Carlos Renteria

In this paper we introduce the relative generalized minimum distance function (RGMDF for short) and it allows us to give an algebraic approach to the relative generalized Hamming weights of the projective Reed--Muller--type codes. Also we introduce the relative generalized footprint function and it gives a tight lower bound for the RGMDF which is much easier to compute.

A novel symmetric cryptosystem of the substitution permutation network type is presented for image encryption in 14 rounds. An algorithm is developed to...

Let K be a finite field. Let X* be a subset of the affine space Kn, which is parameterized by odd cycles. In this paper we give an explicit Gr\"obner basis for the vanishing ideal, I(X*), of X*. We give an explicit formula for the regularity of I(X*) and finally if X* is parameterized by an odd cycle of length k, we show that the Hilbert function of the vanishing ideal of X* can be written as linear combination of Hilbert functions of degenerate torus.

In this paper we estimate the main parameters of some evaluation codes which are known as projective parameterized codes. We find the length of these codes and we give a formula for the dimension in terms of the Hilbert function associated to two ideals, one of them being the vanishing ideal of the projective torus. Also we find an upper bound for the minimum distance and, in some cases, we give some lower bounds for the regularity index and the minimum distance. These lower bounds work in several cases, particularly for any projective parameterized code associated to the incidence matrix of uniform clutters and then they work in the case of graphs.

This work proposes a different procedure to encrypt images of 256 grey levels and colour, using the symmetric system Advanced Encryption Standard with a variable permutation in the first round, after the x-or operation. Variable permutation means using a different one for each input block of 128 bits. In this vein, an algorithm is constructed that defines a Bijective function between sets Nm = {n in N, 0 <= n < fac(m)} with n >= 2 and Pm = {pi, pi is a permutation of 0, 1, ..., m-1}. This algorithm calculates permutations on 128 positions with 127 known constants. The transcendental numbers are used to select the 127 constants in a pseudo-random way. The proposed encryption quality is evaluated by the following criteria: Correlation; horizontal, vertical and diagonal, Entropy and Discrete Fourier Transform. The latter uses the NIST standard 800-22. Also, a sensitivity analysis was performed in encrypted figures. Furthermore, an additional test is proposed which considers the distribution of 256 shades of the three colours; red, green and blue for colour images. On the other hand, it is important to mention that the images are encrypted without loss of information because many banking companies and some safety area countries do not allow the figures to go through a compression process with information loss. i.e., it is forbidden to use formats such as JPEG.