Physical Sciences and Mathematics Commons^™

Some Results On Weighing Matrices, Jennifer Seberry, Albert Leon Whiteman

Professor Jennifer Seberry

It is shown that if q is a prime power then there exists a circulant weighing matrix of order q² + q + 1 with q² non-zero elements per row and column.

This result allows the bound N to be lowered in the theorem of Geramita and Wallis that " given a square integer k there exists an integer N dependent on k such that weighing matrices of weight k and order n and orthogonal designs (1, k) of order 2n exist for every n > N".

A Remark On The Excess Of Hadamard Matrices And Orthogonal Designs, J Hammer, R Levingston, Jennifer Seberry

Professor Jennifer Seberry

Some improved upper and lower bounds are given for the excess of Hadamard matrices. The excess of orthogonal designs is defined and discussed.

A Note On Using Sequences To Construct Orthogonal Designs, Peter J. Robinson, Jennifer Seberry

Professor Jennifer Seberry

Several constructions are given which show how to construct orthogonal designs from sequences of commuting variables with zero non-periodic auto-correlation function.

Pseudo-Random Sequence Generators Using Structured Noise, R S. Safavi-Naini, Jennifer Seberry

Professor Jennifer Seberry

Stream ciphers use the output of a Pseudo-Random (PR) generator to mask the information stream. The security of these cipher systems ultimately depends on the structure of the PR generator. There are some minimum necessary criteria such as long period, flat statistical distribution and high linear complexity that the PR generator of a stream cipher system should satisfy to resist the basic cryptanalytic attacks on such systems. We propose a class of PR generators using the coset elements of a Reed-Muller code. The linear Complexity of these generators is analysed and conditions that assure the highest possible linear complexity for …

Key Scheduling Des Type Cryptosystems, Lawrence Brown, Jennifer Seberry

Professor Jennifer Seberry

This paper reviews some possible design criteria for the key schedule in a DES style cryptosystem. The key schedule involves a Key Rotation component. and the permutation PC2. Together these provide for a diffusion of dependency of ciphertext bits on key bits. Some empirical rules which seem to account for the derivation of the key schedule used in the DES are first presented. A number of trials were run with various key schedules. and some further design rules were derived. An alternative form of key schedule was then tested. This used either a null PC2, or one in which permutations …

Orthogonal Designs: Ii, Anthony V. Geramita, Jennifer Seberry

Professor Jennifer Seberry

Orthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show that orthogonal designs of type (1, k) and order n exist for every k < n when n = 2t+2. 3 and n = 2t+2.5 (where t is a positive integer). We also find orthogonal designs that exist in every order 2n and others that exist in every order 4n. Coupled with some results of earlier work, this means that the weighing matrix conjecture 'For every order n = 0 (mod4) there is, for …

Minimal And Maximal Critical Sets In Room Squares, Ghulam Rasool Chaudhry, Jennifer Seberry

Professor Jennifer Seberry

In this paper we introduce critical sets in Room squares. We give the cardinality of the minimal critical sets (min. cs) and maximal critical sets (max. cs) for inequivalence classes of Room squares of side 7,9, and 11. We also describe algorithms to compute min. cs and max. cs and conjecture the lower and upper bounds for min. cs and max. cs.

Search Key Substitution In The Encipherment Of B-Trees, Thomas Hardjono, Jennifer Seberry

Professor Jennifer Seberry

This paper suggests an improvement to the scheme by Bayer and Metzger for the encipherment of B-Trees. Search keys are "disguised" instead of encrypted, and together with the data pointers and tree pointers which remain encrypted, prevents the opponent or attacker from recreating the correct shape of the B-Tree. Combinatorial block designs are used as a method to substitute the search keys contained within the nodes of the B-Tree. The substitution provides advantages in terms of the number of decryptions necessary to traverse the B-Tree, while the use of block designs are advantageous in terms of the small amount of …

Amicable Hadamard Matrices, Jennifer Seberry

Professor Jennifer Seberry

If X is a symmetric Hadamard matrix, Y is a skew-Hadamard matrix, and XYT is symmetric, then X and Y are said to be amicable Hadamard matrices. A construction for amicable Hadamard matrices is given, and then amicable Hadamard matrices are used to generalize a construction for skew-Hadamard matrices.

