TITLE: RSA signature system
QUESTION [6 upvotes]: Alice wants to construct a RSA signature system to sign messages. The
  system is secure if the measure $n$ is a product of two primes, each
  of them has two digits. 

Describe the construction of the keys. 
Describe how Alice signs the message $m$, with $h(m)=3$. 
Describe how Bill, who gets the message and the signature, verifies the signature at the message.


$$$$ 
I have done the following: 

Alice choose two primes $p$, $q$ and calculates $n=p \cdot q$. She calculates $\phi(n)=(p-1)(q-1)$ and she chooses an integer $e$ coprime with $\phi(n)$. The she finds the inverse $d$, $ed \equiv 1 \pmod {\phi(n)}$. 
The public key is $(n, e)$ and the private key is $d$. 
Is this correct?? 
$s=h(m)^d \pmod n$ 
Do we have to find the private key $d$??

REPLY [2 votes]: Secret Key Generation:

Find odd primes $p$ and $q$
Choose a number $d\in{Z}^*_{\phi(pq)}$

Public Key Generation:

Compute $n=pq$
Find a number $e$, such that, $ed\equiv1\pmod{\phi(n)}$

Message Space:

$[0,\dots,n-1]$

Message Signature:

Compute $T=H(M)$
Compute $S=T^d\bmod{n}$

Message Verification:

Compute $T=H(M)$
Assert $S^e\bmod{n}=T$