Higher-order nonlinearity of Kasami functions

Abstract

The rth-order nonlinearity of a Boolean function is an important cryptographic criterion in analysing the security of stream ciphers and block ciphers. In this paper, we compute the lower bounds on the (r=d)th-order nonlinearity of Kasami Boolean function , where k=2^2d −2 ^d +1. We also compare the values of lower bound obtained in a theorem in this paper to the values of general lower bound obtained by Carlet [Recursive lower bounds on the nonlinearity profile of Boolean functions and their applications, IEEE Trans. Inform. Theory 54(3) (2008), pp. 1262–1272]. It is also shown that our lower bound is better than the lower bound obtained by Carlet.

Keywords:

• Boolean functions
• higher-order derivatives
• higher-order nonlinearity
• Kasami functions

2000 AMS Subject Classifications:

• 94A60
• 94C10
• 06E30

Acknowledgements

The authors are thankful to the anonymous referees for many valuable suggestions that improved the technical and editorial quality of the manuscript. The first author thanks Sugata Gangopadhyay for guidance and several helpful discussions to improve the quality of manuscript. The first author is also thankful to the Ministry of Human and Research Development, New Delhi, India, for financial support to carry out the above work.