Williamson Matrices Of Even Order, Jennifer Seberry

Professor Jennifer Seberry

Recent advances in the construction of Hadamard matrices have depended on the existence of Baumert-Hall arrays and Williamson-type matrices. These latter are four (1,-1) matrices A,B,C,D, of order m, which pairwise satisfy (i) MNT = NMT, M,N E (A,B,C,D), and (ii) AAT +BBT +CCT +DDT = 4mIm, where I is the identity matrix. Currently Williamson matrices are known to exist for all orders less than 100 except: 35,39,47,53,59,65,67,70,71,73,76,77,83,89,94. This paper gives two constructions for Williamson matrices of even order, 2n. This is most significant when no Williamson matrices of order n are known. In particular we give matrices for the …

On The Design Of Permutation P In Des Type Cryptosystems, Lawrence Brown, Jennifer Seberry

Professor Jennifer Seberry

This paper reviews some possible design criteria for the permutation P in a DES style cryptosystem. These permutations provide the diffusion component in a substitution-permutation network. Some empirical rules which seem to account for the derivation of the permutation used in the DES are first presented. Then it is noted that these permutations may be regarded as latin-squares which link the outputs of S-boxes to their inputs at the next stage. A subset of these which perform well in a dependency analysis are then presented and suggested for use in future schemes.

Products Of Hadamard Matrices, Williamson Matrices And Other Orthogonal Matrices Using M-Structures, Jennifer Seberry, Mieko Yamada

Professor Jennifer Seberry

The new concept of M-structures is used to unify and generalize a number of concepts in Hadamard matrices including Williamson matrices, Goethals-Seidel matrices, Wallis-Whiteman matrices and generalized quaternion matrices. The concept is used to find many new symmetric Williamson-type matrices, both in sets of four and eight, and many new Hadamard matrices. We give as corollaries "that the existence of Hadamard matrices of orders 4g and 4h implies the existence of an Hadamard matrix of order 8gh" and "the existence of 'Williamson type matrices of orders u and v implies the existence of 'Williamson type matrices of order 2uu". This …

All Directed Bibds With K = 3 Exist, Jennifer Seberry, David Skillicorn

Professor Jennifer Seberry

A directed BIBD with parameters (v,b,r,k,λ*) is a BIBD with parameters (v, b, r, k, 2λ*) in which each ordered pair of varieties occurs together in exactly λ* blocks. It is shown that λ*v(v - 1) = 0 (mod 3) is a necessary and sufficient condition for the existence of a directed (v, b, r, k, λ*) BIBD with k = 3.

A Subliminal Channel In Codes For Authentication Without Secrecy, Jennifer Seberry

Professor Jennifer Seberry

G.J. Simmons has advanced the concept of using authentication in an open channel to actually convey information. We review the use of the knapsack problem for public codes and explore the use of Shamir's method for a signature only knapsack to convey messages.

A Construction For Orthogonal Designs With Three Variables, Jennifer Seberry

Professor Jennifer Seberry

We show how orthogonal designs OD(48p²t;16p²t, 16p²t,16p²t) can be constructed from an Hadamard matrix of order 4p and an OD(4t;t,t,t,t). This allows us to assert that OD(48p²t; 16p²t,16p²t,16p²t) exist for all t,p ≤ 102 except possibly for tє{67,71,73,77,79,83,86,89,91,97}. These designs are new.

Coloured Designs, New Group Divisible Designs And Pairwise Balanced Designs, C A. Rodger, Dinesh G. Sarvate, Jennifer Seberry

Professor Jennifer Seberry

Many new families of group divisible designs, balanced incomplete block designs and pairwise balanced designs can be obtained by using constructions based on coloured designs (CD). This paper gives one such construction in each case together with an existence theorem for coloured designs.

