Pair Of Orthogonal Latin Squares Research Articles

A New Chaotic Image Encryption Algorithm Based on Transversals in a Latin Square

In this paper, a new combinatorial structure is introduced for image encryption, which has an excellent encryption effect on security and efficiency. An n-transversal in a Latin square has the function of classifying all the matrix's positions, and it can provide a pair of orthogonal Latin squares. Employing an n-transversal of a Latin square, we can permutate all the pixels of an image group by group for the first time, then use two Latin squares for auxiliary diffusion based on a chaotic sequence, and finally, make use of a pair of orthogonal Latin squares to perform the second scrambling. The whole encryption process is "scrambling-diffusion-scrambling". The experimental results indicated that this algorithm passed various tests and achieved a secure and fast encryption effect, which outperformed many of the latest papers. The final information entropy was very close to 8, and the correlation coefficient was approximately 0. All these tests verified the robustness and practicability of the proposed algorithm.

On elementary, odd, semimagic and other classes of antilattices

An antilattice is an algebraic structure based on the same set of axioms as a lattice except that the two commutativity axioms for ∧ and ∨ are replaced by anticommutative counterparts. In this paper, we study certain classes of antilattices, including elementary (no nontrivial subantilattices), odd (no subantilattices of order [Formula: see text]), simple (no nontrivial congruences) and irreducible (not expressible as a direct product). In the finite case, odd antilattices are the same as Leech’s Latin antilattices which arise from the construction of semimagic squares from pairs of orthogonal Latin squares.

Recent results on Choi's orthogonal Latin squares

Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei Lih's construction of Choi's Latin squares of order 9 based on the two $3 \times 3$ orthogonal Latin squares. In this paper, we give a new generalization of Choi's orthogonal Latin squares of order 9 to orthogonal Latin squares of size $n^2$ using the Kronecker product including Lih's construction. We find a geometric description of Choi's orthogonal Latin squares of order 9 using the dihedral group $D_8$. We also give a new way to construct magic squares from two orthogonal non-double-diagonal Latin squares, which explains why Choi's Latin squares produce a magic square of order 9.

Mutually orthogonal latin squares based on cellular automata

We investigate sets of Mutually Orthogonal Latin Squares (MOLS) generated by Cellular Automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter $d$ over an alphabet of $q$ elements generates a Latin square of order $q^{d-1}$, we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field $\mathbb{F}_q$ are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree $n$ over $\mathbb{F}_q$. Finally, we present a construction of MOLS generated by LBCA with irreducible polynomials and prove the maximality of the resulting sets, as well as a lower bound which is asymptotically close to their actual number.

Existence of strongly symmetrical weakly pandiagonal graeco-latin squares

Abstract A graeco-latin sauare is a pair of orthogonal latin squares. It is a design of experiment in which the experimental units are grouped in three different ways. In this paper, constructions of a pair of orthogonal latin sauares which are both strongly symmetrical and weakly pandiagonal are investigated. As a result, it is proved that there exists a pair of strongly symmetrical weakly pandiagonal orthogonal latin sauare of order n if and only if n > 4 and n ≡ 0, 1, 3 (mod 4) with only one possible exception for n = 12.

Применение высокопроизводительных вычислений для поиска троек взаимно частично ортогональных диагональных латинских квадратов порядка 10

This paper deals with the search for triples of partially orthogonal Latin squares of order 10. For every known pair of orthogonal Latin squares of order 10 we add a third diagonal Latin square in such a way that the orthogonality condition between it and squares from a considered pair coincides in the maximum possible number of cells. Two approaches are used: the first one is based on reducing an original problem to Boolean satisfiability problem; the second one is based on Brute force method. Several triples of the aforementioned kind with high characteristics were constructed. The experiments were held in the volunteer computing project SAT@home and on a computing cluster.

A polynomial embedding of pairs of orthogonal partial Latin squares

We show that a pair of orthogonal partial Latin squares of order n can be embedded in a pair of orthogonal Latin squares of order at most 16n4 and all orders greater than or equal to 48n4. This paper provides the first direct polynomial order embedding construction for pairs of orthogonal partial Latin squares.

Mutually orthogonal Latin squares based on general linear groups

Given a finite group G, how many squares are possible in a set of mutually orthogonal Latin squares based on G? This is a question that has been answered for a few classes of groups only, and for no nonsoluble group. For a nonsoluble group G, we know that there exists a pair of orthogonal Latin squares based on G. We can improve on this lower bound when G is one of GL(2, q) or SL(2, q), q a power of 2, q ≠ 2, or is obtained from these groups using quotient group constructions. For nonsoluble groups, that is the extent of our knowledge. We will extend these results by deriving new lower bounds for the number of squares in a set of mutually orthogonal Latin squares based on the group GL(n, q), q a power of 2, q ≠ 2.

