# Kasper Green Larsen's research while affiliated with Aarhus University and other places

## Publications (91)

Preprint
Efficiently computing low discrepancy colorings of various set systems, has been studied extensively since the breakthrough work by Bansal (FOCS 2010), who gave the first polynomial time algorithms for several important settings, including for general set systems, sparse set systems and for set systems with bounded hereditary discrepancy. The hered...
Preprint
The Johnson-Lindenstrauss transform allows one to embed a dataset of $n$ points in $\mathbb{R}^d$ into $\mathbb{R}^m,$ while preserving the pairwise distance between any pair of points up to a factor $(1 \pm \varepsilon)$, provided that $m = \Omega(\varepsilon^{-2} \lg n)$. The transform has found an overwhelming number of algorithmic applications,...
Preprint
The classic algorithm AdaBoost allows to convert a weak learner, that is an algorithm that produces a hypothesis which is slightly better than chance, into a strong learner, achieving arbitrarily high accuracy when given enough training data. We present a new algorithm that constructs a strong learner from a weak learner but uses less training data...
Preprint
The seminal Fast Johnson-Lindenstrauss (Fast JL) transform by Ailon and Chazelle (SICOMP'09) embeds a set of $n$ points in $d$-dimensional Euclidean space into optimal $k=O(\varepsilon^{-2} \ln n)$ dimensions, while preserving all pairwise distances to within a factor $(1 \pm \varepsilon)$. The Fast JL transform supports computing the embedding of...
Preprint
Full-text available
The $3$SUM-Indexing problem was introduced as a data structure version of the $3$SUM problem, with the goal of proving strong conditional lower bounds for static data structures via reductions. Ideally, the conjectured hardness of $3$SUM-Indexing should be replaced by an unconditional lower bound. Unfortunately, we are far from proving this, with t...
Preprint
Full-text available
Given a set of points in a metric space, the $(k,z)$-clustering problem consists of finding a set of $k$ points called centers, such that the sum of distances raised to the power of $z$ of every data point to its closest center is minimized. Special cases include the famous k-median problem ($z = 1$) and k-means problem ($z = 2$). The $k$-median an...
Preprint
Full-text available
It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in \c...
Preprint
Explaining the surprising generalization performance of deep neural networks is an active and important line of research in theoretical machine learning. Influential work by Arora et al. (ICML'18) showed that, noise stability properties of deep nets occurring in practice can be used to provably compress model representations. They then argued that...
Preprint
Full-text available
Property-preserving hash functions allow for compressing long inputs $x_0$ and $x_1$ into short hashes $h(x_0)$ and $h(x_1)$ in a manner that allows for computing a predicate $P(x_0, x_1)$ given only the two hash values without having access to the original data. Such hash functions are said to be adversarially robust if an adversary that gets to p...
Preprint
In this paper, we revisit the classic CountSketch method, which is a sparse, random projection that transforms a (high-dimensional) Euclidean vector $v$ to a vector of dimension $(2t-1) s$, where $t, s > 0$ are integer parameters. It is known that even for $t=1$, a CountSketch allows estimating coordinates of $v$ with variance bounded by $\|v\|_2^2... Chapter In this work, we consider the construction of oblivious RAMs (ORAM) in a setting with multiple servers and the adversary may corrupt a subset of the servers. We present an $$\varOmega (\log n)$$ overhead lower bound for any k-server ORAM that limits any PPT adversary to distinguishing advantage at most 1/4k when only one server is corrupted. In oth... Preprint Full-text available Boosting is one of the most successful ideas in machine learning, achieving great practical performance with little fine-tuning. The success of boosted classifiers is most often attributed to improvements in margins. The focus on margin explanations was pioneered in the seminal work by Schapire et al. (1998) and has culminated in the$k$'th margin... Article Sorting extremely large datasets is a frequently occurring task in practice. These datasets are usually much larger than the computer's main memory; thus, external memory sorting algorithms, first introduced by Aggarwal and Vitter, are often used. The complexity of comparison-based external memory sorting has been understood for decades by now; how... Chapter A secret sharing scheme allows a dealer to distribute shares of a secret among a set of n parties such that any authorized subset of parties can reconstruct