International Association
for Cryptologic Research

In this work we consider the task of designing information-theoretic MPC protocols for which the state of a given party can be recovered from a small amount of parties, a property we refer to as \emph{local repairability}. This is useful when considering MPC over dynamic settings where parties leave and join a computation, a scenario that has gained notable attention in recent literature. Thanks to the results of (Cramer \emph{et al.}~EUROCRYPT'00), designing such protocols boils down to constructing a linear secret-sharing scheme (LSSS) with good locality, that is, each share is determined by only a small amount of other shares, that also satisfies the so-called multiplicativity property. Previous constructions that achieve locality (\emph{e.g.}~using locally recoverable codes---LRCs) do not enjoy multiplicativity, and LSSS that are multiplicative (\emph{e.g.}~Shamir's secret-sharing) do not satisfy locality. Our construction bridges this literature gap by showing the existence of an LSSS that achieves both properties simultaneously. Our results are obtained by making use of well known connection between error correcting codes and LSSS, in order to adapt the LRC construction by (Tamo \& Barg, IEEE Transactions on Information Theory 2014) to turn it into a LSSS. With enough care, such coding-theoretic construction yields our desired locality property, but it falls short at satisfying multiplicativity. In order to address this, we perform an extensive analysis of the privacy properties of our scheme in order to identify parameter regimes where our construction satisfies multiplicativity. Finally, since our LSSS satisfies locality, every share is determined by a small amount of shares. However, in an MPC context it is not enough to let the (small set of) parties to send their shares to the repaired party, since this may leak more information than the regenerated share. To obtain our final result regarding MPC with local repairability, we construct a lightweight MPC protocol that performs such repairing process without any leakage. We provide both a passively secure construction (for the \emph{plain} multiplicative regime) and an actively secure one (for \emph{strong} multiplicativity).

We revisit the question of the overhead to achieve full security (i.e., guaranteed output delivery) in secure multiparty computation (MPC). Recent works have closed the gap between full security and semi-honest security, by introducing protocols where the parties first compute the circuit using a semi-honest protocol and then run a verification step with sublinear communication in the circuit size. However, the number of interaction rounds in the verification step is also sublinear in the circuit's size. Unlike communication, the round complexity typically grows with the circuit's \textit{depth} and not its size. Hence, for large but shallow circuits, this may yield a significant overhead. Motivated by this gap, we make the following contributions: (1) We present a new MPC framework to obtain full security, compatible with effectively \emph{any} ring, that has an additive communication overhead of only $O(\log |C|)$, where $|C|$ is the number of multiplication gates in the circuit, and a \textit{constant} number of additional rounds beyond the underlying semi-honest protocol. Our framework works with any linear secret sharing scheme and relies on a new to utilize the machinery of \textit{zero-knowledge fully linear interactive oracle proofs} (zk-FLIOP) in a black-box way. We present several instantiations to the building blocks of our compiler, from which we derive concretely efficient protocols in different settings. (2) We present extensions to the zk-FLIOP primitive for very general settings: one for proving statements over potentially non-commutative rings that only require certain commutative properties of its largest exceptional set; and one for proving statements over Galois Rings. For Galois rings, we present concrete improvements on the current state-of-the-art for the case of constant-round proofs, by making use of \emph{Reverse Multiplication Friendly Embeddings} (RMFEs).

Consider the task of secure multiparty computation (MPC) among n parties with perfect security and guaranteed output delivery, supporting t < n/3 active corruptions. Suppose the arithmetic circuit C to be computed is defined over a finite ring Z/qZ, for an arbitrary q ∈ Z. It is known that this type of MPC over such ring is possible, with communication that scales as O(n|C|), assuming that q scales as Ω(n). However, for constant-size rings Z/qZ where q = O(1), the communication is actually O(n log n|C|) due to the need of the so-called ring extensions. In most natural settings, the number of parties is variable but the “datatypes” used for the computation are fixed (e.g. 64-bit integers). In this regime, no protocol with linear communication exists. In this work we provide an MPC protocol in this setting: perfect security, G.O.D. and t < n/3 active corruptions, that enjoys linear communication O(n|C|), even for constant-size rings Z/qZ. This includes as important particular cases small fields such as F2, and also the ring Z/2k Z. The main difficulty in achieving this result is that widely used techniques such as linear secret-sharing cannot work over constant-size rings, and instead, one must make use of ring extensions that add Ω(log n) over- head, while packing Ω(log n) ring elements in each extension element in order to amortize this cost. We make use reverse multiplication-friendly embeddings (RMFEs) for this packing, and adapt recent techniques in network routing (Goyal et al. CRYPTO’22) to ensure this can be efficiently used for non-SIMD circuits. Unfortunately, doing this naively results in a restriction on the minimum width of the circuit, which leads to an extra additive term in communication of poly(n) · depth(C). One of our biggest technical contributions lies in designing novel techniques to overcome this limitation by packing elements that are distributed across different layers. To the best of our knowledge, all works that have a notion of packing (e.g. RMFE or packed secret-sharing) group gates across the same layer, and not doing so, as in our work, leads to a unique set of challenges and complications.

