p^{e(p)}$, $μ(p)$ are the exponent and the number of cyclic groups which are direct factors of the $p$-primary component of $G$, respectively. In case where $G$ is a cyclic group having a generating system with $M$ elements, a $O(MN^ε)$ time algorithm for the computation of a basis of $G$ is obtained.