More Like this

1. The Capacity of Uncoded Storage Constrained PIR

   https://doi.org/10.1109/ISIT.2018.8437729  

   Attia, Mohamed Adel; Kumar, Deepak; Tandon, Ravi (June 2018, 2018 IEEE International Symposium on Information Theory (ISIT))

   Private information retrieval (PIR) allows a user to retrieve a desired message out of K possible messages from N databases (DBs) without revealing the identity of the desired message. In this work, we consider the problem of PIR from uncoded storage constrained DBs. Each DB has a storage capacity of μKL bits, where L is the size of each message in bits, and μ ∈ [1/N, 1] is the normalized storage. In the storage constrained PIR problem, there are two key challenges: a) construction of communication efficient schemes through storage content design at each DB that allow download efficient PIR; and b characterizing the optimal download cost via information-theoretic lower bounds. The novel aspect of this work is to characterize the optimum download cost of PIR with storage constrained DBs for any value of storage. In particular, for any (N, K), we show that the optimal tradeoff between storage (μ) and the download cost (D(μ)) is given by the lower convex hull of the pairs ([t/N](1+[1/t]+[1/(t 2 )]+...+[1/(t K-1 )])) for t = 1,2, ..., N. The main contribution of this paper is the converse proof, i.e., obtaining lower bounds on the download cost for PIR as a function of the available storage. 

   more » « less
2. On the Optimal Time/Space Tradeoff for Hash Tables

   Bender, Michael; Farach-Colton, Martin; Kuszmaul, John; Kuszmaul, William; Liu, Mingmou (January 2022, STOC)

   For nearly six decades, the central open question in the study of hash tables has been to determine the optimal achievable tradeoff curve between time and space. State-of-the-art hash tables offer the following guarantee: If keys/values are Θ(logn) bits each, then it is possible to achieve constant-time insertions/deletions/queries while wasting only O(loglogn) bits of space per key when compared to the information-theoretic optimum. Even prior to this bound being achieved, the target of O(log log n) wasted bits per key was known to be a natural end goal, and was proven to be optimal for a number of closely related problems (e.g., stable hashing, dynamic retrieval, and dynamically-resized filters). This paper shows that O(log log n) wasted bits per key is not the end of the line for hashing. In fact, for any k ∈ [log∗ n], it is possible to achieve O(k)-time insertions/deletions, O(1)-time queries, and O(log(k) n) = Ologlog···logn 􏰟 􏰞􏰝 􏰠 k wasted bits per key (all with high probability in n). This means that, each time we increase inser- tion/deletion time by an additive constant, we reduce the wasted bits per key exponentially. We further show that this tradeoff curve is the best achievable by any of a large class of hash tables, including any hash table designed using the current framework for making constant-time hash tables succinct. Our results hold not just for fixed-capacity hash tables, but also for hash tables that are dynamically resized (this is a fundamental departure from what is possible for filters); and for hash tables that store very large keys/values, each of which can be up to no(1) bits (this breaks with the conventional wisdom that larger keys/values should lead to more wasted bits per key). For very small keys/values, we are able to tighten our bounds to o(1) wasted bits per key, even when k = O(1). Building on this, we obtain a constant-time dynamic filter that uses n􏰕logε−1􏰖+nloge+o(n) bits of space for a wide choice of 

   more » « less
3. PIR from Storage Constrained Databases - Coded Caching Meets PIR

   https://doi.org/10.1109/ICC.2018.8422198  

   Tandon, Ravi; Abdul-Wahid, Maryam; Almoualem, Firas; Kumar, Deepak (May 2018, 2018 IEEE International Conference on Communications (ICC))

