Can you help me understand this (old) school question related to RSA usage?

Question

An old question I am preparing for a security / cryptography class has an example of an RSA chiper.

This is almost the same as "RSA by hand - did I do something wrong? (c = m on encryption)" but with some other question following up.

Let $p=3$, $q=5$ giving $N=15$ and $\phi=8$
Let $e=7$. Find $d$.

Fair enough, $e\times d = k\times \phi+1$ can be rather quickly evaluated to $d=7$. $k$ would be $6$ in this case.

Encryption of messages $m$ in the range $1 < m < n$ work fine with $e=d=7$. Not very helpful to have $e=d$, but it works. Well, kind of, as some messages get encoded to the same value (4, 5, 6 and 9 for example). So the target of encryption is reached only partly but the answer for the test $d=7$ is correct and complete.

This was the easier part.

Now… could you have chosen $5$ for $e$? And follow-up question: Why not? The solution provided has

no

and

because $5$ has a common divisior with $n$

Our lecture script and every page I found regarding RSA demands $e$ and $d$ to not share a divisior (except $1$, of course) with $\phi$. I never found a remark regarding sharing with $n$.

Problem is this: What happens if you use $e=5$ (and in turn get $d=13$) is that you get no encrytion at all. $c = m^e\ mod\ n$ stays $= m$ for all $m$.

So the mathematics "work", no information is messed up, but the desired encryption is not reached at all. Is there a reason why $5$ is kind of "neutral exponent"? Does such a thing also happen for "real" rsa values?

In this example $e=5$ is equal to $q$, is that what makes the encryption become an "Involution Mathematics" (wikipedia)?

How would one word his reasoning why $5$ is an unsuitable value for $e$? More important, how can one see this in a exam situation without excel spread sheet to quickly run some numbers (without any calculator, in fact)?

Answer 1

As in the other questions: Such low numbers can lead to curious-and-misleading effects. For example in this case: you choose $q=5$, and it happens that this is equal to $\lambda(15) + 1 = 5$. And that is a coincidence of this example, not a general relation.

Now… could you have chosen 5 for e? And follow-up question: Why not? The solution provided has

no

and

because 5 has a common divisior with n

Which question ans answer are you quoting here? Because it's not the one you linked to.

And I would actually say, there is no reason that $e$ has to be coprime to $N$ - except that it would be really terrible if anyone just tried to calculate $gcd(e,N)$. However, the encryption and decryption still work in the sense that you can retrieve the plaintexts (injective function).

Regarding the coincidence in th example: For every modulus, you have that $m^{\lambda(N)} = 1 \mod N$ with Carmichael function $\lambda$, and in turn $m^{\lambda(N) + 1} = m \mod N$.

And then, for two different primes $p,q$ this can be calculated as $\lambda(pq) = lcm(p-1,q-1)$. Alternatively, this is $\lambda(N) = (p-1)(q-1) / g$ for $g = ggt(p-1,q-1)$.

From that, we can see that if $\lambda(N) + 1 = q$, then we know immediately $p-1$ is a divisor of $q-1$.

So the mathematics "work", no information is messed up, but the desired encryption is not reached at all. Is there a reason why 5 is kind of "neutral exponent"? Does such a thing also happen for "real" rsa values?

Yes, it does, but not for either $p$ or $q$. It happens for $k \lambda(N)+1$ for any arbitrary integer $k$.

In this example e=5 is equal to q, is that what makes the encryption become an "Involution Mathematics" (wikipedia)?

Since this is taken from the linked question - this is something similar, but not the same. An involution happens whenever $e=d \mod \lambda(N)$, because then you have that $ed = e^2 = 1 \mod \lambda(N)$. And again, it's pure coincidence in this number example, that it is equal to $q$. That is wrong in general.