516 research outputs found
Latin Squares and Their Applications to Cryptography
A latin square of order-n is an n x n array over a set of n symbols such that every symbol appears exactly once in each row and exactly once in each column. Latin squares encode features of algebraic structures. When an algebraic structure passes certain latin square tests , it is a candidate for use in the construction of cryptographic systems. A transversal of a latin square is a list of n distinct symbols, one from each row and each column. The question regarding the existence of transversals in latin squares that encode the Cayley tables of finite groups is far from being resolved and is an area of active investigation. It is known that counting the pairs of permutations over a Galois field ��pd whose point-wise sum is also a permutation is equivalent to counting the transversals of a latin square that encodes the addition group of ��pd. We survey some recent results and conjectures pertaining to latin squares and transversals. We create software tools that generate latin squares and count their transversals. We confirm previous results that cyclic latin squares of prime order-p possess the maximum transversal counts for 3 ≤ p ≤ 9. Furthermore, we create a new algorithm that uses these prime order-p cyclic latin squares as building blocks to construct super-symmetric latin squares of prime power order-pd with d \u3e 0; using this algorithm we accurately predict that super-symmetric latin squares of order-pd possess the confirmed maximum transversal counts for 3 ≤ pd ≤ 9 and the estimated lower bound on the maximum transversal counts for 9 \u3c pd ≤ 17. Also, we give some conjectures regarding the number of transversals in a super-symmetric latin square. Lastly, we use the super-symmetric latin square for the additive group of the Galois field (��32, +) to create a simplified version of Grøstl, an iterated hash function, where the compression function is built from two fixed, large, distinct permutations
On the number of transversals in latin squares
The logarithm of the maximum number of transversals over all latin squares of
order is greater than
On the number of transversals in a class of Latin squares
Denote by the Latin square of order formed by the Cayley table of the additive group , where is an odd prime and is a positive integer. It is shown that for each there exists such that for all sufficiently large , the number of transversals in exceeds
Parity of transversals of Latin squares
We introduce a notion of parity for transversals, and use it to show that in
Latin squares of order , the number of transversals is a multiple of
4. We also demonstrate a number of relationships (mostly congruences modulo 4)
involving , where is the number of diagonals of a given
Latin square that contain exactly different symbols.
Let denote the matrix obtained by deleting row and column
from a parent matrix . Define to be the number of transversals
in , for some fixed Latin square . We show that for all and . Also, if has odd order then the
number of transversals of equals mod 2. We conjecture that for all .
In the course of our investigations we prove several results that could be of
interest in other contexts. For example, we show that the number of perfect
matchings in a -regular bipartite graph on vertices is divisible by
when is odd and . We also show that for all , when is an integer matrix of odd
order with all row and columns sums equal to
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