dissertacao.pdf

Definition 2. [Diffie-Hellman concept of public key cryptosystem] A
cryptosystem where the key consists of a pair of public/private keys, where the
encryption function uses the public key and the decryption function uses the
private key, such that:
it should be easy to create pairs of public/private keys
it should be easy do encrypt messages knowing the public key
it should be easy do decrypt messages knowing the private key
it should be hard to compute the private key from the public key
For a public key cryptosystem to be safe, it should be hard to invert the
encryption function without knowing the private key, and it should be hard
to deduce the private key from the public key alone. This kind of functions,
which are easy to compute but difficult to invert without knowing some extra
parameters are called trapdoor one-way functions.
Regarding cryptosystems used nowadays this inversion is not impossible:
rather it is extremely time consuming, making it useless for Marvin to try it
when the information being transmitted is only relevant for a short period of
time.
1.3 The RSA cryptosystem
The RSA cryptosystem is the first ever published public key cryptosystem, de-
veloped by Rivest, Shamir and Adleman, first presented in their 1978 article[38],
and based on the Diffie-Hellman proposal. Its implementation depends on
an a priori choice of two large prime numbers p and q, that are multiplied
to obtain the RSA modulus N = pq and a subsequent choice of a public and a
private integer parameters, e and d, satisfying ed = 1 + k(p − 1)(q − 1) for some
integer k. These two computations are actually a trapdoor one way function of
RSA: while it is easy to compute N = pq, we will show that it is hard to factor
it. As for the equation which allows us to define the exponents, it will be shown
that it is hard to deduce d from e and N without knowing p and q.
Since its first description by the three computer scientists in 1976, RSA has
been thoroughly analysed and many attacks against it have been found. These
attacks, which aim to recover an encrypted message or to deduce the private
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