dissertacao.pdf

Algorithm 3. The implementation of a RSA session with a n bit modulus:
1. Generate two large different prime numbers p, q. This can be efficiently
done by generating random positive integers of size n
2
and checking their
primality (as described in chapter 2) until we have two primes[38].
2. Compute N = pq and φ(N) = (p − 1)(q − 1).
3. Choose e : (e, φ(N)) = 1 and compute d ∼ = e −1 (mod φ(N)). To choose e
in such a conditions, all one has to do is to choose a prime e ′ > max{p, q}
and reduce it modulo N [38]. To compute d we can then use the algorithm
described in Chapter 1 for computing modular inverses.
4. Make available for everyone the public key < e, N > and keep completely
in secret the private key < p, q, d, N >.
All of the parameters of the RSA will influence its security level and the
running time of the operations used by RSA. We usually require the size of N
to be large, usually 1024, 2048 or larger. Because we want that it will also be
hard to factor, it is recommended to have p and q as large as possible given the
size of the modulus. The usual procedure to ensure this is for a n-bits modulus
is to generate p and q as n
2 -bits primes, like described in step 1, which will be
called balanced primes as they have the same bit size.
In step 3 we can compute the exponents in reverse order: we usually chose
to compute first the exponent for which we need some restraints. These can
be for example e = 3 in order to reduce encryption costs or d > N 1
4 , which
prevents some attacks like shown in Section 3.
1.7 RSA Safety
The security of RSA depends on two of its trapdoor one way functions. The
encryption function is one of them, and the problem of inverting it is called the
RSA problem:
Definition 13. (The RSA Problem) Given a cypher text c = m e (mod N)
encrypted using the public key < e, N >, compute the plain text m knowing
only < e, N >.
This problem is, mathematically, to compute e-th roots modulo N. As it is
an open question to know whether there is a polynomial time algorithm which
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