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
Creative Commons Attribution
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},
      url = {https://eprint.iacr.org/2020/002}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.