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Full-Text Articles in Discrete Mathematics and Combinatorics
G-Perfect Nonlinear Functions, James A. Davis, Laurent Poinsot
G-Perfect Nonlinear Functions, James A. Davis, Laurent Poinsot
Department of Math & Statistics Faculty Publications
Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with G-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of G-perfect nonlinear functions, both Z2 -valued and Za …
Some Recent Developments In Difference Sets, James A. Davis, Jonathan Jedwab
Some Recent Developments In Difference Sets, James A. Davis, Jonathan Jedwab
Department of Math & Statistics Faculty Publications
There are five known parameter families for (v, k, λ, n)- difference sets satisfying gcd(v, n)>1: the Hadamard, McFarland, Spence, Davis-Jedwab, and Chen families. The authors recently gave a recursive unifying construction for difference sets from the first four families which relies on relative difference sets. We give an overview of this construction and show that, by modifying it to use divisible difference sets in place of relative difference sets, the recent difference set discoveries of Chen can be brought within the unifying framework. We also demonstrate the recursive use of an auxiliary construction for …
Hadamard Difference Sets In Nonabelian 2-Groups With High Exponent, James A. Davis, Joel E. Iiams
Hadamard Difference Sets In Nonabelian 2-Groups With High Exponent, James A. Davis, Joel E. Iiams
Department of Math & Statistics Faculty Publications
Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Hadamard difference sets. In the abelian case, a group of order 22t + 2 has a difference set if and only if the exponent of the group is less than or equal to 2t + 2. In a previous work (R. A. Liebler and K. W. Smith, in “Coding Theory, Design Theory, Group Theory: Proc. of the Marshall Hall Conf.,” Wiley, New York, 1992), the authors constructed a difference set in a nonabelian group of order …
A Nonexistence Result For Abelian Menon Difference Sets Using Perfect Binary Arrays, K. T. Arasu, James A. Davis, Jonathan Jedwab
A Nonexistence Result For Abelian Menon Difference Sets Using Perfect Binary Arrays, K. T. Arasu, James A. Davis, Jonathan Jedwab
Department of Math & Statistics Faculty Publications
A Menon difference set has the parameters (4N2, 2N2-N, N2-N). In the abelian case it is equivalent to a perfect binary array, which is a multi-dimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. Suppose that the abelian group H×K×Zpα contains a Menon difference set, where p is an odd prime, |K|=pα, and pj≡−1 (mod exp (H)) for some j. Using the viewpoint of perfect binary arrays we prove that K must be cyclic. A …
Research Announcement: Recursive Construction For Families Of Difference Sets, James A. Davis, Jonathan Jedwab
Research Announcement: Recursive Construction For Families Of Difference Sets, James A. Davis, Jonathan Jedwab
Department of Math & Statistics Faculty Publications
A (v, k, λ) difference set is a k-element subset D of a group G of order v for which the multiset {d1d2-1 : d1, d2, ∈ D} contains each nonzero element of G exactly λ times; n = k-λ.
New Constructions Of Menon Difference Sets, K. T. Arasu, James A. Davis, Jonathan Jedwab, Surinder K. Sehgal
New Constructions Of Menon Difference Sets, K. T. Arasu, James A. Davis, Jonathan Jedwab, Surinder K. Sehgal
Department of Math & Statistics Faculty Publications
Menon difference sets have parameters (4N2, 2N2 − N, N2 − N). These have been constructed for N = 2a3b, 0 ⩽ a,b, but the only known constructions in abelian groups require that the Sylow 3-subgroup be elementary abelian (there are some nonabelian examples). This paper provides a construction of difference sets in higher exponent groups, and this provides new examples of perfect binary arrays.
Difference Sets In Abelian 2-Groups, James A. Davis
Difference Sets In Abelian 2-Groups, James A. Davis
Department of Math & Statistics Faculty Publications
Examples of difference sets are given for large classes of abelian groups of order 22d + 2. This fills in the gap of knowledge between Turyn's exponent condition and Dillon's rank condition. Specifically, it is shown thatℤ/(2d)×ℤ/(2d+2) andℤ/(2d+1)×Z/(2d+1) both admit difference sets, and these have many implications.
A Note On Intersection Numbers Of Difference Sets, K. T. Arasu, James A. Davis, Dieter Jungnickel, Alexander Pott
A Note On Intersection Numbers Of Difference Sets, K. T. Arasu, James A. Davis, Dieter Jungnickel, Alexander Pott
Department of Math & Statistics Faculty Publications
We present a condition on the intersection numbers of difference sets which follows from a result of Jungnickel and Pott [3]. We apply this condition to rule out several putative (non-abelian) difference sets and to correct erroneous proofs of Lander [4] for the nonexistence of (352, 27, 2)- and (122, 37, 12)-difference sets.