A complete classification of de mersenne´s primes and its iomplications for computing

A study of Mersenne’s primes is carried out using the multiplicative group modulo 360 and a complete classification is obtained by its residual classes. This allows the search for Mersenne’s primes to be classified into four subgroups mutually exclusive (disjoint) and contributes to the ordered selection of exponents to be computationally tested. According to this idea, Mersenne’s trapeze is presented with the purpose of giving a geometric representation for this classification. Finally, from one of the theorems presented and proven for primes in modulo 360, a conjecture is established that could be solved by computing.

The Number of Non-cyclic Sylow Subgroups of the Multiplicative Group Modulo n

For each positive integer n, let $U(\mathbf {Z}/n\mathbf {Z})$ denote the group of units modulo n, which has order $\phi (n)$ (Euler’s function) and exponent $\lambda (n)$ (Carmichael’s function). The ratio $\phi (n)/\lambda (n)$ is always an integer, and a prime p divides this ratio precisely when the (unique) Sylow p-subgroup of $U(\mathbf {Z}/n\mathbf {Z})$ is noncyclic. Write W(n) for the number of such primes p. Banks, Luca, and Shparlinski showed that for certain constants $C_1, C_2>0$ , $$ \begin{align*} C_1 \frac{\log\log{n}}{(\log\log\log{n})^2} \le W(n) \le C_2 \log\log{n} \end{align*} $$ for all n from a sequence of asymptotic density 1. We sharpen their result by showing that W(n) has normal order $\log \log {n}/\log \log \log {n}$ .

ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

The closure of a braid in a closed orientable surface Ʃ is a link in Ʃ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

A Parallel Processing Technique Based on GMO and BCS for Medical Image Encryption

Image encryption is a technique that provides security to an image and their data from unauthorized access in which there is the lightweight process (LWP) that can be parallelized resulting in the reduction of computation time. In this paper, parallel lossless image encryption, as well as the decryption technique, is proposed. The method is a parallel implementation of group modulo operation (GMO) based bit circular shift (BCS) of pixel bit-plane values. The backbone of this technique is circular bit rotation based on some modulo group key value. The key value used here is the result of group modulo operation. The binary bit values of pixels of the initial Image are rotated circularly to generate a new binary bit value of pixels encrypted image. The enhancement of this GMO and BCS based encryption are also given here by using the parallel implementation of the algorithm. The given results show the parallel implementation technique has the same level of encryption standard but has a better level of the time standard. As discussed in the result section, this technique can be used for medical image encryption as well as in multimedia applications where the transfer of image data is required over a network.

Normal subgroups of invertibles and of unitaries in a C*-algebra

We investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity of a {\mathrm{C}^{*}}-algebra. By relating normal subgroups to closed two-sided ideals we obtain a “sandwich condition” describing all the closed normal subgroups both in the invertible and in the unitary case. We use this to prove a conjecture by Elliott and Rørdam: in a simple \mathrm{C}^{*}-algebra, the group of approximately inner automorphisms induced by unitaries in the connected component of the identity is topologically simple. Turning to non-closed subgroups, we show, among other things, that in a simple unital \mathrm{C}^{*}-algebra the commutator subgroup of the group of invertibles in the connected component of the identity is a simple group modulo its center. A similar result holds for unitaries under a mild extra assumption.

Randomized Key-based GMO-BCS Image Encryption for Securing Medical Image

An essential security requirement while transmitting and receiving medical images is to maintain confidentiality and authorization of these medical images. This paper contains a proposal of an enhanced lossless image encryption algorithm that provides security to Digital Imaging and Communications in Medicine (DICOM) images by producing a random key with using enhanced group modulo based bit circular shift (GMO-BCS) technique. Random key production is the backbone of this technique to provide robust security of medical images that transfer over a network. In the encryption process, we randomly generate a key for each and every pixel of the DICOM image. Group theory is used in this process to create circular shifting in 8-bit pixel values while the security enhancement employs the random key for encryption. This technique is more suitable for medical image encryption either by direct transmission or multimedia app-based transmission under telemedicine and others