Bounds and Constructions for Generalized Batch Codes

Private information retrieval (PIR) codes and batch codes are two important types of codes that are designed for coded distributed storage systems and private information retrieval protocols. These codes have been the focus of much attention in recent years, as they enable efficient and secure storage and retrieval of data in distributed systems. In this paper, we introduce a new class of codes called \emph{$(s,t)$-batch codes}. These codes are a type of storage codes that can handle any multi-set of $t$ requests, comprised of $s$ distinct information symbols. Importantly, PIR codes and batch codes are special cases of $(s,t)$-batch codes. The main goal of this paper is to explore the relationship between the number of redundancy symbols and the $(s,t)$-batch code property. Specifically, we establish a lower bound on the number of redundancy symbols required and present several constructions of $(s,t)$-batch codes. Furthermore, we extend this property to the case where each request is a linear combination of information symbols, which we refer to as \emph{functional $(s,t)$-batch codes}. Specifically, we demonstrate that simplex codes are asymptotically optimal functional $(s,t)$-batch codes, in terms of the number of redundancy symbols required, under certain parameter regime.Comment: 25 page

분산 컴퓨팅과 캐시를 접목한 정보 검색에서의 보안 및 프라이버시

학위논문(박사)--서울대학교 대학원 :공과대학 전기·정보공학부,2020. 2. 이정우.많은 양의 데이터 저장이나 데이터 계산을 위해서는 분산 시스템이 필수적이다. 이러한 분산 시스템의 데이터 저장과 계산의 효율의 높이는 반면, 데이터의 보안과 프라이버시에 대한 위험도 증가시킨다. 본 논문에서는 데이터 저장과 데이터 계산을 위한 분산 시스템에서 데이터에 대한 보안과 프라이버시를 고려한다. 특히, 이러한 시스템에 대하여 보안과 프라이버시를 보장하는 부호화 기법을 제안한다. 우선, 유저가 사전에 캐시에 일정량의 데이터를 저장하고 있는 cache-aided PIR을 제안한다. 제안하는 기법은 기존 PIR 문제의 최적 기법을 기반으로 한다. 제안하는 기법에서, 캐시에 저장된 데이터는 부가정보로 이용되며, 이는 캐시가 없을 때 대비 다운로드양의 감소로 이어진다. 두 번째로, 부호화된 분산 컴퓨팅 시스템에서 마스터의 프라이버시를 고려한다. 이 시스템에서 워커들과 마스터는 각각 고유한 데이터를 가지며, 워커들의 데이터는 라이브러리 형태로 이루어진다. 마스터는 자신의 데이터와 데이터 라이브러리 내 특정 데이터의 함수를 계산해야 한다. 이 때 마스터의 프라이버시는 워커들이 마스터가 라이브러리 안의 어떤 데이터를 원하는지 모르는 것을 뜻한다. 이러한 시스템을 private coded computation이라 하며, 제안하는 기법을 private polynomial codes라 한다. 제안하는 기법에서는 기존의 polynomial codes에서는 고려되지 않았던 비동기적 기법이 도입된다. 이로 인하여 제안하는 기법은 변형된 최적의 RPIR 기법대비 더 빠른 계산시간을 달성한다. 끝으로, 부호화된 분산 컴퓨팅 시스템에서 마스터의 프라이버시와 데이터 보안을 동시에 고려한다. 데이터 보안은 마스터의 고유한 데이터를 워커들로부터 보호하는 것을 의미한다. 이러한 시스템을 private secure coded computation이라 하며, 제안하는 기법을 private secure polynomial codes라 한다. Private polynomial codes를 변형하여 private secure polynomial codes와 private polynomial codes를 계산시간과 통신량 측면에서 비교한다. 그 결과, 같은 양의 통신량에 대하여, private secure polynomial codes가 더 빠른 계산 시간을 달성한다.As a major format of data changes from the text to the videos, the amount of memory for storing data increases exponentially, as well as the amount of computation for handling the data. As a result, to alleviate these burdens of storage and computations, the distributed systems are actively studied. Meanwhile, since low latency is one of the main objectives in 5G communications, recent techniques such as edge computing or federated learning in machine learning become important. Since the decentralized systems are fundamental characteristics of these techniques, the distributed systems which include the decentralized systems also become important. In this dissertation, I consider the distributed systems for storage and computation. For the data storage, large-scale data centers collectively store a library of files where the size of each file is tremendous. When a user needs a specific file, it can be downloaded from distributed data centers. In this system, minimizing the amount of download is a significant concern. The user's privacy in this system implies that the user should conceal the index of its desired file against the databases. This kind of problem is referred to as private information retrieval (PIR) problem. The goal of PIR problem is to minimize the amount of download from the databases while ensuring the user's privacy. Meanwhile, for a large amount of computation, the user can divide the whole computation into sub-computations and distribute them to external workers who constitute a distributed system. There can be three cases for the computation. Firstly, the user may own all of the data to be computed and sends both of its data and instructions for the computation to the workers. Secondly, the workers collectively own all of the data and the user just sends instructions for the data selection and computation to the workers. Thirdly, the user and the workers have their own data respectively and the user sends the data and instructions for the data selection and computation to the workers. Since