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

Chapter 1: Introduction 8
Chapter 5: In addition to the weaknesses of RSA due to weak keys, it may also be
vulnerable due to leaked information about the RSA primes. In this chapter, we
discuss the factorization of the RSA modulus N by reconstructing the primes from
randomly known bits. In Sections 5.1 and 5.2, we analyze the fact that N = pq
can be factored in reasonable time complexity when a few bits of the primes are
known from the least significant halves. In Section 5.3, we analyze the same when
a few bits are known from the most significant halves of the primes. The materials
of this chapter are based on our publication [82].
Chapter 6: At times, one may not obtain explicit bitwise information about
the primes, but have some implicit knowledge regarding those. An attempt at
RSA factorization based on this implicit knowledge is of interest as well. This
chapter deals with the analysis of a situation when one can factor RSA moduli
N 1 = p 1 q 1 ,N 2 = p 2 q 2 ,...,N k = p k q k in polynomial time if p 1 ,p 2 ,...,p k share a
few bits. This problem is called the implicit factorization problem, introduced by
May and Ritzenhofen in [86]. In Section 6.1, we present the implicit factorization
strategy for two or three large integers when they share MSBs and/or LSBs. Next,
in Section 6.2, we analyze the same problem when the primes p 1 ,p 2 share a (contiguous)
portion of bits at the middle. The materials of this chapter are based on
our publications [113,116].
Chapter 7: In this chapter, we present two generalizations, Extended Partially
Approximate Common Divisor Problem (EPACDP) and Extended General Approximate
Common Divisor Problem (EGACDP), of the ‘approximate common
divisor problem’ introduced by Howgrave-Graham [61]. We also propose two applications
of ‘approximate common divisor problem’. In Sections 7.4 and 7.6, we
propose two methods to solve EPACDP, and in Section 7.7, we discuss the solution
of EGACDP. Most importantly, continuing from Chapter 6, we discuss the applications
of EPACDP for implicit factorization when p 1 ,p 2 ,...,p k share some MSBs
andorLSBs. Thematerialsofthischapterarebasedonourpublications[112,116].
Chapter 8: This chapter concludes the thesis. Here we present a comprehensive
summary of our work that has been discussed throughout the thesis. We analyze
andcompareourworkwiththecontemporaryadvancesinthefieldofcryptography
and also discuss open problems which might be interesting for further investigation
along this line of research.