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Circulant matrices are of interest on their own as well as for their relevance to several areas of mathematics. In this note vector-circulant matrices over commutative rings are introduced and related results are given. Vector-circulant based additive codes are presented over a particular finite ring. [ABSTRACT FROM AUTHOR]

Considering Z n the ring of integers modulo n, the classical Fermat-Euler theorem establishes the existence of a specific natural number φ (n) satisfying the following property: x φ (n) = 1 for all x belonging to the group of units of Z n. In this manuscript, this result is extended to a class of rings that satisfies some mild conditions. [ABSTRACT FROM AUTHOR]

In the present work, a procedure for determining idempotents of a commutative ring having a sequence of ideals with certain properties is presented. As an application of this procedure, idempotent elements of various commutative rings are determined. Several examples are included illustrating the main results. [ABSTRACT FROM AUTHOR]

In 1988 Lachaud introduced the class of projective Reed–Muller codes, defined by evaluating the space of homogeneous polynomials of a fixed degree d on the points of P n (F q) . In this paper we evaluate the same space of polynomials on the points of a higher dimensional scroll, defined from a set of rational normal curves contained in complementary linear subspaces of a projective space. We determine a formula for the dimension of the codes, and the exact value of the dimension and the minimum distance in some special cases. [ABSTRACT FROM AUTHOR]

Electronic voting protocols proposed to date meet their properties based on Public Key Cryptography (PKC), which offers high flexibility through key agreement protocols and authentication mechanisms. However, when PKC is used, it is necessary to implement Certification Authority (CA) to provide certificates which bind public keys to entities and enable verification of such public key bindings. Consequently, the components of the protocol increase notably. An alternative is to use Identity-Based Encryption (IBE). With this kind of cryptography, it is possible to have all the benefits offered by PKC, without neither the need of certificates nor all the core components of a Public Key Infrastructure (PKI). Considering the aforementioned, in this paper we propose an electronic voting protocol, which meets the privacy and robustness properties by using bilinear maps. [ABSTRACT FROM AUTHOR]