Sbibd(4k², 2k² + K, K² + K) And Hadamard Matrices Of Order 4k² With Maximal Excess Are Equivalent, Jennifer Seberry

Professor Jennifer Seberry

We show that an SBIBD(4k², 2k² + k, k² + k) is equivalent to a regular Hadamard matrix of order 4k² which is equivalent to an Hadamard matrix of order 4k² with maximal excess. We find many new SBIBD(4k², 2k² + k, k² + k) including those for even k when there is an Hadamard matrix of order 2k (in particular all 2k ≤ 210) and k є {1, 3, 5,...,29,33,...,41,45,51,53,61,...69,75,81,83,89,95,99,625,32m,25. 32m, m ≥ 0}.

Shared Cryptographic Bits Via Quantized Quandrature Phase Amplitudes Of Light, Yi Mu, Jennifer Seberry, Yuliang Zheng

Professor Jennifer Seberry

We propose a novel quantum cryptographic protocol without using polarized photons. The protocol consists of an optical coupler and four nonorthogonal coherent states which are analyzed by means of quadrature phase amplitudes of quantized light field.

Kronecker Products And Bibds, Jennifer Seberry

Professor Jennifer Seberry

Recursive constructions are given which permit, under conditions described in the paper, a (v, b, r, k, lambda)-configuration to be used to obtain a (v', b', r', k, lambda)-configuration. Although there are many equivalent definitions we will mean by a (v, b, r, k, lambda)-configuration or BIBD that (0, 1)-matrix A of size v x b with row sum r and column sum k satisfying AAT = (r - lambda)I + lambdaJ where, as throughout the remainder of this paper, I is the identity matrix and J the matrix with every element +1 whose sizes should be determined from the …

"Albert Leon Whiteman (1915-1995)", S. Golomb, T. Harris, Jennifer Seberry

Professor Jennifer Seberry

The Mathematical Family Tree of Hans Rademacher (with A. Whiteman branch)

Hadamard Matrices Of Order 28m, 36m, And 44m, Jennifer Seberry

Professor Jennifer Seberry

We show that if four suitable matrices of order m exist then there are Hadamard matrices of order 28 m, 36 m, and 44 m. In particular we show that Hadamard matrices of orders 14(q + 1), 18(q + 1), and 22(q + 1) exist when q is a prime power and q = l(mod 4). Also we show that if n is the order of a conference matrix there is an Hadamard matrix of order 4mn. As a consequence there are Hadamard matrices of the following orders less than 4000: 476, 532, 836, 1036, 1012, 1100, 1148, 1276, 1364, …

The Weighing Matrices Of Order 4n And Weight 4n-2 And 2n-1, Marc Gysin, Jennifer Seberry

Professor Jennifer Seberry

We give algorithms and constructions for mathematical and computer searches which allow us to establish the existence of W(4n, 4n - 2) and W (4n, 2n - 1) for many orders 4n less than 4000. We compare these results with the orders for which W(4n, 4n) and W(4n, 2n) are known. We use new algorithms based on the theory of cyclotomy to obtain new T-matrices of order 43 and JM-matrices which yield W(4n, 4n - 2) for n = 5,7,9,11,13,17,19,25,31,37,41,43,61,71,73,157.

