On Walsh Spectrum of Cryptographic Boolean Function

Shashi Kant Pandey; B. K. Dass

doi:10.14429/dsj.67.10638

Shashi Kant Pandey Department of Mathematics, University of Delhi, Delhi – 110 007
B. K. Dass Department of Mathematics, University of Delhi, Delhi – 110 007

DOI: https://doi.org/10.14429/dsj.67.10638

Keywords: Generalised bent function, Regular bent function, Diophantine equations

Abstract

Walsh transformation of a Boolean function ascertains a number of cryptographic properties of the Boolean function viz, non-linearity, bentness, regularity, correlation immunity and many more. The functions, for which the numerical value of Walsh spectrum is fixed, constitute a class of Boolean functions known as bent functions. Bent functions possess maximum possible non-linearity and therefore have a significant role in design of cryptographic systems. A number of generalisations of bent function in different domains have been proposed in the literature. General expression for Walsh transformation of generalised bent function (GBF) is derived. Using this condition, a set of Diophantine equations whose solvability is a necessary condition for the existence of GBF is also derived. Examples to demonstrate how these equations can be utilised to establish non-existence and regularity of GBFs is presented.

Author Biographies

Shashi Kant Pandey, Department of Mathematics, University of Delhi, Delhi – 110 007

Mr Shashi Kant Pandey has obtained his Master of science (Mathematics) in 2009, from Banaras Hindu University, Varanasi and currently pursuing his Ph.D. from Department of Mathematics, Delhi University. His area of interest include: Cryptography, Boolean function, Information theory, Combinatorics and discrete mathematics.

B. K. Dass, Department of Mathematics, University of Delhi, Delhi – 110 007

Dr B.K. Dass is a retired professor and Head, Department of Mathematics and dean, Faculty of Mathematical sciences of University of Delhi. He earned two research degrees viz. Ph.D. in 1975 and D.Sc. in 1983. His research interests include: Coding theory, Information theory, Cryptography, Applied algebra and discrete mathematics. He has published over 100 research papers and edited 7 books.