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I'm having a hard time creating a functioning RSA algorithm for some reason even though I have all the steps right (or at least I think I do). So I have the following :

I picked prime numbers, $p$ and $q$ as:

$$p = 13691$$ $$q = 29387$$

I picked $n$ as $p \cdot q$ $$n = 402337417$$

So $$\phi(n) = 402294340$$

I picked a random $e$ between $1$ and $\phi(n)$ $e = 46117$

My message was $M=3$

I got $d$ by the Extended Euclidean Algorithm as the following: $d= 7795$

When I do the encryption using $M^e \,\text{mod}\, n$ I get: $c= 399797630$

When I do the decryption using $C^d \,\text{mod}\, n$, I get $243069037$, which is not $M = 3$?

Any idea what can be the reason behind this? My guess is that $d$ is incorrect.

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Your $d$ is wrong, double check your EEA calculation. You picked up the wrong Bezout coefficient from $7795\times 402294340 - 67998447\times 46117 =1$. You should use $d= -67998447 \equiv 334295893 \pmod {\phi(n)}.$

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You computed $d$ incorrectly; without knowing exactly what you did, I can't say what you did wrong.

Try $d = 334295893$; in addition, $d = 133148723$ also works as a decryption exponent.

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  • $\begingroup$ How is it possible to have 2 d's that works for the decryption? $\endgroup$
    – rullzing
    Oct 17, 2017 at 22:44
  • $\begingroup$ @rullzing: actually, there are an infinite number of $d$ values that all work for decryption; the two values listed are only the smallest ones. $\endgroup$
    – poncho
    Oct 18, 2017 at 2:16
  • $\begingroup$ @rullzing: specifically, adding or subtracting any multiple of lcm(13690,29386)=201147170 works $\endgroup$ Oct 18, 2017 at 3:27

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