Some results on uniform mixing on abelian Cayley graphs

In the past few decades, quantum algorithms have become a popular research area of both mathematicians and engineers. Among them, uniform mixing provides a uniform probability distribution of quantum information over time which attracts a special attention. However, there are only a few known examples of graphs which admit uniform mixing. In this paper, a characterization of abelian Cayley graphs having uniform mixing is presented. Some concrete constructions of such graphs are provided. Specifically, for cubelike graphs, it is shown that the Cayley graph ${\rm Cay}(\mathbb{F}_2^{2k};S)$ has uniform mixing if the characteristic function of $S$ is bent. Moreover, a difference-balanced property of the eigenvalues of an abelian Cayley graph having uniform mixing is established. Furthermore, it is proved that an integral abelian Cayley graph exhibits uniform mixing if and only if the underlying group is one of the groups: $\mathbb{Z}_2^d, \mathbb{Z}_3^d$, $\mathbb{Z}_4^d$ or $\mathbb{Z}_2^{r}\otimes \mathbb{Z}_4^d$ for some integers $r \geq 1, d\geq 1$. Thus the classification of integral abelian Cayley graphs having uniform mixing is completed.Comment: 33 page

ON DILLON\u27S CLASS H OF BENT FUNCTIONS, NIHO BENT FUNCTIONS AND O-POLYNOMIALS

One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family H. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well- known Maiorana-McFarland class. We first notice that H can be extended to a slightly larger class that we denote by H. We observe that the bent functions constructed via Niho power functions, which four examples are known, due to Dobbertin et al. and to Leander-Kholosha, are the univariate form of the functions of class H. Their restrictions to the vector spaces uF2n=2 , u 2 F? 2n, are linear. We also characterize the bent functions whose restrictions to the uF2n=2 \u27s are affine. We answer to the open question raised by Dobbertin et al. in JCT A 2006 on whether the duals of the Niho bent functions introduced in the paper are Niho bent as well, by explicitely calculating the dual of one of these functions. We observe that this Niho function also belongs to the Maiorana-McFarland class, which brings us back to the problem of knowing whether H (or H) is a subclass of the Maiorana-McFarland completed class. We then show that the condition for a function in bivariate form to belong to class H is equivalent to the fact that a polynomial directly related to its definition is an o-polynomial and we deduce eight new cases of bent functions in H which are potentially new bent functions and most probably not affine equivalent to Maiorana-McFarland functions