In the context of secure multiparty computation (MPC) protocols with guaranteed output delivery (GOD) for the honest majority setting, the state-of-the-art in terms of communication is the work of (Goyal et al. CRYPTO'20), which communicates O(n|C|) field elements, where |C| is the size of the circuit being computed and n is the number of parties. Their round complexity, as usual in secret-sharing based MPC, is proportional to O(depth(C)), but only in the optimistic case where there is no cheating. Under attack, the number of rounds can increase to \Omega(n^2) before honest parties receive output, which is undesired for shallow circuits with depth(C) << n^2. In contrast, other protocols that only require O(depth(C) rounds even in the worst case exist, but the state-of-the-art from (Choudhury and Patra, Transactions on Information Theory, 2017) still requires \Omega(n^4|C|) communication in the offline phase, and \Omega(n^3|C|) in the online (for both point-to-point and broadcast channels). We see there exists a tension between efficient communication and number of rounds. For reference, the recent work of (Abraham et al., EUROCRYPT'23) shows that for perfect security and t<n/3, protocols with both linear communication and O(depth(C)) rounds exist. We address this state of affairs by presenting a novel honest majority GOD protocol that maintains O(depth(C)) rounds, even under attack, while improving over the communication of the most efficient protocol in this setting by Choudhury and Patra. More precisely, our protocol has point-to-point (P2P) online communication of O(n|C|), accompanied by O(n|C|) broadcasted (BC) elements, while the offline has O(n^3|C|) P2P communication with O(n^3|C|) BC. This improves over the previous best result, and reduces the tension between communication and round complexity. Our protocol is achieved via a careful use of packed secret-sharing in order to improve the communication of existing verifiable secret-sharing approaches, although at the expense of weakening their robust guarantees: reconstruction of shared values may fail, but only if the adversary gives away the identities of many corrupt parties. We show that this less powerful notion is still useful for MPC, and we use this as a core building block in our construction. Using this weaker VSS, we adapt the recent secure-with-abort Turbopack protocol (Escudero et al. CCS'22) to the GOD setting without significantly sacrificing in efficiency.

In this work we present a novel actively secure dishonest majority MPC protocol, \textsc{SuperPack}, whose efficiency improves as the number of \emph{honest} parties increases. Concretely, let $0<\epsilon<1/2$ and consider an adversary that corrupts $t<n(1-\epsilon)$ out of $n$ parties. \textsc{SuperPack} requires $6/\epsilon$ field elements of online communication per multiplication gate across all parties, assuming circuit-dependent preprocessing, and $10/\epsilon$ assuming circuit-independent preprocessing. In contrast, most of previous works such as SPDZ (Damg\aa rd \emph{et al}, ESORICS 2013) and its derivatives perform the same regardless of whether there is only one honest party, or a constant (non-majority) fraction of honest parties. The only exception is due to Goyal \emph{et al} (CRYPTO 2022), which achieves $58/\epsilon + 96/\epsilon^2$ field elements assuming circuit-independent preprocessing. Our work improves this result substantially by a factor of at least $25$ in the circuit-independent preprocessing model. Practically, we also compare our work with the best concretely efficient online protocol Turbospeedz (Ben-Efraim \emph{et al}, ACNS 2019), which achieves $2(1-\epsilon)n$ field elements per multiplication gate among all parties. Our online protocol improves over Turbospeedz as $n$ grows, and as $\epsilon$ approaches $1/2$. For example, if there are $90\%$ corruptions ($\epsilon=0.1$), with $n=50$ our online protocol is $1.5\times$ better than Turbospeedz and with $n=100$ this factor is $3\times$, but for $70\%$ corruptions ($\epsilon=0.3$) with $n=50$ our online protocol is $3.5\times$ better, and for $n=100$ this factor is $7\times$. Our circuit-dependent preprocessing can be instantiated from OLE/VOLE. The amount of OLE/VOLE correlations required in our work is a factor of $\approx \epsilon n/2$ smaller than these required by Le Mans (Rachuri and Scholl, CRYPTO 2022) leveraged to instantiate the proprocesing of Turbospeedz. Our dishonest majority protocol relies on packed secret-sharing and leverages ideas from the honest majority \textsc{TurboPack} (Escudero \emph{et al}, CCS 2022) protocol to achieve concrete efficiency for any circuit topology, not only SIMD. We implement both \textsc{SuperPack} and Turbospeedz and verify with experimental results that our approach indeed leads to more competitive runtimes in distributed environments with a moderately large number of parties.