Computer Viruses An Introduction, Jeffrey Horton, Jennifer Seberry

Professor Jennifer Seberry

Computer viruses pose a considerable problem for users of personal computers. The recent emergence of macro viruses as a problem of some importance may heighten virus awareness in general. Yet most people have little or no understanding of common anti-virus measures, the varieties of viruses that exist today, and the strategies which they use to accomplish infection and to defeat anti-viruses. It is well-known that the virus problem is most severe for users of IBM PCs and compatibles; however, users of other platforms, such as the Macintosh, should not become complacent - viruses exist for many platforms in varying numbers. …

New Weighing Matrices, Christos Koukouvinos, Jennifer Seberry

Professor Jennifer Seberry

New weighing matrices and skew weighing matrices are given for many orders 4t ≤ 100. These are constructed by finding new sequences with zero autocorrelation. These results enable us to determine for the first time that for 4t ≤ 84 a W{4t,k) exists for all k = 1, ... ,4t -1 and also that there exists a skew-weighing matrix (also written as an OD(4t;1,k)) for 4t ≤ 80, t odd, k = a2 + b2 + c2,a,b,c integers except k = 4t - 2 must be the sum of two squares.

Reduction Of Angular Momentum Expressions By Matrix Arithmetic, D. J. Newman, Jennifer Seberry

Professor Jennifer Seberry

In perturbation calculations using basis states defined in terms of spherically symmetric potentials it is often necessary to simplify complicated expressions involving n-j symbols. A well known graphical technique can be used to aid in this process. We represent the graphs by their incidence matrices, so that the algebraic manipulations can be carried out by matrix arithmetic. It is shown that the sequence of operations required to simplify a given graph can be determined from structural considerations based on the properties of certain polynomials in the adjacency matrix. This provides a method of performing complete perturbation calculations of this type …

Selected Papers In Combinatorics - A Volume Dedicated To R.G. Stanton, Jennifer Seberry, Brendan Mckay, Scott Vanstone

Professor Jennifer Seberry

Professor Stanton has had a very illustrious career. His contributions to mathematics are varied and numerous. He has not only contributed to the mathematical literature as a prominent researcher but has fostered mathematics through his teaching and guidance of young people, his organizational skills and his publishing expertise. The following briefly addresses some of the areas where Ralph Stanton has made major contributions.

Bose's Method Of Differences Applied To Construct Bhaskar Rao Designs, Jennifer Seberry

Professor Jennifer Seberry

In this paper we show that BIBD(v,b,r,k,λ) where v = pq or pq + 1, when written in the notation of Bose's method of differences may often be used to find generalized Bhaskar Rao designs GBRD(p,b',r',k,λ;G) where G is a group of order q and vice versa. This gives many new GBRDs including a GBRD(9,5,5;Z5) and a GBRD(13,7,7;Z7).

Cyclotomy, Hadamard Arrays And Supplementary Difference Sets, David C. Hunt, Jennifer Seberry

Professor Jennifer Seberry

A 4n x 4n Hadamard array, H, is a square matrix of order 4n with elements ± A, ± B, ± C, ± D each repeated n times in each row and column. Assuming the indeterminates A, B, C, D commute, the row vectors of H must be orthogonal. These arrays have been found for n = 1 (Williamson, 1944), n = 3 (Baumert-Hall, 1965), n = 5 (Welch, 1971), and some other odd n < 43 (Cooper, Hunt, Wallis). The results for n = 25, 31, 37, 41 are presented here, as is a result for n = 9 not …

A Note On Supplementary Difference Sets, Jennifer Seberry

Professor Jennifer Seberry

Let S₁, S₂,···, S_n be subsets of G, a finite abelian group of order v, containing k₁, k₂,...,kn elements respectively. Write T_i for the totality of all differences between elements of S_i (with repetitions), and T for the totality of elements of all the T_i. We will denote this by T= T₁ & T₂ & ... & T_n. If T contains each non-zero element of G a fixed number of times, lambda say, then the sets S₁, S₂, ..., …

Constructions Of Balanced Ternary Designs Based On Generalized Bhaskar Rao Designs, Dinesh G. Sarvate, Jennifer Seberry

Professor Jennifer Seberry

New series of balanced ternary designs and partially balanced ternary designs are obtained. Some of the designs in the series are non-isomorphic solutions for design parameters which were previously known or whose solution was obtained by trial and error, rather than by a systematic method.