Bounds on the Length of Functional PIR and Batch Codes

Abstract

A functional k-PIR code of dimension s consists of n servers storing linear combinations of s linearly independent information symbols. Any linear combination of the s information symbols can be recovered by k disjoint subsets of servers. The goal is to find the smallest number of servers for given k and s. We provide lower bounds on the number of servers and constructions which yield upper bounds on this number. For k ≤ 4, exact bounds on the number of servers are proved. Furthermore, we provide some asymptotic bounds. The problem coincides with the well known private information retrieval problem based on a coded database to reduce the storage overhead, when each linear combination contains exactly one information symbol.If any multiset of size k of linear combinations from the linearly independent information symbols can be recovered by k disjoint subset of servers, then the servers form a functional k-batch code. A functional k-batch code is a functional k-PIR code, where all the k linear combinations in the multiset are equal. We provide some bounds on the number of servers for functional k-batch codes. In particular we present a random construction and a construction based on simplex codes, WOM codes, and RIO codes.

On some batch code properties of the simplex code

“…The binary k × n matrix G (as well as the binary linear code generated by G) is (i) a t-PIR code, (ii) a t-batch code, (iii) a t-odd batch code, or (iv) a t-functional batch code if G can serve any request sequence of length t consisting of (i) the t-fold repetition of a unit vector in E k , (ii) unit vectors in E k only, (iii) vectors in F k 2 of odd weight only, or (iv) nonzero vectors in F k 2 , respectively. The notions of t-PIR code and t-batch code are well known (but note that some authors employ a more general definition and refer to these codes as multiset primitive), and together with t-functional batch codes are defined, for example, in [15,16]. For a recent overview of these and related types of codes, see [7].…”

“…In [9] (see also [10]) it was shown that G k is a 2 k−1 -batch code, but the proof is somewhat cumbersome. Recently, it was conjectured that G k is even a 2 k−1 -functional batch code, and it was shown that G k is a t-functional batch code for t = 2 k−2 + 2 k−4 + ⌊2 k/2 / √ 24⌋, again with a rather involved proof [15,16]. After completion of this paper, we learned that this result was further improved in [13] and [12], where it was shown that G k is a t-functional batch code for t = ⌊(2/3) • 2 k−1 ⌋ and t = ⌊(5/6) • 2 k−1 ⌋ − k, respectively.…”

“…(For precise definitions of this and other used notions, we refer to Section 2. ) In this paper, we give a simple, algorithmic proof of a result that falls halfway between the known result for the simplex code in [9,10] and the conjecture in [15,16]. Our approach is to relate the required properties of the simplex code to certain additive problems in finite abelian groups.…”

“…The proof of that result is somewhat cumbersome, and the algorithm resulting from the proof requires to store and use a database containing all the solutions for the cases where k ≤ 7. More recently, in [15,16] the authors conjecture that the k-dimensional simplex code is even a 2 k−1 -functional batch code. (For precise definitions of this and other used notions, we refer to Section 2.…”

Multi-Server Weakly-Private Information Retrieval

“…The case of maximum distance separable (MDS) coded servers was considered in [11], [12], while the case of arbitrary linear coded servers was studied in [13]- [15]. The concept of PIR has also been extended to several other relevant scenarios, which include colluding servers [11], [13], [15]- [19], robust PIR [16], PIR with Byzantine servers [20], optimal upload cost of PIR, i.e., the minimum required amount of query information [21], access complexity of PIR, i.e., the number of symbols accessed across all servers for the retrieval of a single file [22], single-server PIR with private side information [23], PIR on graph-based replication systems [24], PIR with secure storage [25], functional PIR codes [26], and private proximity retrieval codes [27].…”