Partial graph design embeddings and related problems

An embedding of a partial G-design (X, P) is a G-design (V, B) with the propertynthat X C V and P C B. In 1976, R.M. Wilson [66] showed, among other things,nthat any partial G-design can be finitely embedded; however, the order of Wilson'snembedding is exponentially large with respect to the order of the partial design.nThe problem of finding small embeddings of partial G-designs for various graphsnG has received much attention in recent years, particularly for the case of partialnSteiner triple systems and partial m-cycle systems with m g 4. The work of Andersen,nHilton, Mendelsohn, Hoffman, Lindner, Rodger, Bryant and others has seennconsiderable progress made on this problem. In contrast to this, the search for evenna polynomial embedding of partial (n, k, 1)-BIBDs, for any given k g 4, has beennunsuccessful. The closest result of interest to this problem is Hoffman and Lindner'sn8n + 16rn + 82 embedding for partial (K4 K2)-designs [26]. While the graphnK4 K2 differs from K4 (a block of size 4) by only one edge, Hoffman and Lindner'snembedding relies heavily on the fact that K4 K2 is tripartite. Since K4 is notntripartite, a small embedding for partial K4-designs (that is, partial (n, 4, 1) BIBDs)nappears to be beyond the reach of the current methods.n n In this thesis, we consider the problem of finding small embeddings of partialnG-designs for various graphs G, as well as several closely related problems.n n Given the attention the embedding problem has received for the case of cycles,nthe initial focus of this thesis is on constructing small embeddings for certain cycle-relatedngraphs G. In Chapter 2, we consider the case when G is either D6 (a diagonalncycle of order 6) or a (4, 1)-kite; note that both of these graphs are bipartite andndiffer from an even cycle by only one edge. Chapter 3 presents an embedding ofnpartial (m, k)-kite designs for any given values of m and k.n n In Chapter 4, a technique is presented for obtaining a cubic embedding of anrestricted class of partial K4-designs. This embedding is valid for any partial K4-ndesign which has the property that every copy of K4 contains at least two verticesnwhich do not occur in any other copy of K4. n The concept of irreducibility is introduced in Chapter 5, and is applied to 2-nfold m-cycle systems. Specifically, it is shown that for any pair of integers (m, n)nwith 4 l m l n, if there exists an m-cycle system of order n, then there exists annirreducible 2-fold m-cycle system of order n, except when (m, n) = (5, 5). A similarnresult has already been established by Kramer [35] for the case of 3 -cycles.n n Chapter 6 describes further results on irreducible l-fold m-cycle systems when lg 2. We prove the existence of irreducible 3 -fold m-cycle systems of various orders,nand show that for any l g 2, there exists an irreducible A-fold m-cycle system ofnorder n for a sufficiently large integer n.n n The problem of immersing partial Steiner triple systems is the focus of Chapter 7.nAn immersing of a partial Steiner triple system (V, P) of order n is a l-fold triplensystem (V, B) of order n such that P C B. It is shown that a partial Steiner triplensystem of order n can always be immersed in a 6-fold triple system of order n if nnis odd, and a 12 -fold triple system of order n if n is even.n n Finally, in Chapter 8 we turn our attention to pairs of orthogonal latin squares.nHeinrich and Zhu [24] have shown that a pair of orthogonal latin squares of ordernn can be embedded in a pair of orthogonal latin squares of order t if t g 3n,nthe bound of 3n being best possible. Obtaining an analogous result for pairs ofnpartial orthogonal latin squares has proved to be an extremely challenging problem.nAlthough Lindner has proved that a pair of partial orthogonal latin squares cannalways be finitely embedded, there is no known method which obtains an embeddingnof polynomial order (with respect to the order of the partial arrays). The lessndifficult problem of embedding a single partial latin square in a pair of orthogonalnlatin squares is investigated in this chapter.

On the orthogonal Latin squares polytope

Since 1782, when Euler addressed the question of existence of a pair of orthogonal Latin squares (OLS) by stating his famous conjecture, these structures have remained an active area of research. In this paper, we examine the polyhedral aspects of OLS. In particular, we establish the dimension of the OLS polytope, describe all cliques of the underlying intersection graph and categorize them into three classes. Two of these classes are shown to induce facet-defining inequalities of Chvátal rank two. For each such class, we provide a polynomial separation algorithm of the lowest possible complexity.

An LP-based proof for the non-existence of a pair of orthogonal Latin squares of order 6

This paper presents an alternative proof for the non-existence of orthogonal Latin squares of order 6. Our method is algebraic, rather than enumerative, and applies linear programming in order to obtain appropriate dual vectors. The proof is achievable only after extending previously known results for symmetry elimination.

