## CryptoDB

### Chaoping Xing

#### Publications

**Year**

**Venue**

**Title**

2022

CRYPTO

More Efficient Dishonest Majority Secure Computation over $\mathbb{Z}_{2^k}$ via Galois Rings
Abstract

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.

2021

CRYPTO

Asymptotically-Good Arithmetic Secret Sharing over Z/p^{\ell}Z with Strong Multiplication and Its Applications to Efficient MPC
📺
Abstract

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.

2021

ASIACRYPT

Improved single-round secure multiplication using regenerating codes
📺
Abstract

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.

2020

EUROCRYPT

Blackbox Secret Sharing Revisited: A Coding-Theoretic Approach with Application to Expansionless Near-Threshold Schemes
📺
Abstract

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.

2020

TCC

On the Complexity of Arithmetic Secret Sharing
📺
Abstract

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.

2020

ASIACRYPT

Asymptotically Good Multiplicative LSSS over Galois Rings and Applications to MPC over Z/p^k Z
📺
Abstract

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.

2019

PKC

Reducing the Key Size of McEliece Cryptosystem from Automorphism-induced Goppa Codes via Permutations
Abstract

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.

2018

CRYPTO

Amortized Complexity of Information-Theoretically Secure MPC Revisited
📺
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 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).

2018

CRYPTO

SPD$\mathbb {Z}_{2^k}$: Efficient MPC mod $2^k$ for Dishonest Majority
📺
Abstract

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.

2017

EUROCRYPT

2011

CRYPTO

#### Program Committees

- Asiacrypt 2021
- PKC 2020
- Asiacrypt 2020

#### Coauthors

- Mark Abspoel (2)
- Ignacio Cascudo (3)
- Hao Chen (1)
- Ronald Cramer (11)
- Ivan Damgård (4)
- Daniel Escudero (4)
- Oriol Farràs (1)
- Kwok-Yan Lam (1)
- Zhe Li (1)
- Carles Padró (2)
- Matthieu Rambaud (2)
- Peter Scholl (1)
- Zhenghong Wei (1)
- An Yang (1)
- Sze Ling Yeo (1)
- Chen Yuan (5)