Cryptanalysis of RSA Factorization - Library(ISI Kolkata) - Indian ...

Chapter 2
Mathematical Preliminaries
This chapter is dedicated to provide the reader with a comprehensive overview
of the mathematical framework one may need to read this thesis. Due to the
requirement for complexity analysis of the algorithms we discuss in this thesis, let
us start with a brief overview of the basic asymptotic notations in Section 2.1.
In Section 2.2, we discuss the RSA cryptosystem in detail and also describe the
different variants of RSA used in practice. Next, we move on to our basic focus,
cryptanalysis of RSA, in Section 2.3 and present a comprehensive summary of
attacks on the RSA cryptosystem proposed in the last few decades. Most of the
works in this thesis, as well as many partial key exposure attacks on RSA depend
on lattice based polynomial solving techniques. Thus we study the basic properties
of lattices in Section 2.4, and discuss the existing techniques to solve modular and
integer polynomials in Sections 2.5 and 2.6 respectively.
2.1 Asymptotic Notation
In mathematics, computer science, and other fields related to computation, the
time and space requirement for an algorithm to run is generally represented as a
function of the size of the input(s). In this context, the asymptotic notations are
used to describe the limiting behavior of these functions, in an asymptotic sense,
as the input size tends to a specific value or towards infinity. Bachmann-Landau
notations [27] are a family of asymptotic notations used to compare computational
complexity (time or space) of algorithms. This family comprises of notations ‘Big
O’ (O), ‘Small o’ (o), ‘Big Omega’ (Ω), ‘Big Theta’ (Θ) and ‘Small Omega’ (ω),
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