234 research outputs found
Exterior splashes and linear sets of rank 3
In \PG(2,q^3), let be a subplane of order that is exterior to
\li. The exterior splash of is defined to be the set of
points on \li that lie on a line of . This article investigates
properties of an exterior \orsp\ and its exterior splash. We show that the
following objects are projectively equivalent: exterior splashes, covers of the
circle geometry , Sherk surfaces of size , and
\GF(q)-linear sets of rank 3 and size . We compare our construction
of exterior splashes with the projection construction of a linear set. We give
a geometric construction of the two different families of sublines in an
exterior splash, and compare them to the known families of sublines in a
scattered linear set of rank 3
The tangent splash in \PG(6,q)
Let B be a subplane of PG(2,q^3) of order q that is tangent to .
Then the tangent splash of B is defined to be the set of q^2+1 points of
that lie on a line of B. In the Bruck-Bose representation of
PG(2,q^3) in PG(6,q), we investigate the interaction between the ruled surface
corresponding to B and the planes corresponding to the tangent splash of B. We
then give a geometric construction of the unique order--subplane determined
by a given tangent splash and a fixed order--subline.Comment: arXiv admin note: substantial text overlap with arXiv:1303.550
Projective Aspects of the AES Inversion
We consider the nonlinear function used in the Advanced Encryption
Standard (AES). This nonlinear function is essentially inversion in
the finite field \GF (2^8), which is most naturally considered as a
projective transformation. Such a viewpoint allows us to demonstrate
certain properties of this AES nonlinear function. In particular, we
make some comments about the group generated by such transformations,
and we give a characterisation for the values in the AES
{\em Difference} or XOR {\em Table} for the AES nonlinear function and
comment on the geometry given by this XOR Table
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