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.
