A Greedy De ning Set is a set of entries in a Latin square with the property that when the square is systematically lled in with a greedy algorithm, the greedy algorithm succeeds. Let g(n) be the smallest Greedy De ning Set for any Latin square of order n. We give theorems on the upper bounds of g(n) and a table listing upper bounds of g(n) for small values of n. For a circulant Latin square, w...

This talk’s focus is on the computational complexity of completing partial Latin squares. Our first goal in this talk is to show that for many special classes of partial Latin squares, we can construct completions for these objects in polynomial time. From here, we will prove the fairly surprising claim that completing an arbitrary partial Latin square is an NP-complete task; so while it is “ea...

Hardware-Software co-simulation of a multiple image encryption technique shall be described in this paper. Our proposed multiple image encryption technique is based on the Latin Square Image Cipher (LSIC). First, a carrier image that is based on the Latin Square is generated by using 256-bits of length key. The XOR operation is applied between an input image and the Latin Square Image to genera...

Hardware-Software co-simulation of a multiple image encryption technique shall be described in this paper. Our proposed multiple image encryption technique is based on the Latin Square Image Cipher (LSIC). First, a carrier image that is based on the Latin Square is generated by using 256-bits of length key. The XOR operation is applied between an input image and the Latin Square Image to genera...

A k-plex is a selection of kn entries of a latin square of order n in which each row, column and symbol is represented precisely k times. A transversal of a latin square corresponds to the case k = 1. A k-plex is said to be indivisible if it contains no c-plex for any 0 < c < k. We prove that if n = 2km for integers k ≥ 2 and m ≥ 1 then there exists a latin square of order n composed of 2m disj...

A latin square of order n is an n×n array of n symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of n entries containing no pair of entries that share the same row, column or symbol. Transversals are closely related to the notions of complete mappings and orthomorphisms in (quasi)groups, and are fundamental to the concept of mutuall...

The branch group of a strongly controllable group code is a shift group. We show that a shift group can be characterized in a very simple way. In addition it is shown that if a strongly controllable group code is labeled with Latin squares, then the shift group is solvable. Moreover the mathematical structure of a Latin square (as a translation net) and of a shift group are shown to be closely ...

A critical set in an n× n array is a set C of given entries, such that there exists a unique extension of C to an n× n Latin square and no proper subset of C has this property. For a Latin square L, scs(L) denotes the size of the smallest critical set of L, and scs(n) is the minimum of scs(L) over all Latin squares L of order n. We find an upper bound for the number of partial Latin squares of ...