**On secret sharing with nonlinear product reconstruction**

*Ignacio Cascudo and Ronald Cramer and Diego Mirandola and Carles Padro and Chaoping Xing*

**Abstract: **Multiplicative linear secret sharing is a fundamental notion in the area of secure multi-party computation (MPC) and,
since recently, in the area of two-party cryptography as well. In a nutshell, this notion guarantees that
``the product of two secrets is obtained as a linear function of the vector consisting of the
coordinate-wise product of two respective share-vectors''. This paper focuses on the following foundational question, which is novel to the best of our knowledge. Suppose we {\em abandon the latter linearity condition} and instead require that this product is obtained by {\em some},
not-necessarily-linear ``product reconstruction function''. {\em Is the resulting notion equivalent to
multiplicative linear secret sharing?} We show the (perhaps somewhat counter-intuitive) result that this relaxed notion is strictly {\em more general}.
Concretely, fix a finite field $\FF_q$ as the base field
over which linear secret sharing is considered.
Then we show there exists an (exotic) linear secret sharing
scheme with an unbounded number of players $n$
such that it has $t$-privacy with $t = \Omega(n)$
and such that it does admit a product reconstruction function,
yet this function is {\em necessarily} nonlinear.
In addition, we determine the minimum number of
players for which those exotic schemes exist.
Our proof is based on
combinatorial arguments involving quadratic forms. It extends to similar separation results for important variations,
such as strongly multiplicative secret sharing.

**Category / Keywords: **foundations / (arithmetic) secret sharing

**Original Publication**** (in the same form): **To appear in SIAM J. Discrete Mathematics.

**Date: **received 15 Aug 2013, last revised 9 Apr 2015

**Contact author: **ignacio at cs au dk

**Available format(s): **PDF | BibTeX Citation

**Note: **Updated publication info.

**Version: **20150409:093536 (All versions of this report)

**Short URL: **ia.cr/2013/503

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