6,527 research outputs found
Restricted linear congruences
In this paper, using properties of Ramanujan sums and of the discrete Fourier
transform of arithmetic functions, we give an explicit formula for the number
of solutions of the linear congruence ,
with (), where
() are arbitrary integers. As a consequence, we derive necessary and
sufficient conditions under which the above restricted linear congruence has no
solutions. The number of solutions of this kind of congruence was first
considered by Rademacher in 1925 and Brauer in 1926, in the special case of
. Since then, this problem has been studied, in
several other special cases, in many papers; in particular, Jacobson and
Williams [{\it Duke Math. J.} {\bf 39} (1972), 521--527] gave a nice explicit
formula for the number of such solutions when . The problem is very well-motivated and has found intriguing
applications in several areas of mathematics, computer science, and physics,
and there is promise for more applications/implications in these or other
directions.Comment: Journal of Number Theory, to appea
On an almost-universal hash function family with applications to authentication and secrecy codes
Universal hashing, discovered by Carter and Wegman in 1979, has many
important applications in computer science. MMH, which was shown to be
-universal by Halevi and Krawczyk in 1997, is a well-known universal
hash function family. We introduce a variant of MMH, that we call GRDH,
where we use an arbitrary integer instead of prime and let the keys
satisfy the
conditions (), where are
given positive divisors of . Then via connecting the universal hashing
problem to the number of solutions of restricted linear congruences, we prove
that the family GRDH is an -almost--universal family of
hash functions for some if and only if is odd and
. Furthermore, if these conditions are
satisfied then GRDH is -almost--universal, where is
the smallest prime divisor of . Finally, as an application of our results,
we propose an authentication code with secrecy scheme which strongly
generalizes the scheme studied by Alomair et al. [{\it J. Math. Cryptol.} {\bf
4} (2010), 121--148], and [{\it J.UCS} {\bf 15} (2009), 2937--2956].Comment: International Journal of Foundations of Computer Science, to appea
The dependency diagram of a mixed integer linear programme
The Dependency Diagram of a Linear Programme (LP) shows how the successive inequalities of an LP depend on former inequalities, when variables are projected out by Fourier- Motzkin Elimination. This is explained in a paper referenced below. The paper, given here, extends the results to the Mixed Integer case (MILP). It is shown how projection of a MILP leads to a finite disjunction of polytopes. This is expressed as a set of inequalities (mirroring those in the LP case) augmented by correction terms with finite domains which are subject to linear congruences
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