Secure multiparty computation protocols with dynamic parties, which assume that honest parties do not need to be online throughout the whole execution of the protocol, have recently gained a lot of traction for computations of large scale distributed protocols, such as blockchains. More specifically, in Fluid MPC, introduced in (Choudhuri et al. CRYPTO 2021), parties can dynamically join and leave the computation from round to round. The best known Fluid MPC protocol in the honest majority setting communicates O(n^2) elements per gate where n is the number of parties online at a time. While Le Mans (Rachuri and Scholl, CRYPTO 2022) extends Fluid MPC to the dishonest majority setting with preprocessing, it still communicates O(n^2) elements per gate. In this work we present alternative Fluid MPC solutions that require O(n) communication per gate for both the information-theoretic honest majority setting and the information-theoretic dishonest majority setting with preprocessing. Our solutions also achieve maximal fluidity where parties only need to be online for a single communication round. Additionally, we show that a protocol in the information-theoretic dishonest majority setting with sub-quadratic o(n^2) overhead per gate requires for each of the N parties who may ever participate in the (later) execution phase, \Omega(N) preprocessed data per gate.

In the recent work of (Cheon \& Lee, Eurocrypt'22), the concept of a \emph{degree-$D$ packing method} was formally introduced, which captures the idea of embedding multiple elements of a smaller ring into a larger ring, so that element-wise multiplication in the former is somewhat ``compatible'' with the product in the latter. Then, several optimal bounds and results are presented, and furthermore, the concept is generalized from one multiplication to degrees larger than two. These packing methods encompass several constructions seen in the literature in contexts like secure multiparty computation and fully homomorphic encryption. One such construction is the concept of reverse multiplication-friendly embeddings (RMFEs), which are essentially degree-2 packing methods. In this work we generalize the notion of RMFEs to \emph{degree-$D$ RMFEs} which, in spite of being ``more algebraic'' than packing methods, turn out to be essentially equivalent. Then, we present a general construction of degree-$D$ RMFEs by generalizing the ideas on algebraic geometry used to construct traditional degree-$2$ RMFEs which, by the aforementioned equivalence, leads to explicit constructions of packing methods. Furthermore, our theory is given in a unified manner for general Galois rings, which include both rings of the form $\mathbb{Z}_{p^k}$ and fields like $\mathbb{F}_{p^k}$, which have been treated separately in prior works. We present multiple concrete sets of parameters for degree-$D$ RMFEs (including $D=2$), which can be useful for future works. Finally, we discuss interesting applications of our RMFEs, focusing in particular on the case of non-interactively generating high degree correlations for secure multiparty computation protocols. This requires the use of Shamir secret sharing for a large number of parties, which requires large-degree Galois ring extensions. Our RMFE enables the generation of such preprocessing data over small rings, without paying for the multiplicative overhead incurred by using Galois ring extensions of large degree. For our application we also construct along the way, as a side contribution of potential independent interest, a pseudo-random secret-sharing solution for non-interactive generation of packed Shamir-sharings over Galois rings with structured secrets, inspired by the PRSS solutions from (Benhamouda \emph{et al}, TCC 2021).

