In 2018, Pott et al. investigated vectorial functions with maximal number of bent components. They found one class of binomial functions attaining the upper bound. They also proposed an open problem regarding monomial functions that have the maximal number of bent components. In this paper, we solve this open problem. Specifically, we prove that if$k\geq 2$, then$x^{s(2^{k}+1)}$are the only monomial functions over$\mathbb {F}_{2^{2k}}$that have the maximal number of bent components, where$s\in \{1, 2, 2^{2}, \ldots, 2^{k-1}\}$. As a consequence, we also solve an open problem of Ness and Helleseth about the cross-correlation function between two sequences in 2006.

In this article, we propose secure plaintext image storage protocols in the cloud environment for image owners managing and controlling their outsourced images. To solve control and privacy issues, conventional schemes suggest outsourcing images in the encrypted form. However, encrypted images lose their usability and visibility. For example, image owners cannot quickly find their stored images in the cloud. The proposed protocols encourage plaintext image storage in the cloud with copyright protection of images, visible display, lossless retrieval and controlled deletion via techniques of homomorphic encryption and digital watermarking. We allow the image owner to embed and remove the watermark in a privacy-preserving way without compromising the security of the original data and the computed results. Moreover, the image owner can securely detect the cloud's dishonest act of not completely deleting an image as required or leaking an image without permission. Compared with existing works, our work achieves more functions. We prove that the proposed work achieves the controllable management of the outsourced plaintext images without privacy leakage to unauthorized parties and demonstrate the utility and the efficiency of our protocols through experimental evaluation.