Publicly accessible databases are an indispensable resource for retrieving up to date information. But they also pose a significant risk to the privacy of the user, since a curious database operator can follow the user's queries and infer what the user is after. Indeed, in cases where the users ' intentions are to be kept secret, users are often cautious about accessing the database. It can be shown that when accessing a single database, to completely guarantee the privacy of the user, the whole database should be downloaded, namely n bits should be communicated (where n is the number of bits in the database). In this work, we investigate whether by replicating the database, more efficient solutions to the private retrieval problem can be obtained. We describe schemes that enable a user to access k replicated copies of a database (k * 2) and privately retrieve information stored in the database. This means that each individual database gets no information on the identity of the item retrieved by the user. Our schemes use the replication to gain substantial saving. In particular, we have ffl A two database scheme with communication complexity of O(n1=3). ffl A scheme for a constant number, k, of databases with communication complexity O(n1=k). ffl A scheme for 13 log2 n databases with polylogarithmic (in n) communication complexity.

Alice wants to query a database but she does not want the database to learn what she is querying. She can ask for the entire database. Can she get her query answered with less communication? One model of this problem is Private Information Retrieval, henceforth PIR. We survey results obtained about the PIR model including partial answers to the following questions. (1) What if there are k non-communicating copies of the database but they are computationally unbounded? (2) What if there is only one copy of the database and it is computationally bounded? 1

There have been several papers over the last ten years that consider the number of queries needed to compute a function as a measure of its complexity. The following function has been studied extensively in that light: F A a (x 1 ; : : : ; x a ) = A(x 1 ) 1 1 1 A(x a ): We are interested in the complexity (in terms of the number of queries) of approximating F A a . Let b a and let f be any function such that F A a (x 1 ; : : : ; x a ) and f(x 1 ; : : : ; x a ) agree on at least b bits. For a general set A we have matching upper and lower bounds that depend on coding theory. These are applied to get exact bounds for the case where A is semirecursive, A is superterse, and (assuming P 6= NP) A = SAT. We obtain exact bounds when A is the halting problem using different methods. 1 Introduction The complexity of a function can be measured by the number of queries (to some oracle) needed to compute it. This notion has been studied in both a 3 Dept. of Computer Sc...

The multicovering radii of a code are a recently introduced natural generalizations of the covering radius measuring the smallest radius of balls around codewords that cover all m-tuples of vectors. In this paper we prove a new identity relating the multicovering radii of a code to a relativized notion of ordinary covering radius. This identity is used to prove new bounds on the multicovering radii of particular codes. 1 Keywords: Covering radius, multicovering radius, coding theory, Hamming codes. AMS Classication: 94B65 (94B75, 94B05, 05B40) Running head: Multicovering Bounds from Relative Covering Radii 1 Introduction and Denitions The concept of multicovering radius was introduced by Klapper [5] in the context of studying the existence of stream ciphers secure against a large class of attacks. Let C be a code of length n and m be a positive integer. The m-covering radius of C is the smallest integer r such that every set of m vectors in F n is contained in at least one bal...

: In this work construction methods for so called mixed covering codes are developed. There has been considerable recent growth in the interest in covering codes. They have the property that all words in the space are within a given Hamming distance, called the covering radius, from some codeword. Traditionally, mainly spaces where all coordinates have the same arity (usually 2 or 3, that is, they are binary or ternary) have been discussed. In this work we consider mixed spaces F n 1 q 1 F n 2 q 2 \Delta \Delta \Delta F nm q m , where the coordinates can be of varying arities. The approach is very general, no restrictions are set upon m and the arities q i . The construction methods consist of generalizations of known constructions for covering codes and some completely new constructions. They are divided into three classes; direct constructions, constructions of new codes from old, and the matrix method. Through these constructions upper bounds for the minimal number of codewords...

The minimum number of codewords in a code with t ternary and b binary coordinates and covering radius R is denoted by K(t; b; R). In this paper, necessary and sucient conditions for K(t; b; R) = M are given for all M 5. By the help of generalized s-surjective codes, we develop new methods for nding bounds on K(t; b; R). These results are used to prove the equality K(9; 0; 5) = 6 as well as some new lower bounds such as K(2; 7; 3) 7, K(3; 6; 3) 8, K(5; 3; 3) 8, and K(9; 0; 4) 9. Some bounds for (nonmixed) quaternary codes are also obtained.

Alice wants to query a database but she does not want the database to learn what she is querying. She can ask for the entire database. Can she get her query answered with less communication? One model of this problem is Private Information Retrieval, henceforth PIR. We survey results obtained about the PIR model including partial # University of Maryland, Dept. of Computer Science and Institute for Advanced Computer Studies, College Park, MD 20742. gasarch@cs.umd.edu answers to the following questions. (1) What if there are k non-communicating copies of the database but they are computationally unbounded? (2) What if there is only one copy of the database and it is computationally bounded? 1

A covering code in a Hamming space is a set of codewords with the property that any word in the space is within a specified Hamming distance, the covering radius, from at least one codeword. In this thesis, efficient construction methods for such codes are considered. The constructions work, with some exceptions, for codes over alphabets consisting of any number of symbols. Codes over mixed alphabets are also discussed. Most of the methods are developed in order to determine values of K q (n; R), the minimumnumber of codewords in a q-ary code of length n and covering radius R. Codes obtained by the constructions prove upper bounds on this function. In many of the constructions simulated annealing, a probabilistic optimization method, has turned out to perform very well. Simulated annealing cannot be used to prove optimality of codes found; in that case, the problem is viewed and solved as a set covering problem. For larger codes, a direct approach is not generally feasible; it is sho...

ABSTRACT: We describe schemes that enable a user to accesskreplicated copies of a database (k2) and privately retrieve information stored in the database. This means that each individual database gets no information on the identity of the item retrieved by the user. For a single database, achieving this type of privacy requires communicating the whole database, ornbits (wherenis the number of bits in the database). Our schemes use the replication to gain substantial saving. In particular, we have A two database scheme with communication complexity of O(n1=3). A scheme for a constant number,k, of databases with communication complexityO(n1=k). A scheme for13log2ndatabases with polylogarithmic (inn) communication complexity. 1