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First, let's have the operational definition of terms:

$A \rightarrow B$ is an encryption operation from the original message (which is a base 10 number) $A$ to the encrypted message (also a base 10 number) $B$

$p,q$ are integers such that $n=pq$ is the modulus,

$e,d$ are the public and private keys, respectively.

Problem proper (pick your poison)

(a) How many triples of $n,e,d$ are there, such that for a given A and B, $A \rightarrow B$ using those triples? Any related mathematical expression?

More contextual version:

How many triples are there such that $12345 \rightarrow 67890$?

(b) How many triples of $n,e,d$, can also satisfy the same problem such that for $k$ operations?

More contextual version:

How many triples are there such that $12345 \rightarrow 67890$, $100 \rightarrow 1000$? ($k = 2$ in this case)

(c)

i. What is/are the value(s) of $k$ such that there exists only one triple satisfying the system of operations? (Wait, is it even dependent on $k$?)

ii. What is the minimum value of $k$ such that there exists only no triple satisfying the system of operations?

I was just inspired by this xkcd comic I saw when I created by this problem, and when I tried solving it myself, I thought that I lacked enough mathematical rigor and experience to solve this. I have not taken a formal NT subject, so yeah...

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