On single server private information retrieval in a coding theory perspective

In this paper, we present a new perspective of single server private information retrieval (PIR) schemes by using the notion of linear error-correcting codes. Many of the known single server schemes are based on taking linear combinations between database elements and the query elements. Using the theory of linear codes, we develop a generic framework that formalizes all such PIR schemes. Further, we describe some known PIR schemes with respect to this code-based framework, and present the weaknesses of the broken PIR schemes in a generic point of view

Bounds for Coding Theory over Rings

Coding theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. It has been well established that, with the generalization of the algebraic structure to rings, there is a need to also generalize the underlying metric beyond the usual Hamming weight used in traditional coding theory over finite fields. This paper introduces a generalization of the weight introduced by Shi, Wu and Krotov, called overweight. Additionally, this weight can be seen as a generalization of the Lee weight on the integers modulo 4 and as a generalization of Krotov’s weight over the integers modulo 2s for any positive integer s. For this weight, we provide a number of well-known bounds, including a Singleton bound, a Plotkin bound, a sphere-packing bound and a Gilbert–Varshamov bound. In addition to the overweight, we also study a well-known metric on finite rings, namely the homogeneous metric, which also extends the Lee metric over the integers modulo 4 and is thus heavily connected to the overweight. We provide a new bound that has been missing in the literature for homogeneous metric, namely the Johnson bound. To prove this bound, we use an upper estimate on the sum of the distances of all distinct codewords that depends only on the length, the average weight and the maximum weight of a codeword. An effective such bound is not known for the overweight

Generalization of the Ball-Collision Algorithm

In this paper we generalize the ball-collision algorithm by Bernstein, Lange, Peters from the binary field to a general finite field. We also provide a complexity analysis and compare the asymptotic complexity to other generalized information set decoding algorithms

Generic Decoding of Restricted Errors

Several recently proposed code-based cryptosystems base their security on a slightly generalized version of the classical (syndrome) decoding problem. Namely, in the so-called restricted (syndrome) decoding problem, the error values stem from a restricted set. In this paper, we propose new generic decoders, that are inspired by subset sum solvers and tailored to the new setting. The introduced algorithms take the restricted structure of the error set into account in order to utilize the representation technique efficiently. This leads to a considerable decrease in the security levels of recently published code-based cryptosystems

On the Hardness of the Lee Syndrome Decoding Problem

In this paper we study the hardness of the syndrome decoding problem over finite rings endowed with the Lee metric. We first prove that the decisional version of the problem is NP-complete, by a reduction from the 3-dimensional matching problem. Then, we study the actual complexity of solving the problem, by translating the best known solvers in the Hamming metric over finite fields to the Lee metric over finite rings, as well as proposing some novel solutions. For the analyzed algorithms, we assess the computational complexity in both the finite and asymptotic regimes.Comment: Part of this work appeared as preliminary results in arXiv:2001.0842