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).

A recent line of works on zero-knowledge (ZK) protocols with a vector oblivious linear function evaluation (VOLE)-based offline phase provides a new paradigm for scalable ZK protocols featuring fast proving and small prover memory. Very recently, Baum et al. (Crypto'23) proposed the VOLE-in-the-head technique, allowing such protocols to become publicly verifiable. Many practically efficient protocols for proving circuit satisfiability over any Galois field are implemented, while protocols over rings $\mathbb{Z}_{2^k}$ are significantly lagging behind, with only a proof-of-concept pioneering work called Appenzeller to Brie (CCS'21) and a first proposal called Moz$\mathbb{Z}_{2^k}$arella (Crypto'22). The ring $\mathbb{Z}_{2^{32}}$ or $\mathbb{Z}_{2^{64}}$, though highly important (it captures computation in real-life programming and the computer architectures such as CPU words), presents non-trivial difficulties because, for example, unlike Galois fields $\mathbb{F}_{2^{k}}$, the fraction of units in $\mathbb{Z}_{2^{k}}$ is $1/2$. In this work, we first construct ZK protocols over a high degree Galois ring extension of $\mathbb{Z}_{2^{k}}$ (fraction of units close to $1$) and then convert them to $\mathbb{Z}_{2^k}$ efficiently using amortization techniques. Our results greatly change the landscape of ZK protocols over~$\mathbb{Z}_{2^k}$. (1) We propose a competing ZK protocol that has many advantages over the state-of-the-art Moz$\mathbb{Z}_{2^k}$arella. We remove the undesirable dependence of communication complexity on the security parameter, and achieve communication complexity {\em strictly} linear in the circuit size. Furthermore, our protocol has better concrete efficiency. For $40,80$ bits soundness on circuits over $\mathbb{Z}_{2^{32}}$ and $\mathbb{Z}_{2^{64}}$, we offer $1.15\times$--$2.9\times$ improvements in communication. (2) Inspired by the recently proposed interactive message authentication code technique (Weng et al., CCS'22), we construct a constant round ZK protocol over $\mathbb{Z}_{2^k}$ with sublinear (in the circuit size) communication complexity, which was previously achieved only over fields. (3) We show that the pseudorandom correlation generator approach can be adapted to efficiently implement VOLE over Galois rings, with analysis of the hardness of underlying LPN assumptions over Galois rings. (4) We adapt the VOLE-in-the-head technique to make it work for $\mathbb{Z}_{2^k}$, yielding {\em publicly verifiable} non-interactive ZK protocols over $\mathbb{Z}_{2^k}$ which preserve most of the efficiency metrics of the VOLE-based ZK protocols.

Studies of vector oblivious linear evaluation (VOLE)-based zero-knowledge (ZK) protocols flourish in recent years. Such ZK protocols feature optimal prover computation and a flexibility for handling arithmetic circuits over arbitrary fields. However, most of them have linear communication, which constitutes a bottleneck for handling large statements in a slow network. The pioneer work AntMan (CCS'22), achieved sublinear communication for the first time within VOLE-based ZK, but lost the advantage of fast proving. In this work, we propose two new VOLE-based ZK constructions that achieve sublinear communication and linear computation, simultaneously. Let $\mathcal{C}$ be a circuit with size $S$, input size $n$, and depth $d$. In particular, our first ZK, specialized for layered circuits, has communication $\bigO{n+d\log{S}}$, while our second ZK can be used to prove general circuits and has communication $\bigO{n+d\log{S}+d^2}$. Our results are obtained by introducing the powerful sum-check techniques from the mature line of works on interactive proofs into the context of VOLE-based ZK for the first time. Reminiscent of the {\em non-interactive} line-point zero-knowledge proof system (ITC'21), we introduce an {\em interactive line-point zero-knowledge} (ILPZK) proof system, which serves as a bridge to VOLE-based ZK protocols. In addition, our works also enrich the studies of ZK based on interactive proofs, with new interesting features (e.g., having information-theoretic UC-security, naturally supporting any field) achieved.