Permutation Polynomials Modulo 2w

We give an exact characterization of permutation polynomials modulo n=2w, w≥2: a polynomial P(x)=a0+a1x +···+adxd with integral coefficients is a permutation polynomial modulo n if and only if a1 is odd, (a2+a4+a6+···) is even, and (a3+a5+a7+···) is even. We also characterize polynomials defining latin squares modulo n=2w, but prove that polynomial multipermutations (that is, a pair of polynomials defining a pair of orthogonal latin squares) modulo n=2wdo not exist.

The Non-Existence of Finite Projective Planes of Order 10

A finite projective plane of order n, with n > 0, is a collection of n2+ n + 1 lines and n2+ n + 1 points such that1. every line contains n + 1 points,2. every point is on n + 1 lines,3. any two distinct lines intersect at exactly one point, and4. any two distinct points lie on exactly one line.It is known that a plane of order n exists if n is a prime power. The first value of n which is not a prime power is 6. Tarry [18] proved in 1900 that a pair of orthogonal latin squares of order 6 does not exist, which by Bose's 1938 result [3] implies that a projective plane of order 6 does not exist.

Existence of orthogonal latin squares with aligned subsquares

It is shown that for both v and n even, v > n > 0, there exists a pair of orthogonal latin squares of order v with an aligned subsquare of order n if and only if v⩾3 n, v≠6, n≠2, 6. This is the final case in showing that the above result is true for all v≠6 and for all n≠2, 6. When n=6, the analogous result is obtained for incomplete arrays; the case n=2 having been completed in earlier work.

A short proof of the nonexistence of a pair of orthogonal latin squares of order six

A pair of orthogonal Latin squares of order six do not exist.

Orthogonal latin squares with subsquares

Denote by LS( v, n) a pair of orthogonal latin squares of side v with orthogonal subsquares of side n. It is proved by using a generalized singular direct product that for every odd integer n ⩾ 304 or every even integer n ⩾ 304 in some infinite families, an LS( v, n) exists if and only if v ⩾ 3 n. It is also proved that for every integer n ⩾ 304, an LS( v, n) exists if v > 3 n + 6.

Doubly diagonal orthogonal latin squares

A pair of doubly diagonal orthogonal latin squares of order n, DDOLS( n), is a pair of orthogonal latin squares of order n with the property that each square has a transversal on both the front diagonal (the cells {( i, i):1⩽ i⩽ n}) and the back diagonal (the cells {( i, n + 1− i): 1⩽ i⩽ n}). We show that for all n except n = 2, 3, 6, 10, 12, 14, 15, 18 and 26, there exists a pair of DDOLS( n). Obbviously these do not exist when n = 2, 3 and 6.

Some results on construction of orthogonal Latin squares by the method of sum composition

A method of sum composition for construction of orthogona Latin squares was introduced by A. Hedayat and E. Seiden [1]. In this paper we exhibit procedures for constructing a pair of orthogonal Latin squares of size p α + 4 for primes of the form 4 m + 1 or p ≡ 1, 2, 4 mod 7. We also show that for any p > 2 n and n even one can construct and orthogonal pair of Latin squares of size p α + n using the method of sum composition. We observe that the restriction xy = 1 used by Hedayat and Seiden is sometimes necessary.

Sub-latin squares and incomplete orthogonal arrays

The main result of this paper is that for any pair of orthogonal Latin squares of side k, there will exist for all sufficiently large n a pair of orthogonal Latin squares with the first pair as orthogonal sub-squares. The orthogonal array corresponding to a set of pairwise orthogonal Latin squares, minus the sub-array corresponding to orthogonal sub-squares is called an incomplete orthogonal array; this concept is generalized slightly.

Construction of doubly diagonalized orthogonal latin squares

A transversal T of a latin square is a collection of n cells no two in the same row or column and such that each of the integers 1, 2, …, n appears in exactly one of the cells of T. A latin square is doubly diagonalized provided that both its main diagonal and off-diagonal are transversals. Although it is known that a doubly diagonalized latin square of every order n ≥ 4 exists and that a pair of orthogonal latin squares of order n exists for every n ≠ 2 or 6, it is still an open question as to what the spectrum is for pairs of doubly diagonalized orthogonal latin squares. The best general result seems to be that pairs of orthogonal doubly diagonalized latin squares of order n exist whenever n is odd or a multiple of 4, except possibly when n is a multiple of 3 but not of 9. In this paper we give a new construction for doubly diagonalized latin squares which is used to enlarge the known class for doubly diagonalized orthogonal squares. The construction is based on Sade's singular direct product of quasigroups.