4,450 research outputs found
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
Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square
An alternating sign matrix, or ASM, is a -matrix where the
nonzero entries in each row and column alternate in sign. We generalize this
notion to hypermatrices: an hypermatrix is an
{\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by
fixing one of the three indices, is an ASM. Several results concerning ASHMs
are shown, such as finding the maximum number of nonzeros of an ASHM, and properties related to Latin squares. Moreover, we
investigate completion problems, in which one asks if a subhypermatrix can be
completed (extended) into an ASHM. We show several theorems of this type.Comment: 39 page
A Novel Latin Square Image Cipher
In this paper, we introduce a symmetric-key Latin square image cipher (LSIC)
for grayscale and color images. Our contributions to the image encryption
community include 1) we develop new Latin square image encryption primitives
including Latin Square Whitening, Latin Square S-box and Latin Square P-box ;
2) we provide a new way of integrating probabilistic encryption in image
encryption by embedding random noise in the least significant image bit-plane;
and 3) we construct LSIC with these Latin square image encryption primitives
all on one keyed Latin square in a new loom-like substitution-permutation
network. Consequently, the proposed LSIC achieve many desired properties of a
secure cipher including a large key space, high key sensitivities, uniformly
distributed ciphertext, excellent confusion and diffusion properties,
semantically secure, and robustness against channel noise. Theoretical analysis
show that the LSIC has good resistance to many attack models including
brute-force attacks, ciphertext-only attacks, known-plaintext attacks and
chosen-plaintext attacks. Experimental analysis under extensive simulation
results using the complete USC-SIPI Miscellaneous image dataset demonstrate
that LSIC outperforms or reach state of the art suggested by many peer
algorithms. All these analysis and results demonstrate that the LSIC is very
suitable for digital image encryption. Finally, we open source the LSIC MATLAB
code under webpage https://sites.google.com/site/tuftsyuewu/source-code.Comment: 26 pages, 17 figures, and 7 table
- …