In this work we present a novel actively secure multiparty computation protocol in the dishonest majority setting, where the computation domain is a ring of the type $\mathbb{Z}_{2^k}$. Instead of considering an ``extension ring'' of the form $\mathbb{Z}_{2^{k+\kappa}}$ as in SPD$\mathbb{Z}_{2^k}$ (Cramer et al, CRYPTO 2018) and its derivatives, we make use of an actual ring extension, or more precisely, a Galois ring extension $\mathbb{Z}_{p^k}[\mathtt{X}]/(h(\mathtt{X}))$ of large enough degree, in order to ensure that the adversary cannot cheat except with negligible probability. These techniques have been used already in the context of honest majority MPC over $\mathbb{Z}_{p^k}$, and to the best of our knowledge, our work constitutes the first study of the benefits of these tools in the dishonest majority setting. Making use of Galois ring extensions requires great care in order to avoid paying an extra overhead due to the use of larger rings. To address this, reverse multiplication-friendly embeddings (RMFEs) have been used in the honest majority setting (e.g.~Cascudo et al, CRYPTO 2018), and more recently in the dishonest majority setting for computation over $\mathbb{Z}_2$ (Cascudo and Gundersen, TCC 2020). We make use of the recent RMFEs over $\mathbb{Z}_{p^k}$ from (Cramer et al, CRYPTO 2021), together with adaptations of some RMFE optimizations introduced in (Abspoel et al, ASIACRYPT 2021) in the honest majority setting, to achieve an efficient protocol that only requires in its online phase $12.4k(n-1)$ bits of amortized communication complexity and one round of communication for each multiplication gate. We also instantiate the necessary offline phase using Oblivious Linear Evaluation (OLE) by generalizing the approach based on Oblivious Transfer (OT) proposed in MASCOT (Keller et al, CCS 2016). To this end, and as an additional contribution of potential independent interest, we present a novel technique using Multiplication-Friendly Embeddings (MFEs) to achieve OLE over Galois ring extensions using black-box access to an OLE protocol over the base ring $\mathbb{Z}_{p^k}$ without paying a quadratic cost in terms of the extension degree. This generalizes the approach in MASCOT based on Correlated OT Extension. Finally, along the way we also identify a bug in a central proof in MASCOT, and we implicitly present a fix in our generalized proof.

Compressed $\Sigma$-Protocol Theory (CRYPTO 2020) presents an ``alternative'' to Bulletproofs that achieves the same communication complexity while adhering more elegantly to existing $\Sigma$-protocol theory, which enables their techniques to be directly applicable to other widely used settings in the context of ``plug \& play'' algorithmics. Unfortunately, their techniques are restricted to arithmetic circuits over \emph{prime} fields, which rules out the possibility of using more machine-friendly moduli such as powers of $2$, which have proven to improve efficiency in applications. In this work we show that such techniques can be generalized to the case of arithmetic circuits modulo \emph{any} number. This enables the use of powers of $2$, which can prove to be beneficial for efficiency, but it also facilitates the use of other moduli that might prove useful in different applications. In order to achieve this, we first present an instantiation of the main building block of the theory of compressed $\Sigma$-protocols, namely compact vector commitments. Our construction, which may be of independent interest, is homomorphic modulo \emph{any} positive integer $m$, a result that was not known in the literature before. Second, we generalize Compressed $\Sigma$-Protocol Theory from finite fields to $\mathbb{Z}_m$. The main challenge here is ensuring that there are large enough challenge sets as to fulfill the necessary soundness requirements, which is achieved by considering certain ring extensions. Our techniques have direct application for example to verifiable computation on homomorphically encrypted data.

We construct the first efficient MPC protocol that only requires black-box access to a non-commutative ring $R$. Previous results in the same setting were efficient only either for a constant number of corruptions or when computing branching programs and formulas. Our techniques are based on a generalization of Shamir's secret sharing to non-commutative rings, which we derive from the work on Reed Solomon codes by Quintin, Barbier and Chabot (\textit{IEEE Transactions on Information Theory, 2013}). When the center of the ring contains a set $A = \{\alpha_0, \ldots, \alpha_n\}$ such that $\forall i \neq j, \alpha_i - \alpha_j \in R^*$, the resulting secret sharing scheme is strongly multiplicative and we can generalize existing constructions over finite fields without much trouble. Most of our work is devoted to the case where the elements of $A$ do not commute with all of $R$, but they just commute with each other. For such rings, the secret sharing scheme cannot be linear ``on both sides" and furthermore it is not multiplicative. Nevertheless, we are still able to build MPC protocols with a concretely efficient online phase and black-box access to $R$. As an example we consider the ring $\mathcal{M}_{m\times m}(\mathbb{Z}/2^k\mathbb{Z})$, for which when $m > \log(n+1)$, \enote{maybe adapt/simplify the following claim as the comparison requires some nuances} we obtain protocols that require around $\lceil\log(n+1)\rceil/2$ less communication and $2\lceil\log(n+1)\rceil$ less computation than the state of the art protocol based on Circuit Amortization Friendly Encodings (Dalskov, Lee and Soria-Vazquez, \textit{ASIACRYPT 2020}). In this setting with a ``less commutative" $A$, our black-box preprocessing phase has a less practical complexity of $\poly(n)$. Due to this, we additionally provide specialized, concretely efficient preprocessing protocols for $R = \mathcal{M}_{m\times m}(\mathbb{Z}/2^k\mathbb{Z})$ that exploit the structure of the matrix ring.

