# (Please check working) Given RSA encoding function $E: x\to x^{11} \pmod{3737}$ find the decoding function $D$

Question:
Given RSA encoding function $E: x\to x^{11} \pmod{3737}$ find the decoding function $D$

My working:
$\phi(3737) = \phi(37) \times \phi(101)$
$= 36 \times 100 = 3600$

Using euclid's algorithm (Skipping the fairly long gory details) we show:

$1 = 4\times3600 - 1309 \times 11$

Taking the coefficient of 11 (power of $x$), we get $-1309$ but we want this in positive congruent mod $3600$ so $3600-1309 = 2291$ and finally we have

$D:x\to x^{2291}\pmod{3737}$

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Seems fine. Note that once you have (maybe) found the decoding number, you can check it easily: We have $(11)(2291)=25201$, which is easily congruent to $1$ modulo $3600$. –  André Nicolas Sep 25 '12 at 5:11