This exercise is about the RSA cryptosystem, named after its discoverers Ron Rivest, Adi Shamir, and Leonard Adleman.
Here's how it works: You need two prime numbers $p$ and $q$, compute their product $m = pq$, find a number $b$ that is relatively prime to $\phi(m) = (p1)(q1)$, and compute an inverse $c$ of $b$ modulo $\phi(m)$, i.e., $bc \equiv 1 \bmod \phi(m)$.
You keep all of this private except for the numbers $m$ and $b$ which you make public (in particular, your friends know $m$ and $b$). To send you a message $d$, your friend encodes it as
\[
e = d^b \bmod m \, .
\]
You can decode your friend's message by computing
\[
d = e^c \bmod m \, .
\]
Explain why this decoding works.
What makes this cryptosystem safe? How could you make it safer? What would one need to break it?
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0 ) {
print '
Reference Author 
Reference Title 
Reference URL 



';
} ?>