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

5 1.3 Asymmetric Key Cryptography
this communication. Now, in case of such a dispute, Bob will have to prove
to an unbiased arbitrator (may be the judge or jury at the court) that the
message was in fact sent by Alice. Now, the arbitrator can not verify who
actually encrypted/signed that particular plaintext, as both Alice and Bob
are capable of encrypting/signing it using the same (shared) key. This problem
of authentication is known as repudiation. This problem can not be
solved using symmetric key systems, as the concept of non-repudiation using
signatures can not be accomplished.
To overcome these problems, Diffie and Hellman [33] introduced the notion of
‘asymmetric key’ cryptosystems in 1976. Invention of asymmetric key cryptosystems
is arguably the most celebrated breakthrough in modern day cryptography.
However, one should note that public key cryptosystems are much slower than
symmetric key cryptosystems in general.
1.3 Asymmetric Key Cryptography
In asymmetric key cryptography, the encryption and the decryption keys are different,
but they are related by some mathematical relation. These type of cryptosystems
are also called ‘public key’ cryptosystems, as the encryption key is made
public while the decryption key is kept secret. Hence, it is required for the security
of the system that, finding the decryption key from the encryption key is
computationally infeasible.
The public key cryptosystems are most often based on some computationally
hard mathematical problem. A list of well known hard problems is as follows.
Integer Factorization Problem (IFP). The problem is to find a proper factor
of a given positive integer N > 1. The first usable public key cryptosystem
RSA [110] is based on the hardness of integer factorization problem. Rabin
cryptosystem [104] is also based on integer factorization problem.
Quadratic Residue Problem (QRP). The problem is to decide whether an
element x ∈ Z N has a modular square root where N is a composite number.
It can be proved that if factorization of N is known then QRP in Z N is no
longer hard. Goldwasser-Micali [45] cryptosystem is based on QRP.