some of the workers can be slow for various reasons, the user may use a coding technique, e.g., an erasure code, to avoid the delaying effect caused by the slow workers. This kind of technique is referred to as coded computation. In these systems, speeding up the computation process is a significant concern. In this dissertation, I focus on the third system. In the considered system, the privacy is similar to that of distributed systems for storage. On the other hand, the security implies that the user should conceal the content of its own data against the workers so that the workers do not have any information about the user's own data. In this dissertation, I consider the user's privacy in distributed systems for storage, and both of the privacy and security in distributed systems for the computation. In case of the distributed systems for storage, since the user does not have its own data, the data security on the user's data cannot be considered. Particularly, I propose some achievable schemes that ensure the privacy and security in these systems. To begin with, as a new variation of PIR problem, I consider a user's cache that has some pre-stored data of databases' library. I refer to this problem as cache-aided PIR problem. By introducing the user's cache in the PIR problem, the amount of download from the databases is significantly reduced. The achievable scheme is based on the optimal scheme for conventional PIR problem. In the achievable scheme, the pre-store cache was exploited as an side information, which reduces the amount of download, compared to the PIR problem without cache. Secondly, I consider the master's privacy in coded computation. In the system model, the workers have their own data, as well as the master. The workers' data constitutes a library of several files. The master should compute a function of its own data and a specific file in the library. The master's privacy implies that the workers' should not know which file in the library is desired by the user. I refer to this problem as private coded computation and propose an achievable scheme of private coded computation, namely private polynomial codes. The private polynomial codes are based on the polynomial codes which were proposed in the conventional coded computation system. In the achievable scheme, the workers are grouped for the privacy and asynchronous scheme is considered, which was not considered in the conventional polynomial codes. Due to the asynchronous scheme, the proposed scheme achieves the faster computation time, compared to the modified optimal RPIR scheme. Lastly, I consider the data security in coded computation, as well as the master's privacy. The system model is similar to that of private coded computation. The data security implies that the master should protect its own data against the workers. I refer to this problem as private secure coded computation and propose an achievable scheme, namely private secure polynomial codes. The private secure polynomial codes are based on the polynomial codes which were proposed in the conventional coded computation system. By modifying the private polynomial codes, the private secure polynomial codes and private secure polynomial codes are compared in terms of computation time and communication load. As a result, the private secure polynomial codes achieves faster computation time for the same communication load.1. Introduction 1 1.1 Related work 3 1.1.1 Private information retrieval 3 1.1.2 Coded computation 4 1.2 Contributions and Organization 5 2. Cache-aided Private Information Retrieval 8 2.1 Introduction 8 2.2 System model 9 2.3 Main results : 12 2.4 Achievable scheme 17 2.4.1 Cacheless phase 17 2.4.2 Cache-assisted phase 21 2.4.3 Cache-aided PIR 24 2.5 Tightness of achievable scheme 29 2.6 Conclusions and follow-up works 30 3. Private Coded Computation 32 3.1 Introduction 32 3.2 System model 37 3.3 Main results 41 3.4 Private polynomial codes 42 3.4.1 First example 42 3.4.2 Second example 48 3.4.3 General description 52 3.4.4 Privacy proof 56 3.4.5 Performance analysis 59 3.4.6 Special cases 61 3.5 Simulation results 62 3.5.1 Computation time 62 3.5.2 Communication load 68 3.6 Conclusion 69 4. Private Secure Coded Computation 71 4.1 Introduction 71 4.2 Main results 75 4.3 Private secure polynomial codes 76 4.3.1 Illustrative example 76 4.3.2 General description 80 4.3.3 Performance analysis 83 4.3.4 Privacy and security proof 84 4.4 Simulation results 85 4.4.1 Computation time 86 4.4.2 Communication load 90 4.5 Conclusion 91 5 Conclusion 93 5.1 Summary 93 5.2 Future directions 94 국문초록 105 Acknowledgement 107Docto

