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Recent Advances in Function and Homomorphic Secret Sharing

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Book cover Progress in Cryptology – INDOCRYPT 2017 (INDOCRYPT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10698))

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Abstract

Function Secret Sharing (FSS) and Homomorphic Secret Sharing (HSS) are two extensions of standard secret sharing, which support rich forms of homomorphism on secret shared values.

  • An m-party FSS scheme for a given function family \({\mathcal {F}}\) enables splitting a function \(f: \{0,1\}^n \rightarrow \mathbb {G}\) from \({\mathcal {F}}\) (for Abelian group \(\mathbb {G}\)) into m succinctly described functions \(f_1,\dots ,f_m\) such that strict subsets of the \(f_i\) hide f, and \(f(x) = f_1(x) + \cdots + f_m(x)\) for every input x.

  • An m-party HSS is a dual notion, where an input x is split into shares \(x^1,\dots ,x^m\), such that strict subsets of \(x^i\) hide x, and one can recover the evaluation P(x) of a program P on x given homomorphically evaluated share values \(\mathsf{Eval}(x^1,P),\dots ,\mathsf{Eval}(x^m,P)\).

In the last few years, many new constructions and applications of FSS and HSS have been discovered, yielding implications ranging from efficient private database manipulation and secure computation protocols, to worst-case to average-case reductions.

In this treatise, we introduce the reader to the background required to understand these developments, and give a roadmap of recent advances (up to October 2017).

Supported in part by ISF grant 1861/16, AFOSR Award FA9550-17-1-0069, and ERC Grant no. 307952.

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Notes

  1. 1.

    Function vs. program: Note that in FSS we will consider simple classes of functions where each function has a unique description, whereas in HSS we consider functions with many programs computing it. For this reason we refer to “function” for FSS and “program” for HSS.

  2. 2.

    \(f_i\) vs. \(x^i\): We maintain the subscript/superscript conventions of existing works (primarily [6, 11]). Note that superscript notation is used in HSS where one can consider shares of multiple inputs, \(x_j \mapsto (x_j^1,\dots ,x_j^m)\).

  3. 3.

    Namely, for each new x, the parties will first use their shared randomness to coordinately rerandomize the garbled circuit of f and input labels, respectively.

  4. 4.

    In this case, we think of the function F and all HSS algorithms \(\mathsf{Share},\mathsf{Eval},\mathsf{Dec}\) as implicitly receiving a description of \(\mathbb {G}\) as an additional input.

  5. 5.

    In particular: \(\lambda + n(\lambda + 2)\) for \(\lambda \)-bit outputs, and \(\lambda + n(\lambda +2) - \lfloor \log \lambda \rfloor \) for 1-bit outputs.

  6. 6.

    Recall in HSS the secret share size scales with input size and not function description size.

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Acknowledgements

Tremendous thanks to my FSS/HSS partners in crime, Niv Gilboa and Yuval Ishai, and to additional coauthors on the presented works: Geoffroy Couteau, Huijia Rachel Lin, Michele Orrù, and Stefano Tessaro.

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  1. IDC Herzliya, Herzliya, Israel

    Elette Boyle

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  1. Elette Boyle
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Correspondence to Elette Boyle .

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Editors and Affiliations

  1. Indian Institute of Science (IISc), Bengaluru, India

    Arpita Patra

  2. Department of Computer Science, University of Bristol, Bristol, United Kingdom

    Nigel P. Smart

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Boyle, E. (2017). Recent Advances in Function and Homomorphic Secret Sharing. In: Patra, A., Smart, N. (eds) Progress in Cryptology – INDOCRYPT 2017. INDOCRYPT 2017. Lecture Notes in Computer Science(), vol 10698. Springer, Cham. https://doi.org/10.1007/978-3-319-71667-1_1

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