K. J. Horadam

RMIT University, Melbourne, Victoria, Australia

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Publications (34)14.22 Total impact

  • K.J. Horadam, M.A. Nyblom
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    ABSTRACT: We construct a new family of normalised metrics for measuring the dissimilarity of finite sets in terms of the sizes of the sets and of their intersection. The family normalises a set-based analogue of the Minkowski metric family. It is parametrised by a real variable p≥1p≥1, is monotonic decreasing in pp, equals the normalised set difference metric when p=1p=1 and equals the normalised maximum difference metric in the limit p→∞p→∞. These metrics are suitable for comparison of finite sets in any context. Several applications to comparison of finite graphs are described.
    Discrete Applied Mathematics 01/2013; · 0.72 Impact Factor
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    ABSTRACT: This paper presents an automatic retina verification framework based on the biometric graph matching (BGM) algorithm. The retinal vasculature is extracted using a family of matched filters in the frequency domain and morphological operators. Then, retinal templates are defined as formal spatial graphs derived from the retinal vasculature. The BGM algorithm, a noisy graph matching algorithm, robust to translation, non-linear distortion, and small rotations, is used to compare retinal templates. The BGM algorithm uses graph topology to define three distance measures between a pair of graphs, two of which are new. A support vector machine (SVM) classifier is used to distinguish between genuine and imposter comparisons. Using single as well as multiple graph measures, the classifier achieves complete separation on a training set of images from the VARIA database (60% of the data), equaling the state-of-the-art for retina verification. Because the available data set is small, kernel density estimation (KDE) of the genuine and imposter score distributions of the training set are used to measure performance of the BGM algorithm. In the one dimensional case, the KDE model is validated with the testing set. A 0 EER on testing shows that the KDE model is a good fit for the empirical distribution. For the multiple graph measures, a novel combination of the SVM boundary and the KDE model is used to obtain a fair comparison with the KDE model for the single measure. A clear benefit in using multiple graph measures over a single measure to distinguish genuine and imposter comparisons is demonstrated by a drop in theoretical error of between 60% and more than two orders of magnitude.
    IEEE Transactions on Image Processing 01/2013; 22(9):3625-3635. · 3.20 Impact Factor
  • J. Jeffers, S.A. Davis, K.J. Horadam
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    ABSTRACT: The lack of large public retina image databases means it is difficult to judge the relative merits of the retina biometric itself, of different scoring functions, or of potential biocryptographic constructs. We derive conservative theoretical genuine and imposter score distributions for feature point based retina templates by normal kernel density estimation. We base them on 147 images from the VARIA database and use 7 scoring functions. This allows us to infer EERs in the range 0.3%-1.3% and, for FNMRs of less than 10%, entropy estimates between 65 bits and 200 bits.
    Biometrics (ICB), 2012 5th IAPR International Conference on; 01/2012
  • D. G. Farmer, K. J. Horadam
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    ABSTRACT: We show by explicit construction that the equivalence classes of multiplicative central (p n , p n , p n , 1)-RDSs relative to \mathbb Zpn{\mathbb Z}_p^n in groups E with E/\mathbb Zpn @ \mathbb ZpnE/{\mathbb Z}_p^n \cong {\mathbb Z}_p^n are in one-to-one correspondence with the strong isotopism classes of presemifields of order p n . We also show there are 1,446 equivalence classes of central (16, 16, 16, 1)-RDS relative to \mathbb Z24{\mathbb Z}_2^4, in groups E for which E/\mathbb Z24 @ \mathbb Z24E/{\mathbb Z}_2^4 \cong {\mathbb Z}_2^4. Only one is abelian. KeywordsRelative difference set–Equivalence class–Presemifield
    Cryptography and Communications 01/2011; 3(1):17-28.
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    A. Arakala, S. A. Davis, K. J. Horadam
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    ABSTRACT: We represent the retina vessel pattern as a spatial relational graph, and match features using error-correcting graph matching. We study the distinctiveness of the nodes (branching and crossing points) compared with that of the edges and other substructures (nodes of degree k, paths of length k). On a training set from the VARIA database, we show that as well as nodes, three other types of graph sub-structure completely or almost completely separate genuine from imposter comparisons. We show that combining nodes and edges can improve the separation distance. We identify two retina graph statistics, the edge-to-node ratio and the variance of the degree distribution, that have low correlation with node match score.
    01/2011;
  • K. J. Horadam
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    ABSTRACT: Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals and splitting relative difference sets (RDSs) and introduce an intermediate relation, canonical equivalence, to aid in distinguishing the classes. We identify very precisely the conditions under which a graph equivalence determines a bundle equivalence, using transversals and extensions. We derive a new and easily computed algebraic measure of nonlinearity for a function f, calculated from the image of its coboundary ∂f. This measure is preserved by bundle equivalence but not by the coarser equivalences. It takes its minimum value if f is a homomorphism, and takes its maximum value if the graph of f contains a splitting RDS. KeywordsEquivalent functions–Graph of function–Finite field polynomial–Linear equivalence–Relative difference set–Nonlinear function–APN function