the secret, yet any unauthorized subset learns nothing about it. The family of all authorized subsets is called the access structure. Classic results show that if contains precisely all subsets... Preprint In a landmark paper, P\v{a}tra\c{s}cu demonstrated how a single lower bound for the static data structure problem of reachability in the butterfly graph, could be used to derive a wealth of new and previous lower bounds via reductions. These lower bounds are tight for numerous static data structure problems. Moreover, he also showed that reachabili... Preprint Support Vector Machines (SVMs) are among the most fundamental tools for binary classification. In its simplest formulation, an SVM produces a hyperplane separating two classes of data using the largest possible margin to the data. The focus on maximizing the margin has been well motivated through numerous generalization bounds. In this paper, we re... Article Full-text available The Johnson-Lindenstrauss lemma is one of the corner stone results in dimensionality reduction. It says that for any set of vectors$X \subset \mathbb{R}^n$, there exists a mapping$f : X \to \mathbb{R}^m$such that$f(X)$preserves all pairwise distances between vectors in$X$to within$(1 \pm \varepsilon)$if$m = O(\varepsilon^{-2} \lg N). Muc... Preprint Boosting is one of the most successful ideas in machine learning. The most well-accepted explanations for the low generalization error of boosting algorithms such as AdaBoost stem from margin theory. The study of margins in the context of boosting algorithms was initiated by Schapire, Freund, Bartlett and Lee (1998) and has inspired numerous boosti... Preprint We consider the following problem, which is useful in applications such as joint image and shape alignment. The goal is to recovern$discrete variables$g_i \in \{0, \ldots, k-1\}$(up to some global offset) given noisy observations of a set of their pairwise differences$\{(g_i - g_j) \bmod k\}$; specifically, with probability$\frac{1}{k}+\delt...
Chapter
We prove a lower bound on the communication complexity of unconditionally secure multiparty computation, both in the standard model with $$n=2t+1$$ parties of which t are corrupted, and in the preprocessing model with $$n=t+1$$. In both cases, we show that for any $$g \in \mathbb {N}$$ there exists a Boolean circuit C with g gates, where any secure...
Article
We develop a new algorithm for the turnstile heavy hitters problem in general turnstile streams, the EXPANDERSKETCH, which finds the approximate top-k items in a universe of size n using the same asymptotic O(k log n) words of memory and O(log n) update time as the COUNTMIN and COUNTSKETCH, but requiring only O(k poly(log n)) time to answer queries...
Conference Paper
Sorting extremely large datasets is a frequently occuring task in practice. These datasets are usually much larger than the computer’s main memory; thus external memory sorting algorithms, first introduced by Aggarwal and Vitter (1988), are often used. The complexity of comparison based external memory sorting has been understood for decades by now...
Preprint
We prove an $\Omega(d \lg n/ (\lg\lg n)^2)$ lower bound on the dynamic cell-probe complexity of statistically $\mathit{oblivious}$ approximate-near-neighbor search ($\mathsf{ANN}$) over the $d$-dimensional Hamming cube. For the natural setting of $d = \Theta(\log n)$, our result implies an $\tilde{\Omega}(\lg^2 n)$ lower bound, which is a quadratic...
Preprint
Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by F\"{u}rer, shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg n \cdot 4^{\lg^*n})$, where $\lg^*n$ is the very slowly growing iterated logarithm. In this work, we prove th...
Article
We study the query complexity of a permutation-based variant of the guessing game Mastermind. In this variant, the secret is a pair (z,π) which consists of a binary string z∈{0,1}n and a permutation π of [n]. The secret must be unveiled by asking queries of the form x∈{0,1}n. For each such query, we are returned the score fz,π(x)≔max{i∈[0..n]∣∀j≤i:...
Preprint
Boosting algorithms produce a classifier by iteratively combining base hypotheses. It has been observed experimentally that the generalization error keeps improving even after achieving zero training error. One popular explanation attributes this to improvements in margins. A common goal in a long line of research, is to maximize the smallest margi...
Preprint
We study the query complexity of a permutation-based variant of the guessing game Mastermind. In this variant, the secret is a pair $(z,\pi)$ which consists of a binary string $z \in \{0,1\}^n$ and a permutation $\pi$ of $[n]$. The secret must be unveiled by asking queries of the form $x \in \{0,1\}^n$. For each such query, we are returned the scor...
