On the number of unit solutions of cubic congruence modulo $ n $

<abstract><p>For any positive integer $ n $, let $ \mathbb Z_n: = \mathbb Z/n\mathbb Z = \{0, \ldots, n-1\} $ be the ring of residue classes module $ n $, and let $ \mathbb{Z}_n^{\times}: = \{x\in \mathbb Z_n|\gcd(x, n) = 1\} $. In 1926, for any fixed $ c\in\mathbb Z_n $, A. Brauer studied the linear congruence $ x_1+\cdots+x_m\equiv c\pmod n $ with $ x_1, \ldots, x_m\in\mathbb{Z}_n^{\times} $ and gave a formula of its number of incongruent solutions. Recently, Taki Eldin extended A. Brauer's result to the quadratic case. In this paper, for any positive integer $ n $, we give an explicit formula for the number of incongruent solutions of the following cubic congruence</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ x_1^3+\cdots +x_m^3\equiv 0\pmod n\ \ \ {\rm with} \ x_1, \ldots, x_m \in \mathbb{Z}_n^{\times}. $\end{document} </tex-math></disp-formula></p> </abstract>

A Common General Access Structure Construction Approach in Secret Image Sharing

(k, n) threshold is a special case of the general access structure (GAS) in secret image sharing (SIS), therefore GAS is more extensive than (k, n) threshold. Most of conventional SIS, including visual secret sharing (VSS), polynomial-based SIS, linear congruence (LC)-based SIS, etc., were proposed with only (k, k) threshold or (k, n) threshold other than GAS. This article introduces a common GAS construction approach in SIS with on pixel expansion from existing (k, k) threshold or (k, n) threshold SIS. The authors input classic SIS methods to test the efficiency and feasibility of the proposed common GAS construction approach. Experiments are presented to indicate the efficiency of the approach by illustrations and analysis.

Adaptive Partial Image Secret Sharing

In contrast to encrypting the full secret image in classic image secret sharing (ISS), partial image secret sharing (PISS) only encrypts part of the secret image due to the situation that, in general, only part of the secret image is sensitive or secretive. However, the target part needs to be selected manually in traditional PISS, which is human-exhausted and not suitable for batch processing. In this paper, we introduce an adaptive PISS (APISS) scheme based on salience detection, linear congruence, and image inpainting. First, the salient part is automatically and adaptively detected as the secret target part. Then, the target part is encrypted into n meaningful shares by using linear congruence in the processing of inpainting the target part. The target part is decrypted progressively by only addition operation when more shares are collected. It is losslessly decrypted when all the n shares are collected. Experiments are performed to verify the efficiency of the introduced scheme.