On the Cryptanalysis of a Latin Cubes-Based Image Cryptosystem

Based on orthogonal Latin cubes, an image cryptosystem with confusion–diffusion–confusion cipher architecture has been proposed recently (Inf. Sci. 2019, 478, 1–14). However, we find that there are four fatal vulnerabilities in this image cryptosystem, which leave open doors for cryptanalysis. In this paper, we propose a reference-validation inference algorithm and design screening-based rules to efficiently break the image cryptosystem. Compared with an existing cryptanalysis algorithm, the proposed method requires fewer pairs of chosen plain-cipher images, and behaves stably since different keys, positions of chosen bits and contents of plain images will not affect the cryptanalysis performance. Experimental results show that our cryptanalysis algorithm only requires 8×H×W3+3pairs of chosen plain-cipher images, where H×W represents the image’s resolution. Comparative studies demonstrate effectiveness and superiority of the proposed cryptanalysis algorithm.     

Orthogonal factor‐pair Latin squares of prime‐power order

We introduce a linear method for constructing factor‐pair Latin squares of prime‐power order and we identify criteria for determining whether two factor‐pair Latin squares constructed using this linear method are orthogonal. Then we show that families of pairwise mutually orthogonal diagonal factor‐pair Latin squares exist in all prime‐power orders.     

n维超立方体的层次结构模型

本文首次提出了 n维超立方体的层次结构模型 HHC,详细讨论了该结构中结点的分布及各结点的连接关系 .并利用 HHC,讨论了超立方体非对称比较模型的最优诊断算法 ,极大独立点集等问题     

Hopf C*-algebras related to the Latin square

A sufficient condition is given for the multiparametric Hopf algebras to be Hopf* -algebras. Then a special subclass of the -algebra related to a Latin square is given. After being completed, its generators are all of norm one.     

Cycles embedding on folded hypercubes with faulty nodes

Let F_Fv$F{F}_v$ be the set of faulty nodes in an n$n$-dimensional folded hypercube F_Qn$F{Q}_n$ with |F_Fv|≤n−2$\left|F{F}_v\right|\le n-2$. In this paper, we show that if n≥3$n\ge 3$, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every even length from 4$4$ to 2ⁿ−2|F_Fv|$2^n-2\left|F{F}_v\right|$, and if n≥2$n\ge 2$ and n$n$ is even, then every edge of F_Qn−F_Fv$F{Q}_n-F{F}_v$ lies on a fault-free cycle of every odd length from n+1$n+1$ to 2ⁿ−2|F_Fv|−1$2^n-2\left|F{F}_v\right|-1$.     

On avoiding odd partial Latin squares and r-multi Latin squares

We show that for any positive integer k?4, if R is a (2k-1)×(2k-1) partial Latin square, then R is avoidable given that R contains an empty row, thus extending a theorem of Chetwynd and Rhodes. We also present the idea of avoidability in the setting of partial r-multi Latin squares, and give some partial fillings which are avoidable. In particular, we show that if R contains at most nr/2 symbols and if there is an n×n Latin square L such that δn of the symbols in L cover the filled cells in R where 0<δ<1, then R is avoidable provided r is large enough.     

On the number of transversals in Cayley tables of cyclic groups

It is well known that if n is even, the addition table for the integers modulo n (which we denote by B_n) possesses no transversals. We show that if n is odd, then the number of transversals in B_n is at least exponential in n. Equivalently, for odd n, the number of diagonally cyclic latin squares of order n, the number of complete mappings or orthomorphisms of the cyclic group of order n, the number of magic juggling sequences of period n and the number of placements of n non-attacking semi-queens on an n×n toroidal chessboard are at least exponential in n. For all large n we show that there is a latin square of order n with at least (3.246)ⁿ transversals.We diagnose all possible sizes for the intersection of two transversals in B_n and use this result to complete the spectrum of possible sizes of homogeneous latin bitrades.We also briefly explore potential applications of our results in constructing random mutually orthogonal latin squares.     

A note on rainbow matchings in strongly edge-colored graphs

The Ryser Conjecture which states that there is a transversal of size $n$ in a Latin square of odd order $n$ is equivalent to finding a rainbow matching of size $n$ in a properly edge-colored $K_{n,n}$ using $n$ colors when $n$ is odd. Let $\delta$ be the minimum degree of a graph. Wang proposed a more general question to find a function $f\left(\delta \right)$ such that every properly edge-colored graph of order $f\left(\delta \right)$ contains a rainbow matching of size $\delta$, which currently has the best bound of $f\left(\delta \right)\le 3.5\delta +2$ by Lo. Babu, Chandran and Vaidyanathan investigated Wang’s question under a stronger color condition. A strongly edge-colored graph is a properly edge-colored graph in which every monochromatic subgraph is an induced matching. Wang, Yan and Yu proved that every strongly edge-colored graph of order at least $2\delta +2$ has a rainbow matching of size $\delta$. In this note, we extend this result to graphs of order at least $2\delta +1$.     

Group-labeled light dual multinets in the projective plane

In this paper we investigate light dual multinets labeled by a finite group in the projective plane $PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$. We present two classes of new examples. Moreover, under some conditions on the characteristic of $\mathbb{K}$, we classify group-labeled light dual multinets with lines of length at least 9.     

Biembeddings of Abelian groups

We prove that, with the single exception of the 2‐group C, the Cayley table of each Abelian group appears in a face 2‐colorable triangular embedding of a complete regular tripartite graph in an orientable surface. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 71–83, 2010