   Private information retrieval (PIR) allows a user to retrieve a desired message out of K possible messages from N databases without revealing the identity of the desired message. There has been significant recent progress on understanding fundamental information-theoretic limits of PIR, and in particular the download cost of PIR for several variations. Majority of existing works however, assume the presence of replicated databases, each storing all the K messages. In this work, we consider the problem of PIR from storage constrained databases. Each database has a storage capacity of μKL bits, where K is the number of messages, L is the size of each message in bits, and μ ∈ [1/N, 1] is the normalized storage. In the storage constrained PIR problem, there are two key design questions: a) how to store content across each database under storage constraints; and b) construction of schemes that allow efficient PIR through storage constrained databases. The main contribution of this work is a general achievable scheme for PIR from storage constrained databases for any value of storage. In particular, for any (N, K), with normalized storage μ = t/N, where the parameter t can take integer values t ∈ {1, 2, ..., N}, we show that our proposed PIR scheme achieves a download cost of (1 + 1/t + 1/2 + ⋯ + 1/t K-1 ). The extreme case when μ = 1 (i.e., t = N) corresponds to the setting of replicated databases with full storage. For this extremal setting, our scheme recovers the information-theoretically optimal download cost characterized by Sun and Jafar as (1 + 1/N + ⋯ +1/N K-1 ). For the other extreme, when μ = 1/N (i.e., t = 1), the proposed scheme achieves a download cost of K. The most interesting aspect of the result is that for intermediate values of storage, i.e., 1/N <; μ <; 1, the proposed scheme can strictly outperform memory-sharing between extreme values of storage. 

   more » « less
4. Information Theoretic Limits of Data Shuffling for Distributed Learning

   https://doi.org/10.1109/GLOCOM.2016.7841903  

   Attia, Mohamed Adel; Tandon, Ravi (December 2016, IEEE Globecom)

   Data shuffling is one of the fundamental building blocks for distributed learning algorithms, that increases the statistical gain for each step of the learning process. In each iteration, different shuffled data points are assigned by a central node to a distributed set of workers to perform local computations, which leads to communication bottlenecks. The focus of this paper is on formalizing and understanding the fundamental information-theoretic tradeoff between storage (per worker) and the worst-case communication overhead for the data shuffling problem. We completely characterize the information theoretic tradeoff for K = 2, and K = 3 workers, for any value of storage capacity, and show that increasing the storage across workers can reduce the communication overhead by leveraging coding. We propose a novel and systematic data delivery and storage update strategy for each data shuffle iteration, which preserves the structural properties of the storage across the workers, and aids in minimizing the communication overhead in subsequent data shuffling iterations. 

   more » « less
5. Impossibility of efficient information-theoretic fuzzy extraction

   https://doi.org/10.1007/s10623-024-01376-z  

   Fuller, Benjamin (March 2024, Designs, Codes and Cryptography)

   Fuzzy extractors convert noisy signals from the physical world into reliable cryptographic keys. Fuzzy min-entropy measures the limit of the length of key that a fuzzy extractor can derive from a distribution (Fuller et al. in IEEE Trans Inf Theory 66(8):5282–5298, 2020). In general, fuzzy min-entropy that is superlogarithmic in the security parameter is required for a noisy distribution to be suitable for key derivation. There is a wide gap between what is possible with respect to computational and information-theoretic adversaries. Under the assumption of general-purpose obfuscation, keys can be securely derived from all distributions with superlogarithmic entropy. Against information-theoretic adversaries, however, it is impossible to build a single fuzzy extractor that works for all distributions (Fuller et al. 2020). A weaker information-theoretic goal is building a fuzzy extractor for each probability distribution. This is the approach taken by Woodage et al. (in: Advances in Cryptology—CRYPTO, Springer, pp 682–710, 2017). Prior approaches use the full description of the probability mass function and are inefficient. We show this is inherent: for a quarter of distributions with fuzzy min-entropy and $2^k$ points there is no secure fuzzy extractor that uses less $$2^{\Theta(k)}$$ bits of information about the distribution. We show an analogous result with stronger parameters for information-theoretic secure sketches. Secure sketches are frequently used to construct fuzzy extractors. 

   more » « less