Preprint
Sorting extremely large datasets is a frequently occuring task in practice. These datasets are usually much larger than the computer's main memory; thus external memory sorting algorithms, first introduced by Aggarwal and Vitter (1988), are often used. The complexity of comparison based external memory sorting has been understood for decades by now...
Preprint
Full-text available
An oblivious data structure is a data structure where the memory access patterns reveals no information about the operations performed on it. Such data structures were introduced by Wang et al. [ACM SIGSAC'14] and are intended for situations where one wishes to store the data structure at an untrusted server. One way to obtain an oblivious data str...
Conference Paper
This paper proves the first super-logarithmic lower bounds on the cell probe complexity of dynamic boolean (a.k.a. decision) data structure problems, a long-standing milestone in data structure lower bounds. We introduce a new approach and use it to prove a Ω(log1.5n) lower bound on the operational time of a wide range of boolean data structure pro...
Preprint
A priority queue is a fundamental data structure that maintains a dynamic set of (key, priority)-pairs and supports Insert, Delete, ExtractMin and DecreaseKey operations. In the external memory model, the current best priority queue supports each operation in amortized $O(\frac{1}{B}\log \frac{N}{B})$ I/Os. If the DecreaseKey operation does not nee...
Conference Paper
The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC’15]. In recent work, Larsen and Williams [SODA’17] attacked the problem from the upper bound side and gave a surprising cell probe data structu...
Preprint
Feature hashing, also known as {\em the hashing trick}, introduced by Weinberger et al. (2009), is one of the key techniques used in scaling-up machine learning algorithms. Loosely speaking, feature hashing uses a random sparse projection matrix $A : \mathbb{R}^n \to \mathbb{R}^m$ (where $m \ll n$) in order to reduce the dimension of the data from...
Article
Full-text available
We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give both conditional and unconditional lower bounds for variants of exact matching with wildcards, inner product, a...
Article
Full-text available
The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC'15]. In recent work, Larsen and Williams [SODA'17] attacked the problem from the upper bound side and gave a surprising cell probe data structu...
Article
In discrepancy minimization problems, we are given a family of sets $\mathcal{S} = \{S_1,\dots,S_m\}$, with each $S_i \in \mathcal{S}$ a subset of some universe $U = \{u_1,\dots,u_n\}$ of $n$ elements. The goal is to find a coloring $\chi : U \to \{-1,+1\}$ of the elements of $U$ such that each set $S \in \mathcal{S}$ is colored as evenly as possib...
Conference Paper
One of the biggest open problems in external memory data structures is the priority queue problem with DecreaseKey operations. If only Insert and ExtractMin operations need to be supported, one can design a comparison-based priority queue performing O((N/B)lgM/BN) I/Os over a sequence of N operations, where B is the disk block size in number of wor...
Article
This paper proves the first super-logarithmic lower bounds on the cell probe complexity of dynamic boolean (a.k.a. decision) data structure problems, a long-standing milestone in data structure lower bounds. We introduce a new method for proving dynamic cell probe lower bounds and use it to prove a $\tilde{\Omega}(\log^{1.5} n)$ lower bound on the...
Article
The $k$-Means clustering problem on $n$ points is NP-Hard for any dimension $d\ge 2$, however, for the 1D case there exist exact polynomial time algorithms. The current state of the art is a $O(kn^2)$ dynamic programming algorithm that uses $O(nk)$ space. We present a new algorithm improving this to $O(kn \log n)$ time and optimal $O(n)$ space. We...
Article
We study regular expression membership testing: Given a regular expression of size $m$ and a string of size $n$, decide whether the string is in the language described by the regular expression. Its classic $O(nm)$ algorithm is one of the big success stories of the 70s, which allowed pattern matching to develop into the standard tool that it is tod...
Article
One of the biggest open problems in external memory data structures is the priority queue problem with DecreaseKey operations. If only Insert and ExtractMin operations need to be supported, one can design a comparison-based priority queue performing $O((N/B)\lg_{M/B} N)$ I/Os over a sequence of $N$ operations, where $B$ is the disk block size in nu...
