In this series of five lectures, we explore some of the many problems
in cryptography where combinatorial designs have played a substantial
role. To set the stage, we review some ideas from classical cryptography
and cryptanalysis, and briefly outline the major advances in public key
cryptography. Using this foundation, we examine problems in authentication,
secret sharing and threshold schemes, key distribution schemes, and visual
cryptography. Our theme is how requirements for balance and regularity
lead to wellstudied combinatorial designs; how the cryptographic problem
leads to challenging combinatorial problems; and, in a few cases, how
combinatorial existence theorems provide useful information in the
application. With this in mind, the focus is on applicable mathematics.
The lectures are accessible to those without substantial background in either cryptography or combinatorial designs.
We give five talks on the mathematics associated with aspects of
computer security including
complexity,
elliptic curves,
secret sharing,
authentication and
bent functions.
The talks start from first principles and move to recent research in the area.
The first three lectures provide a brief introduction to coding theory
and its links to combinatorial designs in the spirit of Chapter ``Codes''
of the CRC Handbook of Combinatorial Designs, with mentioning some new results
that have occurred after the publication of the handbook.
The third lecture discusses a class of codes arising
from graphs, with an application to quantum errorcorrecting
codes. The fifth lecture describes how linear perfect codes are used to
characterize the classical geometric designs by the minimum dimension
of their codes. This characterization generalizes a result by
Hamada and Ohmori (1975) for the binary case, and proves a conjecture
of Assmus.
We consider Hadamard matrices with the quasi3
property, and their implications
for the existence of certain quasisymmetric designs, strongly
regular graphs, and codes meeting the GreyRankin bound.
These questions lead naturally to
an interesting infinite family of parameters,
and we pose some existence questions.
In this talk, we discuss the development of group testing
algorithms, erasurecorrecting codes, and codes for spreadspectrum
optical communications that all arise from triple systems in which
certain configurations of triples are forbidden. Computational methods
and recursive techniques for the construction of triple systems avoiding
the specified configurations are outlined, and some existence theorems
for weakly unionfree systems are established.
Nonexistence of generalized Hadamard Matrices over groups is
investigated by means of a necessary condition of Brock, which may be
found in CRC Handbook of Combinatorial Design. Methods for
establishing the lack of nontrivial solutions to quadratic
diophantine equations are used to obtain sequences { t_{n} } for
which Butson Hadamard Matrices BH(p, pt_{n}) do not exist, where p > 2
is a prime.
In the last decade the theory of signed groups, the theory of
cocyclic designs, and the application of these theories to Hadamard
matrices have been developing independently. In both cases, exciting new
results have been established, and there is great promise for future
workit may not be unreasonable to say that either one of these theories
has the potential to completely revolutionize the way we think about
Hadamard matrices.
Both theories are grounded in the same basic facts in the projective (and linear) representation theory of groups, but with different approaches, language and notation. Therefore, it is natural to ask: to what degree are the two theories equivalent or analogous to each other? To what extent are they different or complementary? In what ways can the two theories contribute to each other, or form a larger, more comprehensive picture?
I shall make what I believe to be a first effort at addressing some of these questions.
In a projective geometry, PG(m,2), of dimension m over GF(2),
we define a
skew karc S to be a set of k points such that there are at most two
points of each line in S, and the diagonals of any 4 points of S do not
meet. I will discuss the relationship between skew karcs and linear
binary codes. Special attention is paid to the situation where the codes
has minimum distance 5.
Any conference matrix or Hadamard matrix may be used in a
particular way to obtain a group divisible design of twice the order.
A conference matrix or Hadamard matrix D is said to be cocyclic if
there is a group G, a 2cocycle
f:G×G®{0,±1}, and a map a:G®{0,±1} such
that D may be written in the form

In both cases the conference matrix or Hadamard matrix is cocyclic over G with cocycle f and extension group E if and only if the automorphism group of the corresponding group divisible design has a regular subgroup isomorphic to E and containing the involution which interchanges the points in each group. Any such regular subgroup yields a relative difference set in the extension group E.
In this paper we completely characterize all relative difference sets associated with Paley conference matrix and the Paley type I Hadamard matrix. We accomplish this by exhibiting a one to one correspondence between the regular subgroups and certain near fields. Our characterization follows from the fact that all finite near fields are known. This allows us to determine all the groups over which the Paley matrices above are cocyclic.
Our arguments apply in several exceptional near fields. We include some examples.
A Hadamard matrix H of order n is an n×n matrix whose
entries are all equal to ±1 and which satisfies

