Cryptology ePrint Archive: Report 2012/093

Recent Results on Balanced Symmetric Boolean Functions

Yingming Guo and Guangpu Gao and Yaqun Zhao

Abstract: In this paper we prove all balanced symmetric Boolean functions of fixed degree are trivial when the number of variables grows large enough. We also present the nonexistence of trivial balanced elementary symmetric Boolean functions except for $n=l\cdot2^{t+1}-1$ and $d=2^t$, where $t$ and $l$ are any positive integers, which shows Cusick's conjecture for balanced elementary symmetric Boolean functions is exactly the conjecture that all balanced elementary symmetric Boolean functions are trivial balanced. In additional, we obtain an integer $n_0$, which depends only on $d$, that Cusick's conjecture holds for any $n>n_0$.

Category / Keywords: foundations / Boolean functions, Balancedness, elementary symmetric functions

Date: received 23 Feb 2012, last revised 23 Apr 2012

Contact author: guoyingming123 at gmail com

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Version: 20120423:091026 (All versions of this report)

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