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I'm trying to work through an encryption assignment and the instructions aren't particularly clear because they don't go hand in hand with the lecture video.

Using encryption key e = 9 in modulo 23, perform the following calculations:

  1. Take the secret message M = 21, encrypt this message as a number E. Note down the value of E.
  2. Let d be the decryption key for this encryption system. Compute d. Complete the following sentence: e × d ≡ a(mod b). Note down the value of a, b and d.
  3. Use the results above (or an alternative approach) to decrypt the secret message M that was encrypted as E = 15. Note down the value of M.

I was sorta able to do the first one but I have my doubts. So e = 9, p = 23(got this from a classmate) and M = 21. The encryption algorithm given in the lecture was E ≡ M^e (mod p) and that turned out to be 17. So the message 21 was encrypted to 17. I am going off of p = 23 from a classmate but I don't quite get she got that value from.

The 2nd issue that I'm having is E = 15 in #2. The value is suppose to be 17.

The 3rd issue I'm having is the value of a. I know the value for e and I can find d by isolating it but I don't know what to make of a.

I'm not asking anyone to do my homework for me. My problem is that this is super unclear and half-assed. I was wondering if those well-versed in encryption can break this down for me and highlight my mistakes.

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  • $\begingroup$ See my answer: $a=1$ and $b=22$ (or $p-1$) or really $\phi(n)$ in general in #2. $\endgroup$ – Henno Brandsma May 28 at 13:57
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This is a toy system that is used in didactic contexts as an introduction to RSA, in a way.

We have some prime $p$ (we are going to do the power operations modulo this prime). If $m$ is the message (an integer in $\{1,\ldots,p-1\}$), we have some encryption exponent $e$, and encryption indeed works as $$E(m) = m^e \pmod{p}$$.

So for $p=23$, $e=9$ we indeed have $E(21)=17$, as $21^{9} = 17 \pmod{23}$.

The decryption exponent $d$ can be found by solving $ e \times d = 1 \pmod{p-1}$ (the $p-1$ is $\phi(p)$ and Euler's/Fermat's theorem tells us that $m^{\phi(p)} = m \pmod{p}$ for all non-zero $m$.) So exponentiation with $d$: $$D(m)=m^d \pmod{p}$$ is the inverse of $E$ and vice versa.

In your number example with $p=23, m=21, e=9$ we have $p-1=22$ and $9 \times 5 = 1\pmod{22}$ and indeed $D(17)=17^{5}=21\pmod{p}$ as expected: we get back our starting value $m$.

In #3 you know $E(m)=15$ so $m=D(15)=15^5 \pmod{23}= 7$. You should distinguish $E = E(m)$ from the exponent $e$.

So this system crucially depends on $e$ being kept secret, otherwise $d$ is pretty trivial to compute via Euclid's algorithm (that's how I found $d=5$ too) and anyone can decrypt. RSA depends on the fact that given $n$ of the form $pq$, with large unknown $p$ and $q$, makes it hard to find $\phi(n)$ and hence $d$ from $e$, as we find $d$ by solving $e \times d = 1 \pmod{\phi(n)}$ where $n$ is the modulus we are using, like $p$ in your case.

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Encryption key "e = 9 in modulo 23" means $e=n\mod{p}$, or $e=9\mod23$. The given encryption algorithm is $E=M^e(\mod{p})$. Do you see how your classmate got $p=23$ now?

The value you get for $E$ is wrong. $E=(21^9)(\mod{23})$. Always check your work with a calculator, don't trust doing math by hand. I got this drilled into me at a young age by my father, who is a mathematician. It's too easy to screw up, and errors propagate, so having a sanity check is always a good idea.

For finding $d$, this is a simplified version of RSA, so $e\times d=1(\mod{p-1})$

The missing information from the question (a bit about how RSA works) is stuff you're probably expected to have read about in a textbook or learned in a lecture. You should be able to figure out the answer from this. If not the other answer has a bit more of the work done.

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