    Journal of Algebraic Combinatorics 01/2011; 35(3):477-496. · 0.63 Impact Factor
  • Security and Communication Networks. 01/2011; 4:563-576.
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    ABSTRACT: Point-pattern matching of minutiae is the most common method used in fingerprint biometrics, but it is generally insufficient by itself. It has particular limitations in matching partial prints or in secure (biocryptographic) matching. Here, we add structure with a new spatial graph represention of a fingerprint, with minutiae as nodes. Using a sample of fingerprint graphs extracted from the FVC2002 database, we show that matching fingerprints using only the edges of the graphs performs almost as well as using only the nodes. Combinations of edges and nodes have superior performance to either individual score.
    2011 International Conference on Digital Image Computing: Techniques and Applications (DICTA), Noosa, QLD, Australia, December 6-8, 2011; 01/2011
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    Alain LeBel, D. L. Flannery, K. J. Horadam
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    ABSTRACT: Theshift actionon the 2-cocycle groupZ2(G,C)of afinite groupGwith coefficients in afinitely generated abelian group C has several useful applications in combinatorics and digital communications, arising from the invariance of a uniform distribution property of cocycles under the action. In this paper we study the shift orbit structure of the coboundary subgroup B2(G,C) of Z2(G,C). The study is placed within a well- known setting involving the Loewy and socle series of a group algebra over G. We prove new bounds on the dimensions of terms in such series. Asymptotic results on the size of shift orbits are also derived; for example, if C is an elementary abelian p-group, then almost all shift orbits in B2(G,C) are maximal-sized for large enough finite p-groups G of certain classes.
    Journal of Pure and Applied Algebra. 07/2010; 214(7).
  • K. J. Horadam
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    ABSTRACT: For cryptographic purposes, we want to find functions with both low differential uniformity and dissimilarity to all linear functions and to know when such functions are essentially different. For vectorial Boolean functions, extended affine equivalence and the coarser Carlet–Charpin–Zinoviev (CCZ) equivalence are both used to distinguish between nonlinear functions. It remains hard to tell when CCZ equivalent functions are EA-inequivalent. This paper presents a framework for solving this problem in full generality, for functions between arbitrary finite groups. This common framework is based on relative difference sets (RDSs). The CCZ and EA equivalence classes of perfect nonlinear (PN) functions are each derived, by quite different processes, from equivalence classes of splitting semiregular RDSs. By generalizing these processes, we obtain a much strengthened formula for all the graph equivalences which define the EA equivalence class of a given function, amongst those which define its CCZ equivalence class. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 260–273, 2010
    Journal of Combinatorial Designs 03/2010; 18(4):260 - 273. · 0.69 Impact Factor
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    Jason Jeffers, Arathi Arakala, K. J. Horadam
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    ABSTRACT: This paper studies the amount of distinctive information contained in a privacy protecting and compact template of a retinal image created from the locations of crossings and bifurcations in the choroidal vasculature, otherwise called feature points. Using a training set of 20 different retina, we build a template generator that simulates one million imposter comparisons and computes the number of imposter retina comparisons that successfully matched at various thresholds. The template entropy thus computed was used to validate a theoretical model of imposter comparisons. The simulator and the model both estimate that 20 bits of entropy can be achieved by the feature point-based template. Our results reveal the distinctiveness of feature point-based retinal templates, hence establishing their potential as a biometric identifier for high security and memory intensive applications.
    20th International Conference on Pattern Recognition, ICPR 2010, Istanbul, Turkey, 23-26 August 2010; 01/2010
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    ABSTRACT: We compare two vessel extraction methods for creation of a retina template, using a database of 20 images of normal retinas. Each vessel in a well defined region is represented by a three dimensional feature, from which a retina template is built. Based on the sample distributions, we propose a preliminary theoretical model to predict the entropy of a retina template. We analyse by experimental and theoretical means the entropy present, and infer that entropy from our retina template compares sufficiently favourably with that of a minutia-based fingerprint template to warrant further study.
    06/2009: pages 1250-1259;
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    A. Arakala, K.J. Horadam, S. Boztas
