Some Results on Combinatorial Batch Codes and Permutation Binomials over Finite Fields

Abstract

In this thesis,we study combinatorial batch codes (CBCs) and permutation binomials (PBs) over �nite �elds of even characteristic. Our primary motivation for considering these problems comes from their importance in cryptography. CBCs are replication based variants of batch codes, which were introduced in [IKOS04a] as a tool for reducing the computational overhead of private information retrieval protocols (a cryptographic primitive). On the other hand, permutation polynomials, with favourable cryptographic properties, have applications in symmetric key encryption schemes, especially in block ciphers. Moreover, these two objects are interesting in their own right, and they have connections with other important combinatorial objects. CBCs are much similar to unbalanced expanders, a much studied combinatorial object having numerous applications in theoretical computer science. On the other hand, the speci�c class of PBs that we consider in this work, are intimately related to orthomorphisms. Orthomorphisms are relevant in the construction of mutually orthogonal latin squares, a classical combinatorial objects having applications in design of statistical experiments. These aspects motivate us to explore theoretical properties of CBCs and PBs over �nite �elds. However, these two objects are inherently widely di�erent; CBCs are purely combinatorial objects, and PBs are algebraic entities. So, we explore these two objects independently in two di�erent parts, where our entire focus lies in exploring theoretical aspects of these objects. In Part I, we consider CBCs. There, we provide bounds on the parameters of CBCs and obtain explicit constructions of optimal CBCs. In Part II, we consider PBs over �nite �elds. There, we obtain explicit characterization and enumeration of subclasses of PBs under certain restrictions. Next, we describe these two parts in more detail.