The privacy-preserving machine learning (PPML) has gained growing importance over the last few years. One of the biggest challenges is to improve the efficiency of PPML so that the communication and computation costs of PPML are affordable for large machine learning models such as deep learning. As we know, linear algebra such as matrix multiplication occupies a significant part of the computation in deep learning such as deep convolutional neural networks (CNN). Thus, it is desirable to propose the MPC protocol specialized for the matrix operations. In this work, we propose a dishonest majority MPC protocol over matrix rings which supports matrix multiplication and addition. Our MPC protocol can be seen as a variant of SPDZ protocol, i.e., the MAC and global key of our protocol are vectors of length m and the secret of our protocol is an $m \times m$ matrix. Compared to the classic SPDZ protocol, our MPC protocol reduces the communication complexity by at least m times to securely compute a matrix multiplication. We also show that the communication complexity of our MPC protocol is asymptotically as good as [16] which also presented a dishonest majority MPC protocol specialized for matrix operations, i.e., the communication complexity of securely computing a multiplication gate is $O(m^2 n^2 log q)$ in the preprocessing phase and $O(m^2 n log q)$ in the online phase. The share size and the number of multiplications of our protocol are reduced by around 50% and 40% of [16], respectively. However, we take a completely different approach. The protocol in [16] uses a variant of BFV scheme to embed a whole matrix into a single ciphertext and then treats the matrix operation as the entry-wise operation in the ciphertext while our approach resorts to a variant of vector linear oblivious evaluation (VOLE) called the subfield VOLE [33] which can securely compute the additive sharing of $v\bm{x}$ for $v \in F_{q^b}, \bm{x}\in F_q^a$ with sublinear communication complexity. Finally, we note that our MPC protocol can be easily extended to small fields.

Reversed multiplication friendly embedding (RMFE) amortization has been playing an active role in the state-of-the-art constructions of MPC protocols over rings (in particular, the ring $\mathbb{Z}_{2^k}$). As far as we know, this powerful technique has NOT been able to find applications in the crown jewel of two-party computation, the non-interactive secure computation (NISC), where the requirement of the protocol being non-interactive constitutes a formidable technical bottle-neck. We initiate such a study focusing on statistical NISC protocols in the VOLE-hybrid model. Our study begins with making the {\em decomposable affine randomized encoding (DARE)} based semi-honest NISC protocol compatible with RMFE techniques, which together with known techniques for forcing a malicious sender Sam to honestly follow DARE already yield a secure amortized protocol, assuming both parties follow RMFE encoding. Achieving statistical security in the full malicious setting is much more challenging, as applying known techniques for enforcing compliance with RMFE incurs interaction. To solve this problem, we put forward a new notion dubbed non-malleable RMFE (NM-RMFE), which is a randomized RMFE such that, once one party deviates from the encoding specification, the randomness injected by the other party will randomize the output, preventing information from being leaked. NM-RMFE simultaneously forces both parties to follow RMFE encoding, offering a desired {\em non-interactive} solution to amortizing NISC. We believe that NM-RMFE is on its own an important primitive that has applications in secure computation and beyond, interactive and non-interactive alike. With an asymptotically good instantiation of our NM-RMFE, we obtain the first {\em statistical} reusable NISC protocols in the VOLE-hybrid model with {\em constant communication overhead} for arithmetic branching programs over $\mathbb{Z}_{2^k}$. As side contributions, we consider computational security and present two concretely efficient NISC constructions in the random oracle model from conventional RMFEs.

Beerliov{\'{a}}{-}Trub{\'{\i}}niov{\'{a}} and Hirt introduced hyper-invertible matrix technique to construct the first perfectly secure MPC protocol in the presence of maximal malicious corruptions $\lfloor \frac{n-1}{3} \rfloor$ with linear communication complexity per multiplication gate\cite{BH08}. This matrix allows MPC protocol to generate correct shares of uniformly random secrets in the presence of malicious adversary. Moreover, the amortized communication complexity of generating each sharing is linear. Due to this prominent feature, the hyper-invertible matrix plays an important role in the construction of MPC protocol and zero-knowledge proof protocol where the randomness needs to be jointly generated. However, the downside of this matrix is that the size of its base field is linear in the size of its matrix. This means if we construct an $n$-party MPC protocol over $\F_q$ via hyper-invertible matrix, $q$ is at least $2n$. In this paper, we propose the ramp hyper-invertible matrix which can be seen as the generalization of hyper-invertible matrix. Our ramp hyper-invertible matrix can be defined over constant-size field regardless of the size of this matrix. Similar to the arithmetic secret sharing scheme, to apply our ramp hyper-invertible matrix to perfectly secure MPC protocol, the maximum number of corruptions has to be compromised to $\frac{(1-\epsilon)n}{3}$. As a consequence, we present the first perfectly secure MPC protocol in the presence of $\frac{(1-\epsilon)n}{3}$ malicious corruptions with constant communication complexity. Besides presenting the variant of hyper-invertible matrix, we overcome several obstacles in the construction of this MPC protocol. Our arithmetic secret sharing scheme over constant-size field is compatible with the player elimination technique, i.e., it supports the dynamic changes of party number and corrupted party number. Moreover, we rewrite the public reconstruction protocol to support the sharings over constant-size field. Putting these together leads to the constant-size field variant of celebrated MPC protocol in \cite{BH08}. We note that although it was widely acknowledged that there exists an MPC protocol with constant communication complexity by replacing Shamir secret sharing scheme with arithmetic secret sharing scheme, there is no reference seriously describing such protocol in detail. Our work fills the missing detail by providing MPC primitive for any applications relying on MPC protocol of constant communication complexity. As an application of our perfectly secure MPC protocol which implies perfect robustness in the MPC-in-the-Head framework, we present the constant-rate zero-knowledge proof with $3$ communication rounds. The previous work achieves constant-rate with $5$ communication rounds \cite{IKOS07} due to the statistical robustness of their MPC protocol. Another application of our ramp hyper-invertible matrix is the information-theoretic multi-verifier zero-knowledge for circuit satisfiability\cite{YW22}. We manage to remove the dependence of the size of circuit and security parameter from the share size.

