We study vectorial bent functions with Dillon-type exponents. These functions have attracted attention because they are hyperbent whenever they are bent, and they achieve the highest possible algebraic degree among all bent functions on the same domain. In low dimensions we determine the simplest possible forms of such functions when they map to GF(4). We prove non-existence results for certain monomial and multinomial bent functions mapping to large codomains.

Kloosterman sums are exponential sums on finite fields that have important applications in cryptography and coding theory. We use Stickelberger's theorem and the Gross-Koblitz formula to determine the value of the binary Kloosterman sum at$a$modulo 64, modulo 128, and modulo 256 in terms of coefficients of the characteristic polynomial of$a$.

Charpin and Gong recently characterized a large class of hyperbent functions defined on fields of order$2^n$, which include the well-known monomial functions with the Dillon exponent as a special case. We give a reformulation of the Charpin-Gong criterion in terms of the number of rational points on certain hyperelliptic curves. We present two applications of our result: The time needed to check the hyperbentness of a specific function is now polynomial in$n$, and hyperbent functions with subfield coefficients can be constructed.

A proper vertex coloring of a graph is called rainbow if, for each vertex$v$, all neighbors of$v$receive distinct colors. A$k$-regular graph$G$is called rainbow (or domatically full) if it admits a rainbow$(k+1)$-coloring. The$d$-dimensional grid graph$G_d$is the graph whose vertices are the points of${\BBZ}^d$and two vertices are adjacent if and only if their$l_1$-distance is$1$. We use a simple construction to prove that$G_d$is rainbow for all$d\ge 1$. We discuss an important application of this result in steganography.

We use Gröbner bases and the Pedersen-Roy-Szpirglas real solution counting method to analyze systems of algebraic equations satisfied by metric invariants of the tetrahedron. The transformation from the geometric to the algebraic setting is effected using the Cayley-Menger determinant formula for the volume of a simplex. We prove that, in general, the four face areas, circumradius and volume together do not uniquely determine a tetrahedron, and that there exist non-regular tetrahedra that are uniquely determined just by the four face areas and circumradius. This settles two open problems posed by the American Mathematical Monthly in 1999.