dissertacao.pdf

1 Introduction
1.1 Cryptography
Cryptography is a science practised for over four thousand years. It regards,
in a rather simple way, the finding and improvement of methods for sending
messages between two chosen parts without risk that an unauthorized third
party can understand the sent messages.
While in the beginning this process was achieved by sending the messages
personally, i.e., from mouth to hear, with the increasing of the communca-
tions’ distance this has become impossible. So cryptographers started devel-
oping methods of sending messages from a sender, Alice, to a receiver, Bob,
without a malicious eavesdropper, Marvin, being able to understand the mes-
sages unless he knows exactly how they were encrypted. Such a method is called
a cryptosystem. Its mathematical description is given below:
Definition 1. Cryptosystem is a tuple < M, C, K, e, d > where:
M is the set of plaintext messages
C is the set of cypher text messages
K is the set of keys
e is the encryption function, e : M × K → C such that e(m|k) = c
d is the decryption function, d : C × K → M such that d(e(m|k)|k) = m
To send a plaintext m, Alice encrypts it with an a-priori defined key k, obtaining
the cypher text c = e(m|k), which she sends to the Bob, who decrypts it using
the same key k obtaining m = d(e(m|k)|k).
The key k should be known only by Alice and Bob, otherwise Marvin could
decrypt c just by knowing the decryption function. For this reason, this kind of
cryptosystems are called Private Key Cryptosystems or Symmetric Key
Cryptosystems, as the same key is used for encryption and decryption.
Throughout the years, numerous private key cryptosystems were created,
with ever increasing complexity. Eventually, they were considered insufficient.
What made them insufficient was the requirement that the secret key had to be
known both to Alice and Bob. This means that before exchanging the encrypted
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