In 2016, Guruswami and Wootters showed Shamir's secret-sharing scheme defined over an extension field has a regenerating property. Namely, we can compress each share to an element of the base field by applying a linear form, such that the secret is determined by a linear combination of the compressed shares. Immediately it seemed like an application to improve the complexity of unconditionally secure multiparty computation must be imminent; however, thus far, no result has been published. We present the first application of regenerating codes to MPC, and show that its utility lies in reducing the number of rounds. Concretely, we present a protocol that obliviously evaluates a depth-$d$ arithmetic circuit in $d + O(1)$ rounds, in the amortized setting of parallel evaluations, with $o(n^2)$ ring elements communicated per multiplication. Our protocol makes use of function-dependent preprocessing, and is secure against the maximal adversary corrupting $t < n/2$ parties. All existing approaches in this setting have complexity $\Omega(n^2)$. Moreover, we extend some of the theory on regenerating codes to Galois rings. It was already known that the repair property of MDS codes over fields can be fully characterized in terms of its dual code. We show this characterization extends to linear codes over Galois rings, and use it to show the result of Guruswami and Wootters also holds true for Shamir's scheme over Galois rings.

In this work we consider information-theoretically secure MPC against an \emph{mixed} adversary who can corrupt $t_p$ parties passively, $t_a$ parties actively, and can make $t_f$ parties fail-stop. With perfect security, it is known that every function can be computed securely if and only if $3t_a + 2t_p + t_f < n$, for statistical security the bound is $2t_a + 2t_p + t_f < n$. These results say that for each given set of parameters $(t_a, t_p, t_f)$ respecting the inequality, there exists a protocol secure against this particular choice of corruption thresholds. In this work we consider a \emph{dynamic} adversary. Here, the goal is a \emph{single} protocol that is secure, no matter which set of corruption thresholds $(t_a, t_p, t_f)$ from a certain class is chosen by the adversary. A dynamic adversary can choose a corruption strategy after seeing the protocol and so is much stronger than a standard adversary. Dynamically secure protocols have been considered before for computational security. Also the information theoretic case has been studied, but only considering non-threshold adversaries, leading to inefficient protocols. We consider threshold dynamic adversaries and information theoretic security. For statistical security we show that efficient dynamic secure function evaluation (SFE) is possible if and only if $2t_a + 2t_p + t_f < n$, but any dynamically secure protocol must use $\Omega(n)$ rounds, even if only fairness is required. Further, general reactive MPC is possible if we assume in addition that $2t_a+2t_f \leq n$, but fair reactive MPC only requires $2t_a + 2t_p + t_f < n$. For perfect security we show that both dynamic SFE and verifiable secret sharing (VSS) are impossible if we only assume $3t_a + 2t_p + t_f < n$ and remain impossible even if we also assume $t_f=0$. In fact even SFE with security with abort is impossible in this case. On the other hand, perfect dynamic SFE with guaranteed output delivery (G.O.D.) is possible when either $t_p = 0$ or $t_a = 0$ i.e. if instead we assume $3t_a+t_f < n$ or $2t_p +t_f < n$. Further, perfect dynamic VSS with G.O.D. is possible under the stronger conditions $3t_a + 3/2t_f \leq n$ or $2t_p + 2t_f \leq n$. These conditions are also sufficient for perfect reactive MPC. On the other hand, because perfect fair VSS only requires $3t_a+2t_p+t_f< n$, perfect reactive MPC is possible whenever perfect SFE is.