Private Information Retrieval Schemes for Coded Data with Arbitrary Collusion Patterns

In Private Information Retrieval (PIR), one wants to download a file from a database without revealing to the database which file is being downloaded. Much attention has been paid to the case of the database being encoded across several servers, subsets of which can collude to attempt to deduce the requested file. With the goal of studying the achievable PIR rates in realistic scenarios, we generalize results for coded data from the case of all subsets of servers of size $t$ colluding, to arbitrary subsets of the servers. We investigate the effectiveness of previous strategies in this new scenario, and present new results in the case where the servers are partitioned into disjoint colluding groups.Comment: Updated with a corrected statement of Theorem

Secure and Private Cloud Storage Systems with Random Linear Fountain Codes

An information theoretic approach to security and privacy called Secure And Private Information Retrieval (SAPIR) is introduced. SAPIR is applied to distributed data storage systems. In this approach, random combinations of all contents are stored across the network. Our coding approach is based on Random Linear Fountain (RLF) codes. To retrieve a content, a group of servers collaborate with each other to form a Reconstruction Group (RG). SAPIR achieves asymptotic perfect secrecy if at least one of the servers within an RG is not compromised. Further, a Private Information Retrieval (PIR) scheme based on random queries is proposed. The PIR approach ensures the users privately download their desired contents without the servers knowing about the requested contents indices. The proposed scheme is adaptive and can provide privacy against a significant number of colluding servers.Comment: 8 pages, 2 figure

On the Asymptotic Capacity of XX-Secure TT-Private Information Retrieval with Graph Based Replicated Storage

The problem of private information retrieval with graph-based replicated storage was recently introduced by Raviv, Tamo and Yaakobi. Its capacity remains open in almost all cases. In this work the asymptotic (large number of messages) capacity of this problem is studied along with its generalizations to include arbitrary $T$-privacy and $X$-security constraints, where the privacy of the user must be protected against any set of up to $T$ colluding servers and the security of the stored data must be protected against any set of up to $X$ colluding servers. A general achievable scheme for arbitrary storage patterns is presented that achieves the rate $(\rho_{\min}-X-T)/N$, where $N$ is the total number of servers, and each message is replicated at least $\rho_{\min}$ times. Notably, the scheme makes use of a special structure inspired by dual Generalized Reed Solomon (GRS) codes. A general converse is also presented. The two bounds are shown to match for many settings, including symmetric storage patterns. Finally, the asymptotic capacity is fully characterized for the case without security constraints $(X=0)$ for arbitrary storage patterns provided that each message is replicated no more than $T+2$ times. As an example of this result, consider PIR with arbitrary graph based storage ($T=1, X=0$) where every message is replicated at exactly $3$ servers. For this $3$-replicated storage setting, the asymptotic capacity is equal to $2/\nu_2(G)$ where $\nu_2(G)$ is the maximum size of a $2$-matching in a storage graph $G[V,E]$. In this undirected graph, the vertices $V$ correspond to the set of servers, and there is an edge $uv\in E$ between vertices $u,v$ only if a subset of messages is replicated at both servers $u$ and $v$