Capacity-Achieving Private Information Retrieval Codes From MDS-Coded Databases With Minimum Message Size

Abstract

We consider constructing capacity-achieving linear codes with minimum message size for private information retrieval (PIR) from N non-colluding databases, where each message is coded using maximum distance separable (MDS) codes, such that it can be recovered from accessing the contents of any T databases. It is shown that the minimum message size (sometimes also referred to as the sub-packetization factor) is significantly, in fact exponentially, lower than previously believed. More precisely, when K > T / gcd(N, T ) where K is the total number of messages in the system and gcd(·, ·) means the greatest common divisor, we establish, by providing both novel code constructions and a matching converse, the minimum message size as lcm(N − T, T ), where lcm(·, ·) means the least common multiple. On the other hand, when K is small, we show that it is in fact possible to design codes with a message size even smaller than lcm(N − T, T ). arXiv:1903.08229v2 [cs.IT] 22 Jan 2020 significantly restricts the possible usage scenarios. Moreover, a large message size also usually implies that the encoding and the decoding functions are more complex, which not only requires more engineering efforts to implement but also hinders the efficiency of the system operation. From a theoretical point of view, a code with a smaller message size usually implies a more transparent coding structure, which can be valuable for related problems; see, e.g., [7] for such an example. Thus codes with a smaller message size are highly desirable in both theory and practice.The capacity-achieving code given in [4] requires L = T N K , which can be extremely large for a system with even a moderate number of messages. The problem of reducing the message size of capacity-achieving codes was recently considered by Xu and Zhang [6], and it was shown that under the assumption that all answers are of the same length, the message size must satisfy L ≥ T (N/ gcd(N, T )) K−1 . These existing results may have left the impression that capacityachieving codes would necessitate a message size exponential in the number of messages.In this work, we show that the minimum message size for capacity-achieving PIR codes can in fact be significantly smaller than previously believed, by providing capacity-achieving linear codes with message size L = lcm(N −T, T ). Two linear code constructions, referred to as Construction-A and Construction-B, respectively, are given. The two constructions have the same download cost and message size, however Construction-B has a better upload cost (i.e., a lower communication cost for the user to send the queries), at the expense of being slightly more sophisticated than Construction-A. The key difference between the two proposed constructions and existing codes in the literature is that the proposed codes reduce the reliance on the so-called variety symmetry [8], which should be distinguished from the asymmetry discussed in [9], and the answers may be of different lengths 1 . We further show that this is in fact the minimum message size whe...

On the Capacity of Leaky Private Information Retrieval

“…• Perfect user privacy and DB privacy [9] ( = 0, δ = 0). By setting = 0, δ = 0 in Theorems 1 and 2, we obtain the SPIR results in [9] where the optimal required shared randomness is given by α * (0, 0) = α 1 (0, 0) = α 2 (0, 0) = 1 N −1 and the optimal download cost is obtained using the bounds in (17) and (20) as D * ( = 0, δ = 0) = D LB ( = 0, δ = 0) = D UB ( = 0, δ = 0) = 1 + 1 N − 1 .…”

“…which proves that δ 1 ( ) must be greater than or equal δ 2 ( ) for any value of ≥ 0. Following that, we can write the multiplicative gap ratio between the upper and lower bounds on D * ( , δ) given in (17) and (20) as follows:…”

On the Capacity of Private Information Retrieval from Coded, Colluding, and Adversarial Servers

“…This is in contrast to the schemes in [2]- [4], [18], where each file is assumed to be subdivided into a number of packets that grows exponentially with the number of files m. It was shown in [18] that an exponential (in m) number of packets per file was necessary for a PIR scheme with optimal download rate, under the assumption that all servers respond to the queries and the responses have the same size. In [19] a scheme was presented that achieves the capacity with only O(n) packets by making a weaker assumption on the size of the responses than in [18].…”

“…Here, we have assumed that all the servers respond with equal size responses. However, by loosening this assumption, improvements for finite m are possible, along the same lines as in [19].…”

On the Information Leakage in Private Information Retrieval Systems

“…For K = 2, the individual leakage is equal to the total leakage. The minimum download cost for the same value of individual and total leakage levels are the same, which can be easily verified from (28) and (30). Similarly, the minimum amount of common randomness are also equal to each other, i.e., ρ s min = ρ w min , which can be verified from (27) and (29).…”