415 research outputs found
Combinatorial characterizations of algebraic manipulation detection codes involving generalized difference families
This paper provides a mathematical analysis of optimal algebraic manipulation detection
(AMD) codes. We prove several lower bounds on the success probability of an adversary and
we then give some combinatorial characterizations of AMD codes that meet the bounds with
equality. These characterizations involve various types of generalized difference families. Constructing
these difference families is an interesting problem in its own right
Hadamard partitioned difference families and their descendants
If is a Hadamard difference set (HDS) in , then
is clearly a partitioned
difference family (PDF). Any -PDF will be said of Hadamard-type
if as the one above. We present a doubling construction which,
starting from any such PDF, leads to an infinite class of PDFs. As a special
consequence, we get a PDF in a group of order and three
block-sizes , and , whenever we have a
-HDS and the maximal prime power divisors of are
all greater than
Disjoint difference families and their applications
Difference sets and their generalisations to difference families arise from the study of designs and many other applications. Here we give a brief survey of some of these applications, noting in particular the diverse definitions of difference families and the variations in priorities in constructions. We propose a definition of disjoint difference families that encompasses these variations and allows a comparison of the similarities and disparities. We then focus on two constructions of disjoint difference families arising from frequency hopping sequences and showed that they are in fact the same. We conclude with a discussion of the notion of equivalence for frequency hopping sequences and for disjoint difference families
Non-disjoint strong external difference families can have any number of sets
Strong external difference families (SEDFs) are much-studied combinatorial
objects motivated by an information security application. A well-known
conjecture states that only one abelian SEDF with more than 2 sets exists. We
show that if the disjointness condition is replaced by non-disjointness, then
abelian SEDFs can be constructed with more than 2 sets (indeed any number of
sets). We demonstrate that the non-disjoint analogue has striking differences
to, and connections with, the classical SEDF and arises naturally via another
coding application
Existence and non-existence results for Strong External Difference Families
We consider strong external difference families (SEDFs); these are external difference
families satisfying additional conditions on the patterns of external differences that occur,
and were first defined in the context of classifying optimal strong algebraic manipulation detection
codes. We establish new necessary conditions for the existence of (n, m, k, �)-SEDFs; in
particular giving a near-complete treatment of the � = 2 case. For the case m = 2, we obtain a
structural characterization for partition type SEDFs (of maximum possible k and �), showing
that these correspond to Paley partial difference sets. We also prove a version of our main
result for generalized SEDFs, establishing non-trivial necessary conditions for their existence
Circular external difference families, graceful labellings and cyclotomy
(Strong) circular external difference families (which we denote as CEDFs and
SCEDFs) can be used to construct nonmalleable threshold schemes. They are a
variation of (strong) external difference families, which have been extensively
studied in recent years. We provide a variety of constructions for CEDFs based
on graceful labellings (-valuations) of lexicographic products , where denotes a cycle of length .
SCEDFs having more than two subsets do not exist. However, we can construct
close approximations (more specifically, certain types of circular algebraic
manipulation detection (AMD) codes) using the theory of cyclotomic numbers in
finite fields
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