17 research outputs found
On A Cryptographic Identity In Osborn Loops
This study digs out some new algebraic properties of an Osborn loop that will
help in the future to unveil the mystery behind the middle inner mappings
of an Osborn loop. These new algebraic properties, will open our eyes
more to the study of Osborn loops like CC-loops which has received a
tremendious attention in this and VD-loops whose study is yet
to be explored. In this study, some algebraic properties of non-WIP Osborn
loops have been investigated in a broad manner. Huthnance was able to deduce
some algebraic properties of Osborn loops with the WIP i.e universal weak
WIPLs. So this work exempts the WIP. Two new loop identities, namely left self
inverse property loop(LSIPL) identity and right self inverse property
loop(RSLPL) are introduced for the first time and it is shown that in an Osborn
loop, they are equivalent. A CC-loop is shown to be power associative if and
only if it is a RSLPL or LSIPL. Among the few identities that have been
established for Osborn loops, one of them is recognized and recommended for
cryptography in a similar spirit in which the cross inverse property has been
used by Keedwell following the fact that it was observed that Osborn loops that
do not have the LSIP or RSIP or 3-PAPL or weaker forms of inverse property,
power associativity and diassociativity to mention a few, will have cycles(even
long ones). These identity is called an Osborn cryptographic identity(or just a
cryptographic identity).Comment: 10 pages, submitted for publicatio
ON MIDDLE UNIVERSAL -INVERSE QUASIGROUPS AND THEIR APPLICATIONS TO CRYPTOGRAPHY
This study presents a special type of middle isotopism under which
-inverse quasigroups are isotopic invariant. A sufficient
condition for an -inverse quasigroup that is specially isotopic
to a quasigroup to be isomorphic to the quasigroup isotope is
established. It is shown that under this special type of middle
isotopism, if is a positive even integer, then, a quasigroup is
an -inverse quasigroup with an inverse cycle of length if
and only if its quasigroup isotope is an -inverse quasigroup with
an inverse cycle of length . But when is an odd positive
integer. Then, if a quasigroup is an -inverse quasigroup with an
inverse cycle of length , its quasigroup isotope is an
-inverse quasigroup with an inverse cycle of length if and
only if the two quasigroups are isomorphic. Hence, they are
isomorphic -inverse quasigroups. Explanations and procedures are
given on how these results can be used to apply -inverse
quasigroups to cryptography, double cryptography and triple
cryptography
ON MIDDLE UNIVERSAL WEAK AND CROSS INVERSE PROPERTY LOOPS WITH EQUAL LENGHT OF INVERES CYCLES
This study presents a special type of middle isotopism under which
the weak inverse property(WIP) is isotopic invariant in loops. A
sufficient condition for a WIPL that is specially isotopic to a loop
to be isomorphic to the loop isotope is established. Cross inverse
property loops(CIPLs) need not satisfy this sufficient condition. It
is shown that under this special type of middle isotopism, if is
a positive even integer, then a WIPL has an inverse cycle of length
if and only if its isotope is a WIPL with an inverse cycle of
length . But, when is an odd positive integer. If a loop or
its isotope is a WIPL with only and inverse cycles of length
, its isotope or the loop is a WIPL with only and inverse
cycles of length if and only if they are isomorphic. So, that
both are isomorphic CIPLs. Explanations and procedures are given on
how these results can be used to apply CIPLs to cryptography
The Quasigroup Block Cipher and its Analysis
This thesis discusses the Quasigroup Block Cipher (QGBC) and its analysis. We first present the basic form of the QGBC and then follow with improvements in memory consumption and security. As a means of analyzing the system, we utilize tools such as the NIST Statistical Test Suite, auto and crosscorrelation, then linear and algebraic cryptanalysis. Finally, as we review the results of these analyses, we propose improvements and suggest an algorithm suitable for low-cost FPGA implementation