This work introduces novel techniques to improve the translation between arithmetic and binary data types in multi-party computation. To this end, we introduce a new approach to performing these conversions, using what we call \emph{extended doubly-authenticated bits} (edaBits), which correspond to shared integers in the arithmetic domain whose bit decomposition is shared in the binary domain. These can be used to considerably increase the efficiency of non-linear operations such as truncation, secure comparison and bit-decomposition. Our eDaBits are similar to the \emph{daBits} technique introduced by Rotaru et al.~(Indocrypt 2019). However, our main observations are that (1) applications that benefit from daBits can also benefit from edaBits in the same way, and (2) we can generate edaBits directly in a much more efficeint way than computing them directly from a set of DaBits. Technically, the second contribution is much more challenging, and involves a novel cut and choose technique that may be of independent interest, and requires taking advantage of natural tamper-resilient properties of binary circuits that occur in our construction to obtain the best level of efficiency. Finally, we show how our eDaBits can be applied to efficiently implement various non-linear protocols of interest, and we thoroughly analyze their correctness for both signed and unsigned integers. The results of this work can be applied to any corruption threshold, although they seem best suited to dishonest majority protocols such as SPDZ. We implement and benchmark our constructions, and experimentally verify that our technique yield a substantial increase in effiency. Our eDaBits save in communication by a factor that lies between $2$ and $170$ for secure comparisons with respect to a purely arithmetic approach, and between $2$ and $60$ with respect to using daBits. Improvements in throughput per second are more subdued but still as high as a factor of $47$. We also apply our novel machinery to the tasks of biometric matching and convolutional neural networks, obtaining a noticeable improvement as well.

We study information-theoretic multiparty computation (MPC) protocols over rings Z/p^k Z that have good asymptotic communication complexity for a large number of players. An important ingredient for such protocols is arithmetic secret sharing, i.e., linear secret-sharing schemes with multiplicative properties. The standard way to obtain these over fields is with a family of linear codes C, such that C, $C^\perp$ and C^2 are asymptotically good (strongly multiplicative). For our purposes here it suffices if the square code C^2 is not the whole space, i.e., has codimension at least 1 (multiplicative). Our approach is to lift such a family of codes defined over a finite field F to a Galois ring, which is a local ring that has F as its residue field and that contains Z/p^k Z as a subring, and thus enables arithmetic that is compatible with both structures. Although arbitrary lifts preserve the distance and dual distance of a code, as we demonstrate with a counterexample, the multiplicative property is not preserved. We work around this issue by showing a dedicated lift that preserves \emph{self-orthogonality} (as well as distance and dual distance), for p > 2. Self-orthogonal codes are multiplicative, therefore we can use existing results of asymptotically good self-dual codes over fields to obtain arithmetic secret sharing over Galois rings. For p = 2 we obtain multiplicativity by using existing techniques of secret-sharing using both C and $C^\perp$, incurring a constant overhead. As a result, we obtain asymptotically good arithmetic secret-sharing schemes over Galois rings. With these schemes in hand, we extend existing field-based MPC protocols to obtain MPC over Z/p^k Z, in the setting of a submaximal adversary corrupting less than a fraction 1/2 - \varepsilon of the players, where \varepsilon > 0 is arbitrarily small. We consider 3 different corruption models, and obtain O(n) bits communicated per multiplication for both passive security and active security with abort. For full security with guaranteed output delivery we use a preprocessing model and get O(n) bits per multiplication in the online phase and O(n log n) bits per multiplication in the offline phase. Thus, we obtain true linear bit complexities, without the common assumption that the ring size depends on the number of players.

At CRYPTO 2018, Cramer et al. introduced a secret-sharing based protocol called SPD$$\mathbb {Z}_{2^k}$$ that allows for secure multiparty computation (MPC) in the dishonest majority setting over the ring of integers modulo $$2^k$$, thus solving a long-standing open question in MPC about secure computation over rings in this setting. In this paper we study this problem in the information-theoretic scenario. More specifically, we ask the following question: Can we obtain information-theoretic MPC protocols that work over rings with comparable efficiency to corresponding protocols over fields? We answer this question in the affirmative by presenting an efficient protocol for robust Secure Multiparty Computation over $$\mathbb {Z}/p^{k}\mathbb {Z}$$ (for any prime p and positive integer k) that is perfectly secure against active adversaries corrupting a fraction of at most 1/3 players, and a robust protocol that is statistically secure against an active adversary corrupting a fraction of at most 1/2 players.

Most multi-party computation protocols allow secure computation of arithmetic circuits over a finite field, such as the integers modulo a prime. In the more natural setting of integer computations modulo $$2^{k}$$, which are useful for simplifying implementations and applications, no solutions with active security are known unless the majority of the participants are honest.We present a new scheme for information-theoretic MACs that are homomorphic modulo $$2^k$$, and are as efficient as the well-known standard solutions that are homomorphic over fields. We apply this to construct an MPC protocol for dishonest majority in the preprocessing model that has efficiency comparable to the well-known SPDZ protocol (Damgård et al., CRYPTO 2012), with operations modulo $$2^k$$ instead of over a field. We also construct a matching preprocessing protocol based on oblivious transfer, which is in the style of the MASCOT protocol (Keller et al., CCS 2016) and almost as efficient.