I'm trying to calculate $d$. I have the following parameters given:
$p = 239$
$q = 181$
$n = p \times q = 43259$
$\phi(n) = (p - 1) \times (q - 1) = 42840$
$e = 11$
For calculating $d$ I use the Euclidean Algorithm:
$a = qb + r$
$42840 = 3894 \times 11 + 6$
$11 = 1 \times 6 + 5$
$6 = 1 \times 5 + 1$
$5 = 5 \times 1 + 0$
This leads me to this table here:
\begin{array}{|r|r|r|r|} \hline a & b & q & r & x & y \\ \hline 42840 & 11 & 3894 & 6 & 2 & -7789 \\ \hline 11 & 6 & 1 & 5 & -1 & 2 \\ \hline 6 & 5 & 1 & 1 & 1 & -1 \\ \hline 5 & 1 & 5 & 0 & 0 & 1 \\ \hline \end{array}
When I check this solution, everything is fine:
$1 = 42840 \times 2 + 11 \times (-7789)$
but my $d$ should be $35051$, because I need $e \times d = 1 \bmod 42840$.
We can check this via encryption and decryption as well. Let's say I encrypt the number $6$:
$E(M) = M^e \bmod n = 6^{11} \bmod 43259 = 27082$
The decryption works fine for $d=35051$
$D(C) = C^d \bmod n = 27082^{35051} \bmod 43259 = 6$
But it's wrong for $-7789$.
So, what did I do wrong to calculate $d$? Can you provide a full path to $d$?