In the mid seventies, Seberry proved that for any odd integer m > 0, there is a Hadamard matrix of order 2^{a}m for any integer a > 2log_{2}m.
More recently, researchers have formed the opinion that Hadamard matrices with additional structure exist. Any such matrix corresponds to or yields (a) a binary 2cocycle with specific combinatorial properties, (b) a group divisible design with a regular group action, (c) a normal relative difference set with specific parameters, and (d) a Hadamard group in the sense of Ito. The discovery of these connections has spurred a new generation of researchers to reexamine the wealth of constructions and ideas developed over the past thirty years with an eye to finding and studying new classes of the related objects.
As part of this process the speaker has proved, with Michael J. Smith, that there is a cocyclic Hadamard matrix of order 2^{a} m for any odd integer m > 0 and any integer a > 8log_{2}m. A corollary to the argument is the existence of a very large class of maximal size normal relative difference sets with forbidden subgroup generated by a central involution. In particular, any group of odd squarefree order m may be embedded as a direct factor into a Hadamard group of order 2^{é8log2mù}m.
Two sequences are said to be compatible if their periodic
autocorrelation functions sum to a constant. We show that a
pair of sequences are compatible if and only if the squared
magnitudes of corresponding terms in their discrete Fourier
transforms (DFTs) also sum to a constant. It follows that
the DFT of any sequence that belongs to a compatible pair is
limited in magnitude. This limit may thus be employed as a
test to reduce the number of sequences for consideration in
the search for compatible pairs. We call this the power
spectral density (PSD) test.
The effectiveness of the PSD test is demonstrated by searching for compatible pairs of 0,1 sequences of odd length l and Hamming weight w = (l+1)/2. Such pairs are equivalent to 2(l;w,w;w) supplementary difference sets and can be used to construct Hadamard matrices of order 2(l+1) with a two block circulant core. Exhaustive searches were performed for l £ 45. For l = 45, the PSD test reduced the number of candidate sequences by a factor of 625. Of those passing the test, only about 1 per 1000 belong to compatible pairs. The number of compatible pairs appears to be growing somewhat less than exponentially with l, growing approximately as 2^{l/3} in the range 29 £ l £ 45. Unfortunately, the probability of finding such pairs by random search appears to be decreasing exponentially. The exhaustive searches confirm that the vast majority of sequences in compatible pairs are uniquely compatible with just one other sequence, not counting cyclic shifts and reversals.
In the seventies it was shown that all possible 3tuples are
the types of orthogonal designs of order 24 except
(4,4,15), (7,7,7) and (7,8,8).
We will present a new method to construct all these together will all
remaining full 4tuples, 18 in all.
A colorful version of Plotkin arrays of order 24, 40 and 56 is shown.
Using these and some new arrays, we show the existence of
an infinite family of new orthogonal designs.
Many codes and sequences designed for robust or secure
communications are built from Hadamard matrices or from
related symmetric block designs or difference sets.
If an alphabet larger than {0,1} is required, the natural extension is to generalised Hadamard matrices (GHM), with entries in a group C: that is, v ×v matrices H satisfying

An abelian GHM which is also cocyclic, is equivalent to a semiregular central relative difference set and to a divisible design with a regular group of automorphisms, class regular with respect to the forbidden central subgroup. Therefore the code and sequence construction techniques for Hadamard matrices are applicable to the general case.
In the first talk I will introduce cocycles and their properties, give some familiar examples of this unfamiliar concept and demonstrate the equivalence of the abovementioned objects.
In the second talk I will present recent results on the theory of cocyclic GHM and their applications to cocyclic Hadamard codes and perfect arrays.
For given integers v,k, and t such that v > k > t, let
P_{tk}^{v} be (v  t) by (v  k) (0,1) matrices whose rows
are indexed by the tsubsets of a vset S, whose columns are indexed by
the ksubsets B of S, and the entry P_{tk}^{v}(A,B) of row A and
column B is 1 if A Ì B and 0 otherwise.
In this lecture, we consider the null space of P_{tk}^{v} over GF(q), denoted by C_{tk}^{v}(q). The elements of C_{tk}^{v}(q) are called trades over GF(q). We discuss the combinatorial properties of these objects and through that we study the code space formed by them. Specifically, by examining the case C_{23}^{v}(3), we characterize the codewords of small weights and for v º 0 (mod 3), the spectrum of weights is obtained. A very simple algorithm for decoding is presented and the automorphism group of these codes are discussed. Also the family of C_{1k}^{v}(2) is completely characterized and some optimal codes are found.
The present paper represents a continuation of work done by
Bömer and Antweiler (``Periodic complementary binary sequences'',
IEEE Trans. on Information Theory, Vol. 36,
pp. 14871494, Nov. 1990).
We give direct and recursive constructions of
aperiodic and periodic complementary sequences.
Using these constructions, many missing entries in
the table of
Bömer and Antweiler can be filled.
Let X denote the set of all ksubsets of a fixed set V with size v,
and give this set some extra structure by defining two ksubsets x and
y to be at distance i if x Çy = ki. A simple tdesign
on v points is just a subset of X satisfying a regular condition. Let
W denote the set of all ntuples (words of length n) of a fixed
alphabet set S of size q, and define two words x and y to be at
distance i if x and y differ in i coordinate positions. A perfect
ecode of length n is just a subset C of W satisfying the
condition that every word in W lies at distance e or smaller from
exactly one codeword in C. If we do view a tdesign and an ecode in
this way then a number of new parameters suggest themselves.
The talk will be an introduction to the Delsarte's theory on tdesigns
and ecodes. It will survey the works of Bannai, Biggs, Delsarte, Godsil,
Ito, Munemasa and others, highlighting the use of association schemes in design
theory and coding theory.
Quasisymmetric circulant weighing matrices are circulant weighing
matrices whose pattern of zeroes is symmetric. We address the existence
question of such matrices and according to the parity of the the order
and the parity of the weight give an answer.
Let D_{2n} be a dihedral group of order 2n and Z
be the rational integer ring. Kimura gave the necessary and
sufficient conditions such that a matrix of order 8n+4 obtained from
the elements
of the group ring Z[D_{2n}] of a dihedral group D_{2n}
becomes a Hadamard matrix, where n is an odd integer.
We show that if p is an odd prime and q = 2p 1 is a prime power,
then there exists a family of Hadamard matrices of dihedral group type.
We prove this theorem by giving the elements of
Z[D_{2p}] concretely. Gauss sum over GF(p) and the
relative Gauss sum over GF(q^{2}) are important to prove the theorem.