Browsing by Author "Keskar, Pradipkumar H."

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Algebraic methods in difference sets and bent functions
(JACODESMATH, 2021) Keskar, Pradipkumar H.
We provide some applications of a polynomial criterion for difference sets. These include counting the difference sets with specified parameters in terms of Hilbert functions, in particular a count of bent functions. We also consider the question about the bentness of certain Boolean functions introduced by Carlet when the $\mathcal{C}$-condition introduced by him doesn't hold.
Descent principle in modular Galois theory
(IAS, 2001-05) Keskar, Pradipkumar H.
We propound a descent principle by which previously constructed equations over GF.qn/.X/ may be deformed to have incarnations over GF.q/.X/ without changing their Galois groups. Currently this is achieved by starting with a vectorial (= additive) q-polynomial of q-degreemwith Galois group GL.m; q/ and then, under suitable conditions, enlarging its Galois group to GL.m; qn/ by forming its generalized iterate relative to an auxiliary irreducible polynomial of degree n. Elsewhere this was proved under certain conditions by using the classification of finite simple groups, and under some other conditions by using Kantor’s classification of linear groups containing a Singer cycle. Now under different conditions we prove it by using Cameron-Kantor’s classification of two-transitive linear groups.
Patterson-Wiedemann Construction Revisited
(Elsevier, 2003-05) Keskar, Pradipkumar H.
In 1983, Patterson and Wiedemann constructed Boolean functions on n = 15 input variables having nonlinearity strictly greater than 2n−1 − 2n−1/2. Construction of Boolean functions on odd number of variables with such high nonlinearity was not known earlier and also till date no other construction method of such functions is known. We note that the Patterson-Wiedemann construction can be understood in terms of interleaved sequences as introduced by Gong in 1995. We show that the Patterson-Wiedemann functions can be described as repetitions of a particular binary string. As example we elaborate the cases for n = 15,21. Under this framework, we map the problem of finding Patterson-Wiedemann functions into a problem of solving a system of linear inequalities over the set of integers and provide proper reasoning about the choice of the orbits. This, in turn, reduces the search space. Similar analysis also reduces the complexity of calculating generalized non-linearity for such functions. In an attempt to understand the above construction from the group theoretic view point, we characterize the group of all G-F(2)-linear transformations of GF(2ab) which acts on PG(2, 2a).
Polynomial Criterion for Abelian Difference Sets
(Springer, 2020-03) Keskar, Pradipkumar H.
Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the special structure of the group ring of an Abelian group to establish a one-to one correspondence of the class of difference sets with specific parameters in that group with the set of all complex solutions of a specified system of polynomial equations. The correspondence also develops some tests for a Boolean function to be a bent function.