Article
For any integers $d, n \geq 2$ and $1/({\min\{n,d\}})^{0.4999} < \varepsilon<1$, we show the existence of a set of $n$ vectors $X\subset \mathbb{R}^d$ such that any embedding $f:X\rightarrow \mathbb{R}^m$ satisfying $$\forall x,y\in X,\ (1-\varepsilon)\|x-y\|_2^2\le \|f(x)-f(y)\|_2^2 \le (1+\varepsilon)\|x-y\|_2^2$$ must have m = \Omega(\varep... Conference Paper We propose a new zero-knowledge protocol applicable to additively homomorphic functions that map integer vectors to an Abelian group. The protocol demonstrates knowledge of a short preimage and achieves amortised efficiency comparable to the approach of Cramer and Damgård from Crypto 2010, but gives a much tighter bound on what we can extract from... Article We consider the \emph{Online Boolean Matrix-Vector Multiplication} (OMV) problem studied by Henzinger et al. [STOC'15]: given an n \times n Boolean matrix M, we receive n Boolean vectors v_1,\ldots,v_n one at a time, and are required to output M v_i (over the Boolean semiring) before seeing the vector v_{i+1}, for all i. Previous know... Article In turnstile \ell_p \varepsilon-heavy hitters, one maintains a high-dimensional x\in\mathbb{R}^n subject to \texttt{update}(i,\Delta) causing x_i\leftarrow x_i + \Delta, where i\in[n], \Delta\in\mathbb{R}. Upon receiving a query, the goal is to report a small list L\subset[n], |L| = O(1/\varepsilon^p), containing every "heavy hitt... Article Full-text available There has been a resurgence of interest in lower bounds whose truth rests on the conjectured hardness of well known computational problems. These conditional lower bounds have become important and popular due to the painfully slow progress on proving strong unconditional lower bounds. Nevertheless, the long term goal is to replace these conditional... Conference Paper Let $${\cal D} =\{d_1,d_2,...,d_D\}$$ be a collection of D string documents of n characters in total. The two-pattern matching problems ask to index $${\cal D}$$ for answering the following queries efficiently. report/count the unique documents containing P 1 and P 2. report/count the unique documents containing P 1, but not P 2. Here P 1 and P 2 r... Article For any n>1 and 0<\varepsilon<1/2, we show the existence of an n^{O(1)}-point subset X of \mathbb{R}^n such that any linear map from (X,\ell_2) to \ell_2^m with distortion at most 1+\varepsilon must have m = \Omega(\min\{n, \varepsilon^{-2}\log n\}). Our lower bound matches the upper bounds provided by the identity matrix and the... Article In the orthogonal range reporting problem, we are to preprocess a set of n points with integer coordinates on a U \times U grid. The goal is to support reporting all k points inside an axis-aligned query rectangle. This is one of the most fundamental data structure problems in databases and computational geometry. Despite the importance of th... Article Full-text available This paper studies the \emph{\varepsilon-approximate range emptiness} problem, where the task is to represent a set S of n points from \{0,\ldots,U-1\} and answer emptiness queries of the form "[a ; b]\cap S \neq \emptyset ?" with a probability of \emph{false positives} allowed. This generalizes the functionality of \emph{Bloom filters} f... Article We say a turnstile streaming algorithm is "non-adaptive" if, during updates, the memory cells written and read depend only on the index being updated and random coins tossed at the beginning of the stream (and not on the memory contents of the algorithm). Memory cells read during queries may be decided upon adaptively. All known turnstile streaming... Article Full-text available We consider NCA labeling schemes: given a rooted tree T, label the nodes of T with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels, the label of their nearest common ancestor. For trees with n nodes we present upper and lower bounds establishing that labels of size (2\pm \epsilon)... Conference Paper We revisit the classic problem of sampling from a discrete distribution: Given n non-negative w-bit integers x1,..,xn, the task is to build a data structure that allows sampling i with probability proportional to xi. The classic solution is Walker's alias method that takes, when implemented on a Word RAM, O(n) preprocessing time, O(1) expected quer... Conference Paper The skyline of a set of points in the plane is the subset of maximal points, where a point (x,y) is maximal if no other point (x',y') satisfies x'>=x and y'>=x. We consider the problem of preprocessing a set P of n points into a space efficient static data structure supporting orthogonal skyline counting queries, i.e. given a query rectangle R to r... Article Range reporting on categorical (or colored) data is a well-studied generalization of the classical range reporting problem in which each of the N input points has an associated color (category). A query then asks to report the set of colors of the points in a given rectangular query range, which may be far smaller than the set of all points in the... Article We study the query complexity of determining a hidden permutation. More specifically, we study the problem of learning a secret (z,π) consisting of a binary string z of length n and a permutation π of [n]. The secret must be unveiled by asking queries x ∈ {0,1}n , and for each query asked, we are returned the score f z,π (x) defined as f_{z,\pi}...
