254 research outputs found
Public Key Cryptography based on Semigroup Actions
A generalization of the original Diffie-Hellman key exchange in
found a new depth when Miller and Koblitz suggested that such a protocol could
be used with the group over an elliptic curve. In this paper, we propose a
further vast generalization where abelian semigroups act on finite sets. We
define a Diffie-Hellman key exchange in this setting and we illustrate how to
build interesting semigroup actions using finite (simple) semirings. The
practicality of the proposed extensions rely on the orbit sizes of the
semigroup actions and at this point it is an open question how to compute the
sizes of these orbits in general and also if there exists a square root attack
in general. In Section 2 a concrete practical semigroup action built from
simple semirings is presented. It will require further research to analyse this
system.Comment: 20 pages. To appear in Advances in Mathematics of Communication
On Integer Images of Max-plus Linear Mappings
Let us extend the pair of operations (max,+) over real numbers to matrices in
the same way as in conventional linear algebra. We study integer images of
max-plus linear mappings. The question whether Ax (in the max-plus algebra) is
an integer vector for at least one x has been studied for some time but
polynomial solution methods seem to exist only in special cases. In the
terminology of combinatorial matrix theory this question reads: is it possible
to add constants to the columns of a given matrix so that all row maxima are
integer? This problem has been motivated by attempts to solve a class of
job-scheduling problems. We present two polynomially solvable special cases
aiming to move closer to a polynomial solution method in the general case
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