## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- AIMS Mathematics
- Conference Publications
- Electronic Research Announcements
- Mathematics in Engineering

### Open Access Journals

AMC

We discuss how to apply Gaudry's index calculus algorithm for abelian varieties to solve the discrete logarithm problem in the trace zero variety of an elliptic curve. We treat in particular the practically relevant cases of field extensions of degree 3 or 5. Our theoretical analysis is compared to other algorithms present in the literature, and is complemented by results from a prototype implementation.

AMC

In this paper we study spread codes: a family of constant-dimension
codes for random linear network coding. In other words, the codewords
are full-rank matrices of size $k\times n$ with entries in a finite
field $\mathbb F_q$. Spread codes are a family of optimal codes with
maximal minimum distance. We give a minimum-distance decoding
algorithm which requires $\mathcal{O}((n-k)k^3)$ operations over
an extension field $\mathbb F_{q^k}$. Our algorithm is more
efficient than the previous ones in the literature, when the dimension $k$ of the codewords is small with respect to $n$.
The decoding algorithm takes advantage of the algebraic structure
of the code, and it uses original results on minors of a matrix
and on the factorization of polynomials over finite fields.

AMC

The paper analyzes CFVZ, a new public key cryptosystem whose security is based on a matrix version of the discrete logarithm problem over an elliptic curve. It is shown that the complexity of solving the underlying problem for the proposed system is dominated by the complexity of solving a fixed number of discrete logarithm problems in the group of an elliptic curve. Using an adapted Pollard rho algorithm it is shown that this problem is essentially as hard as solving one discrete logarithm problem in the group of an elliptic curve. Hence, the CFVZ cryptosystem has no advantages over traditional elliptic curve cryptography and should not be used in practice.

## Year of publication

## Related Authors

## Related Keywords

[Back to Top]