Cryptology ePrint Archive: Report 2020/002

On a Conjecture of O'Donnell

Qichun Wang

Abstract: Let $f:\{-1,1\}^n\rightarrow \{-1,1\}$ be with total degree $d$, and $\widehat{f}(i)$ be the linear Fourier coefficients of $f$. The relationship between the sum of linear coefficients and the total degree is a foundational problem in theoretical computer science. In 2012, O'Donnell Conjectured that \[ \sum_{i=1}^n \widehat{f}(i)\le d\cdot {d-1 \choose \lfloor\frac{d-1}{2}\rfloor}2^{1-d}. \] In this paper, we prove that the conjecture is equivalent to a conjecture on the cryptographic Boolean function. We then prove that the conjecture is true for $d=1,n-1$. Moreover, we count the number of $f$'s such that the upper bound is achieved.

Category / Keywords: foundations / Boolean function, Linear coefficient, Total degree, Resiliency

Date: received 2 Jan 2020

Contact author: qcwang at fudan edu cn

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Version: 20200103:073913 (All versions of this report)

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