Paper 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.
Metadata
- Available format(s)
-
PDF
- Category
- Foundations
- Publication info
- Preprint. MINOR revision.
- Keywords
- Boolean functionLinear coefficientTotal degreeResiliency
- Contact author(s)
- qcwang @ fudan edu cn
- History
- 2020-01-03: received
- Short URL
- https://ia.cr/2020/002
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2020/002,
author = {Qichun Wang},
title = {On a Conjecture of O'Donnell},
howpublished = {Cryptology ePrint Archive, Paper 2020/002},
year = {2020},
note = {\url{https://eprint.iacr.org/2020/002}},
url = {https://eprint.iacr.org/2020/002}
}
