Möbius transforms, coincident Boolean functions and non-coincidence property of Boolean functions

Abstract

Boolean functions and their Möbius transforms are involved in logical calculation, digital communications, coding theory and modern cryptography. So far, little is known about the relations of Boolean functions and their Möbius transforms. This work is composed of three parts. In the first part, we present relations between a Boolean function and its Möbius transform so as to convert the truth table/algebraic normal form (ANF) to the ANF/truth table of a function in different conditions. In the second part, we focus on the special case when a Boolean function is identical to its Möbius transform. We call such functions coincident. In the third part, we generalize the concept of coincident functions and indicate that any Boolean function has the coincidence property even it is not coincident.

Keywords:

• Boolean functions
• Möbius transform
• coincident functions
• h-non-coincident functions
• anti-coincident functions
• non-coincident weight

2000 AMS Subject Classifications :

• 06E30
• 94A60
• 94C10

Acknowledgements

The first author was supported by the Australian Research Council Grant DP0987734. The second author was supported by the Australian Research Council Grant DP0665035, the Singapore National Research Foundation Grant NRF-CRP2-2007-03 and the Singapore Ministry of Education Research Grant T206B2204. All the authors would like to thank the anonymous referees whose comments have helped improve this paper.

Notes

^†Part of this work was presented in Boolean Functions: Cryptography and Application 2007 (BFCA'07) 5.