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    ABSTRACT: We illustrate the steps in building a secure authentication system using a construct called PinSketch where the identifier is a set of points derived from the minutiae extracted from a fingerprint. We define a relative set difference based threshold for matching templates, that takes the sizes of the sets being compared into account and show that it performs better than a fixed threshold for all fingerprints in a database. We describe the process of selecting the quantisation parameters and the design of the authentication system based on the PinSketch construct. The main question asked, and answered, is: At the operating parameters chosen, is the authentication system secure? We determine an empirical measure of the entropy of the quantised minutiae-based template and demonstrate how an attacker can exploit a non uniform distribution of the template elements to masquerade as a genuine user. We finally list desirable requirements for a template if it has to be protected using a set difference based construct like PinSketch.
    Biometrics Symposium, 2008. BSYM '08; 10/2008
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    D. G. FARMER, K. J. HORADAM
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    ABSTRACT: We derive bivariate polynomial formulae for cocycles and coboundaries in Z2(pn,pn), and a basis for the (pn−1−n)-dimensional GF(pn)-space of coboundaries. When p=2 we determine a basis for the -dimensional GF(2n)-space of cocycles and show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form.
    Journal of the Australian Mathematical Society 09/2008; 85(02):177 - 190. · 0.45 Impact Factor
  • Conference Proceeding: Presemifield bundles over GF(p3)
    D.G. Farmer, K.J. Horadam
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    ABSTRACT: We analyse the numbers and representatives of all bundles of presemifields in the isotopism class of GF(p<sup>3</sup>), for application to the search for Hadamard codes and differentially uniform functions.
    Information Theory, 2008. ISIT 2008. IEEE International Symposium on; 08/2008
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    Alain LeBel, K. J. Horadam
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    ABSTRACT: Determining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by Nyberg on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg's result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non-cyclic abelian groups and use it to find all the orthogonal cocycles over Z2t, 2 ≤ t ≤ 4. We conjecture that any orthogonal cocycle over Z2t, t ≥ 2, must be multiplicative. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 173–181, 2008
    Journal of Combinatorial Designs 03/2008; 16(3):173 - 181. · 0.69 Impact Factor
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    K. J. Horadam, D. G. Farmer
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    ABSTRACT: Bundles are equivalence classes of functions derived from equivalence classes of transversals. They preserve measures of resistance to differential and linear cryptanalysis. For functions over GF(2 n ), affine bundles coincide with EA-equivalence classes. From equivalence classes (“bundles”) of presemifields of order p n , we derive bundles of functions over GF(p n ) of the form λ(x)*ρ(x), where λ, ρ are linearised permutation polynomials and * is a presemifield multiplication. We prove there are exactly p bundles of presemifields of order p 2 and give a representative of each. We compute all bundles of presemifields of orders p n ≤ 27 and in the isotopism class of GF(32) and we measure the differential uniformity of the derived λ(x)*ρ(x). This technique produces functions with low differential uniformity, including PN functions (p odd), and quadratic APN and differentially 4-uniform functions (p = 2).
    Designs Codes and Cryptography 01/2008; 49:79-94. · 0.78 Impact Factor
  • K. J. Horadam
    Arithmetic of Finite Fields, 2nd International Workshop, WAIFI 2008, Siena, Italy, July 6-9, 2008, Proceedings; 01/2008
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    Arathi Arakala, Jason Jeffers, K. J. Horadam
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    ABSTRACT: We propose an authentication scheme using fingerprint biometrics, protected by a construct called a Fuzzy Extractor. We look at a new way of quantizing and digitally representing the minutiae measurements so that a construct called PinSketch can be applied to the minutiae. This is converted to a Fuzzy Extractor by tying some random information to the minutiae measurements. We run a matching algorithm at chosen quantization parameters and show that the authentication accuracy is within acceptable limits. We demonstrate that our authentication system succeeds in protecting the users’ identity.
    Advances in Biometrics, International Conference, ICB 2007, Seoul, Korea, August 27-29, 2007, Proceedings; 01/2007
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    K.J. Horadam, A. Rao
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    ABSTRACT: The class of generalised Hadamard transforms includes the Fourier, generalised, discrete Fourier, Walsh-Hadamard, complex Hadamard and reverse jacket transforms. The generalised Hadamard transforms may by partly classified by signal length, by group of entries in the transform matrix and by a recently introduced third parameter, the jacket width of the transform matrix. Here we introduce a weighted trace map, which realises the Fourier transform as an exponential weighted sum of Galois ring traces. We give examples of Fourier transforms with jacket width 0, jacket width 1 and maximum jacket width (half the signal length). We show the Fourier transforms of length 4<sup>k</sup> with entries in {plusmn1, plusmni} obtained using the weighted trace map from the Galois ring GR(4,k) have jacket width 2<sup>k-1</sup>
    Information Theory, 2006 IEEE International Symposium on; 08/2006

Publication Stats

176 Citations
40 Downloads
1k Views
14.22 Total Impact Points

Institutions

  • 2000–2013
    • RMIT University
      • • School of Mathematical and Geospatial Sciences
      • • Department of Mathematical Sciences
      Melbourne, Victoria, Australia
    • University of Vic
      Vic, Catalonia, Spain
  • 1998–2006
    • Melbourne Institute of Technology
      Melbourne, Victoria, Australia