Paper 2014/516

On the Classification of Finite Boolean Functions up to Fairness

Nikolaos Makriyannis

Abstract

Two parties, $P_1$ and $P_2$, wish to jointly compute some function $f(x,y)$ where $P_1$ only knows $x$, whereas $P_2$ only knows $y$. Furthermore, and most importantly, the parties wish to reveal only what the output suggests. Function $f$ is said to be computable with \emph{complete fairness} if there exists a protocol computing $f$ such that whenever one of the parties obtains the correct output, then both of them do. The only protocol known to compute functions with complete fairness is the one of Gordon et al (STOC 2008). The functions in question are finite, Boolean, and the output is shared by both parties. The classification of such functions up to fairness may be a first step towards the classification of all functionalities up to fairness. Recently, Asharov (TCC 2014) identifies two families of functions that are computable with fairness using the protocol of Gordon et al and another family for which the protocol (potentially) falls short. Surprisingly, these families account for almost all finite Boolean functions. In this paper, we expand our understanding of what can be computed fairly with the protocol of Gordon et al. In particular, we fully describe which functions the protocol computes fairly and which it (potentially) does not. Furthermore, we present a new class of functions for which fair computation is outright impossible. Finally, we confirm and expand Asharov's observation regarding the fairness of finite Boolean functions: almost all functions $f:X\times Y\rightarrow \{0,1\}$ for which $|X|\neq |Y|$ are fair, whereas almost all functions for which $|X|= |Y|$ are not.

Metadata
Available format(s)
PDF
Category
Cryptographic protocols
Publication info
Published elsewhere. SCN 2014
Keywords
Complete FairnessSecure Two-Party Computation
Contact author(s)
nikolaos makriyannis @ upf edu
History
2014-07-02: received
Short URL
https://ia.cr/2014/516
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2014/516,
      author = {Nikolaos Makriyannis},
      title = {On the Classification of Finite Boolean Functions up to Fairness},
      howpublished = {Cryptology ePrint Archive, Paper 2014/516},
      year = {2014},
      note = {\url{https://eprint.iacr.org/2014/516}},
      url = {https://eprint.iacr.org/2014/516}
}
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