**Amortized Complexity of Information-Theoretically Secure MPC Revisited**

*Ignacio Cascudo and Ronald Cramer and Chaoping Xing and Chen Yuan*

**Abstract: **A fundamental and widely-applied paradigm due to Franklin and Yung (STOC 1992) on Shamir-secret-sharing based general $n$-player MPC shows how one may trade the adversary threshold $t$ against amortized communication complexity, by using a so-called packed version of Shamir's scheme. For e.g.~the BGW-protocol (with active security), this trade-off means that if $t + 2k -2 < n/3$, then $k$ parallel evaluations of the same arithmetic circuit on different inputs can be performed at the overall cost corresponding to a single BGW-execution.

In this paper we propose a novel paradigm for amortized MPC that offers a different trade-off, namely with the size of the field of the circuit which is securely computed, instead of the adversary threshold. Thus, unlike the Franklin-Yung paradigm, this leaves the adversary threshold unchanged. Therefore, for instance, this paradigm may yield constructions enjoying the maximal adversary threshold $\lfloor (n-1)/3 \rfloor$ in the BGW-model (secure channels, perfect security, active adversary, synchronous communication).

Our idea is to compile an MPC for a circuit over an extension field to a parallel MPC of the same circuit but with inputs defined over its base field and with the same adversary threshold. Key technical handles are our notion of reverse multiplication-friendly embeddings (RMFE) and our proof, by algebraic-geometric means, that these are constant-rate, as well as efficient auxiliary protocols for creating ``subspace-randomness'' with good amortized complexity. In the BGW-model, we show that the latter can be constructed by combining our tensored-up linear secret sharing with protocols based on hyper-invertible matrices á la Beerliova-Hirt (or variations thereof). Along the way, we suggest alternatives for hyper-invertible matrices with the same functionality but which can be defined over a large enough constant size field, which we believe is of independent interest.

As a demonstration of the merits of the novel paradigm, we show that, in the BGW-model and with an optimal adversary threshold $\lfloor (n-1)/3 \rfloor$, it is possible to securely compute a binary circuit with amortized complexity $O(n)$ of bits per gate per instance. Known results would give $n \log n$ bits instead. By combining our result with the Franklin-Yung paradigm, and assuming a sub-optimal adversary (i.e., an arbitrarily small $\epsilon>0$ fraction below 1/3), this is improved to $O(1)$ bits instead of $O(n)$.

**Category / Keywords: **cryptographic protocols / multiparty computation, amortization, information-theoretical security, multiplication-friendly embeddings

**Original Publication**** (with minor differences): **IACR-CRYPTO-2018

**Date: **received 4 May 2018, last revised 6 Jun 2018

**Contact author: **ignacio at math aau dk

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

**Note: **Accepted to CRYPTO 2018. This is the final version submitted by the authors to the IACR and to Springer-Verlag on the 3rd June 2018.

**Version: **20180606:153405 (All versions of this report)

**Short URL: **ia.cr/2018/429

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