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.

The current paper studies information-theoretically secure multiparty computation (MPC) over rings $\Z/p^{\ell}\Z$. This is a follow-up research of recent work on MPC over rings $\Z/p^{\ell}\Z$. In the work of \cite[TCC2019]{tcc}, a protocol based on the Shamir secret sharing over $\Z/p^{\ell}\Z$ was presented. As in the field case, its limitation is that the share size has to grow as the number of players increases. Then several MPC protocols were developed in \cite[Asiacrypt 2020]{asiacrypt} to overcome this limitation. However, the MPC protocols in \cite[Asiacrypt 2020]{asiacrypt} suffer from several drawbacks: (i) the offline multiplication gate has super-linear communication complexity; (ii) the share size is doubled for the most important case, namely over $\Z/2^{\ell}\Z$ due to infeasible lifting of self-orthogonal codes from fields to rings; (iii) most importantly, the BGW model could not be applied via the secret sharing given in \cite[Asiacrypt 2020]{asiacrypt} due to lack of strong multiplication. Our contribution in this paper is three fold. Firstly, we overcome all the drawbacks in \cite{tcc,asiacrypt} mentioned above. Secondly, we establish an arithmetic secret sharing with strong multiplication, which is the most important primitive in the BGW model. Thirdly, we lift Reverse Multiplication Friendly Embeddings (RMFE) from fields to rings, with same (linear) complexity. Note that RMFE has become a standard technique for amortized communication complexity in MPC, as in \cite[CRYPTO'18]{crypto2018} and \cite[CRYPTO'19]{dn19}. To obtain our theoretical results, we use the existence of lifts of curves over rings, then use the known results stating that Riemann-Roch spaces are free modules. To make our scheme practical, we start from good algebraic geometry codes over finite fields obtained from existing computational techniques. Then we present, and implement, an efficient algorithm to Hensel-lift the generating matrix of the code, such that the multiplicative conditions are preserved over rings. Existence of this specific lift is guaranteed by the previous theory. On the other hand, a random lifting of codes over from fields to Galois rings does not preserve multiplicativity in general. (Notice that our indirect method is motivated by the fact that, following the theory instead, would require to ``preprocess'' the curve under a form with ``smooth" equations, in particular with many variables, before lifting it. But computing on these objects over rings is out of the scope of existing research). Finally we provide efficient elementary methods for sharing and (robust) reconstruction of secrets over rings. As a result, arithmetic secret sharing over $\Z/p^{\ell}\Z$ with strong multiplication can be efficiently constructed and practically applied.

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.

A {\em blackbox} secret sharing (BBSS) scheme works in exactly the same way for all finite Abelian groups $G$; it can be instantiated for any such group $G$ and {\em only} black-box access to its group operations and to random group elements is required. A secret is a single group element and each of the $n$ players' shares is a vector of such elements. Share-computation and secret-reconstruction is by integer linear combinations. These do not depend on $G$, and neither do the privacy and reconstruction parameters $t,r$. This classical, fundamental primitive was introduced by Desmedt and Frankel (CRYPTO 1989) in their context of ``threshold cryptography.'' The expansion factor is the total number of group elements in a full sharing divided by $n$. For threshold BBSS with $t$-privacy ($1\leq t \leq n-1$), $t+1$-reconstruction and arbitrary $n$, constructions with minimal expansion $O(\log n)$ exist (CRYPTO 2002, 2005). These results are firmly rooted in number theory; each makes (different) judicious choices of orders in number fields admitting a vector of elements of very large length (in the number field degree) whose corresponding Vandermonde-determinant is sufficiently controlled so as to enable BBSS by a suitable adaptation of Shamir's scheme. Alternative approaches generally lead to very large expansion. The state of the art of BBSS has not changed for the last 15 years. Our contributions are two-fold. (1) We introduce a novel, nontrivial, effective construction of BBSS based on {\em coding theory} instead of number theory. For threshold-BBSS we also achieve minimal expansion factor $O(\log n)$. (2) Our method is more versatile. Namely, we show, for the first time, BBSS that is {\em near-threshold}, i.e., $r-t$ is an arbitrarily small constant fraction of $n$, {\em and} that has expansion factor~$O(1)$, i.e., individual share-vectors of {\em constant} length (``asymptotically expansionless''). Threshold can be concentrated essentially freely across full range. We also show expansion is minimal for near-threshold and that such BBSS cannot be attained by previous methods. Our general construction is based on a well-known mathematical principle, the local-global principle. More precisely, we first construct BBSS over local rings through either Reed-Solomon or algebraic geometry codes. We then ``glue'' these schemes together in a dedicated manner to obtain a global secret sharing scheme, i.e., defined over the integers, which, as we finally prove using novel insights, has the desired BBSS properties. Though our main purpose here is advancing BBSS for its own sake, we also briefly address possible protocol applications.

