Welcome to the resource topic for 2020/002
Title:
On a Conjecture of O’Donnell
Authors: 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.
ePrint: https://eprint.iacr.org/2020/002
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