Conference Paper
In this paper, we study the cell probe complexity of evaluating an n-degree polynomial P over a finite field F of size at least n(1+Omega(1)). More specifically, we show that any static data structure for evaluating P(x), where x is an element of F, must use Omega(lg vertical bar F vertical bar/ lg(Sw/n lg vertical bar F vertical bar)) cell probes...
Article
In this paper, we study the role non-adaptivity plays in maintaining dynamic data structures. Roughly speaking, a data structure is non-adaptive if the memory locations it reads and/or writes when processing a query or update depend only on the query or update and not on the contents of previously read cells. We study such non-adaptive data structu...
Article
Range reporting is a one of the most fundamental topics in spatial databases and computational geometry. In this class of problems, the input consists of a set of geometric objects, such as points, line segments, rectangles etc. The goal is to preprocess the input set into a data structure, such that given a query range, one can efficiently report...
Article
In this paper we present a number of improved lower bounds for range searching in the pointer machine and the group model. In the pointer machine, we prove lower bounds for the approximate simplex range reporting problem. In approximate simplex range reporting, points that lie within a distance of ε ⋅ Diam(s) from the border of a query simplex s, a...
Conference Paper
In this paper, we consider two fundamental problems in the pointer machine model of computation, namely orthogonal range reporting and rectangle stabbing. Orthogonal range reporting is the problem of storing a set of n points in d-dimensional space in a data structure, such that the t points in an axis-aligned query rectangle can be reported effici...
Conference Paper
Full-text available
We consider range queries in arrays that search for low-frequency elements: least frequent elements and α-minorities. An α-minority of a query range has multiplicity no greater than an α fraction of the elements in the range. Our data structure for the least frequent element range query problem requires O(n) space, O(n 3/2) preprocessing time, and...
Conference Paper
Range selection is the problem of preprocessing an input array A of n unique integers, such that given a query (i, j, k), one can report the k'th smallest integer in the subarray A[i], A[i + 1],..., A[j]. In this paper we consider static data structures in the word-RAM for range selection and several natural special cases thereof. The first special...
Conference Paper
In this paper we establish an intimate connection between dynamic range searching in the group model and combinatorial discrepancy. Our result states that, for a broad class of range searching data structures (including all known upper bounds), it must hold that tutq = Ω(disc2/lg n) where tu is the worst case update time, tq the worst case query ti...
Article
In this paper we develop a new technique for proving lower bounds on the update time and query time of dynamic data structures in the cell probe model. With this technique, we prove the highest lower bound to date for any explicit problem, namely a lower bound of tq=Ω((lg n/lg(wtu))²). Here n is the number of update operations, w the cell size, tq...
Article
Motivated by information retrieval applications, we consider the one-dimensional colored range reporting problem in rank space. The goal is to build a static data structure for sets C_1,...,C_m \subseteq {1,...,sigma} that supports queries of the kind: Given indices a,b, report the set Union_{a <= i <= b} C_i. We study the problem in the I/O model,...
Article
We present several new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model for points in rank space: ** We present two data structures for 2-d orthogonal range emptiness. The first achieves O(n lglg n) space and O(lglg n) query time. This imp...
Conference Paper
In this paper we establish an intimate connection between dynamic range searching in the group model and combinatorial discrepancy. Our result states that, for a broad class of range searching data structures (including all known upper bounds), it must hold that tutq = Ω(disc2), where tu is the worst case update time, tq is the worst case query tim...