Since the mid 2000s, asymptotically-good strongly-multiplicative linear (ramp) secret sharing schemes over a fixed finite field have turned out as a central theoretical primitive in numerous constant-communication-rate results in multi-party cryptographic scenarios, and, surprisingly, in two-party cryptography as well. Known constructions of this most powerful class of arithmetic secret sharing schemes all rely heavily on algebraic geometry (AG), i.e., on dedicated AG codes based on asymptotically good towers of algebraic function fields defined over finite fields. It is a well-known open question since the first (explicit) constructions of such schemes appeared in CRYPTO 2006 whether the use of ``heavy machinery'' can be avoided here. i.e., the question is whether the mere existence of such schemes can also be proved by ``elementary'' techniques only (say, from classical algebraic coding theory), even disregarding effective construction. So far, there is no progress. In this paper we show the theoretical result that, (1) {\em no matter whether this open question has an affirmative answer or not}, these schemes {\em can} be constructed explicitly by {\em elementary algorithms} defined in terms of basic algebraic coding theory. This pertains to all relevant operations associated to such schemes, including, notably, the generation of an instance for a given number of players $n$, as well as error correction in the presence of corrupt shares. We further show that (2) the algorithms are {\em quasi-linear time} (in $n$); this is (asymptotically) significantly more efficient than the known constructions. That said, the {\em analysis} of the mere termination of these algorithms {\em does} still rely on algebraic geometry, in the sense that it requires ``blackbox application'' of suitable {\em existence} results for these schemes. Our method employs a nontrivial, novel adaptation of a classical (and ubiquitous) paradigm from coding theory that enables transformation of {\em existence} results on asymptotically good codes into {\em explicit construction} of such codes via {\em concatenation}, at some constant loss in parameters achieved. In a nutshell, our generating idea is to combine a cascade of explicit but ``asymptotically-bad-yet-good-enough schemes'' with an asymptotically good one in such a judicious way that the latter can be selected with exponentially small number of players in that of the compound scheme. This opens the door to efficient, elementary exhaustive search. In order to make this work, we overcome a number of nontrivial technical hurdles. Our main handles include a novel application of the recently introduced notion of Reverse Multiplication-Friendly Embeddings (RMFE) from CRYPTO 2018, as well as a novel application of a natural variant in arithmetic secret sharing from EUROCRYPT 2008.

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.

In this paper, we propose a new general construction to reduce the public key size of McEliece cryptosystems constructed from automorphism-induced Goppa codes. In particular, we generalize the ideas of automorphism-induced Goppa codes by considering nontrivial subsets of automorphism groups to construct Goppa codes with a nice block structure. By considering additive and multiplicative automorphism subgroups, we provide explicit constructions to demonstrate our technique. We show that our technique can be applied to automorphism-induced Goppa codes based cryptosystems to further reduce their key sizes.

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 thresholdt 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 $$t + 2k -2 < n/3$$ t+2k-2<n/3, then kparallel 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 $$\lfloor (n-1)/3 \rfloor $$ ⌊(n-1)/3⌋ 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 $$\lfloor (n-1)/3 \rfloor $$ ⌊(n-1)/3⌋, it is possible to securely compute a binary circuit with amortized complexity O(n) of bits per gate per instance. Known results would give $$n \log n$$ nlogn bits instead. By combining our result with the Franklin-Yung paradigm, and assuming a sub-optimal adversary (i.e., an arbitrarily small $$\epsilon >0$$ ϵ>0 fraction below 1/3), this is improved to O(1) bits instead of O(n).

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.