Conference Paper
Given a set of points with uncertain locations, we consider the problem of computing the probability of each point lying on the skyline, that is, the probability that it is not dominated by any other input point. If each point's uncertainty is described as a probability distribution over a discrete set of locations, we improve the best known exact...
Conference Paper
We consider the problem of automatically cleaning massive sonar data point clouds, that is, the problem of automatically removing noisy points that for example appear as a result of scans of (shoals of) fish, multiple reflections, scanner self-reflections, refraction in gas bubbles, and so on. We describe a new algorithm that avoids the problems of...
Article
Full-text available
We study the LeadingOnes game, a Mastermind-type guessing game first regarded as a test case in the complexity theory of randomized search heuristics. The first player, Carole, secretly chooses a string z ∈ {0, 1} n and a permutation π of [n]. The goal of the second player, Paul, is to identify the secret (z, π) with a small number of queries. A qu...

## Citations

... The special case z = 2 is the widely studied Euclidean k-Means problem. Following a long line of research [HM04, FS05, HK07, Che09, LS10, FL11, FGS + 13, FSS20, BLLM16, BFL + 17, SW18, BLL18, BBC + 19, HV20, BJKW21a,CLSS22], it is now known that Euclidean k-Means admits an ε-coreset S of sizẽ O(kε −2 min{k, ε −2 }) [CSS21,CLSS22], where an ε-coreset means that for every center set C, the cost of P and that of S are within a (1 ± ε)-factor. The most immediate approach to construct a coreset is to sample a subset of the input (and reweight its points appropriately), and the main challenge is to find a sampling distribution that works well. ...
... However, for perfectly secure ORAM, it is not clear whether the lower bound holds. e theme of a lot of works [13,[27][28][29][30][31][32][33][34][35] associated with ORAM is to find a lower bound of bandwidth overhead in different settings. Typically, there are two ways to measure the bandwidth overhead. ...
... However, for undirected unicast networks, it was conjectured in 2004 [1], [2] that network coding has no rate benefit over routing. This conjecture has found significance in theoretical computer science, see, e.g., [3], [4]. Despite its importance, the conjecture has been verified for only a handful of network instances and families of networks. ...
... Lower bounds, merely based on counting arguments, have also been applied to the class of forbidden graph access structures [75] and their generalization known as uniform access structures [1,12,58], respectively, in [10] and [2]. We refer to [56] for another example of lower bounds achieved using simple counting arguments. ...
... In the same paper, an information-theoretic lower bound of Ω( nk δ 2 ) queries is proved. Later, Larsen et al. (2020) focused on the two-cluster case, and proposed an algorithm with query complexity O( n log n δ 2 + log 2 n δ 6 ). Very recently, Peng & Zhang (2021) focused on improving the dependency on δ, and gave a polynomial time algorithm with query complexity O( nk log n δ 2 + k 10 log 2 n δ 4 ), which recovers all clusters of size Ω( k 4 log n δ 2 ). ...
Citing conference paper
... The first cell probe lower bound for cryptographic data structures was presented by Larsen and Nielsen [46] that presented a tight lower bound for oblivious RAMs (ORAMs). This paper led to a line of works proving lower bounds for a variety of different settings and primitives including other oblivious data structures [39], differential private guarantees [56], weaker adversaries that do not see query boundaries [38], non-colluding multiple-server settings [47], oblivious near-neighbor search [45], encrypted search [54] and smaller block sizes [40]. We note that all these lower bounds are for dynamic data structures and utilize either the information transfer [55] or chronogram techniques [29]. ...
... Bar-Ilan and Beaver [13] were the first to suggest a way to evaluate functions in a constant number of rounds, followed by further works that attempt to lower-bound that constant number. Theoretically, two rounds of communication are now known to be optimal for MPC-in the plain or preprocessing model [14,15]. Recent works by [1,16,17] present plain-model protocols that enable MPC in two rounds of communication with perfect passive security against an honest majority. ...
... We suggest to consider replacements for inner-consistency in terms of the centric consistency (Property. 29) and the outer-consistency should be replaced with motion consistency (Def. 39). ...