Finding d given everything else in RSA

Question

I've got everything else (e, p, q, phi(n), n), but cannot see a way to figure out what d should be. P and q are both 300 digits, so phi(n) and n are about 600 digits, but e is only 5 digits. I do already understand that ed is congruent to 1 mod phi(n). Thanks for any suggestions.

score 2 · Answer 1 · answered Mar 20 '16 at 06:26

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Usually you can calculate D with

$ D = (phi(N)*K + 1) / e$

Where

$K =$ Any small value, try 2

This video will help you, a lot: https://www.youtube.com/watch?v=e_auEoqetec

answered Mar 20 '16 at 06:26

AppIns

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How so? I am confused on how important K is in RSA – AppIns Mar 20 '16 at 07:10
@HennoBrandsma: $K<e$, thus $K$ is small when $e$ is. In particular $e=3\implies K=2$. That does not harm security in any known way, when proper RSA padding is used. – fgrieu Mar 20 '16 at 08:51
@fgrieu True. I'll delete the comment. – Henno Brandsma Mar 20 '16 at 09:24

score 1 · Accepted Answer · answered Mar 20 '16 at 06:34

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Well, $d$ is just the inverse of $d$ modulo $\phi(n)$, as you state. So (if you don't want to write a program) go to a site like Wolfram and type the query (with the correct numbers, not 3 and 35 as I did as an example).

Some googling will give you simple python programs that will do it (python has built-in bignums so it's easy to do in that language).

answered Mar 20 '16 at 06:34

Henno Brandsma

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Thanks for the reply! I actually wrote some C# stuff to create and calculate all the other variables. I have this function to work with, but I'm not sure exactly what goes where in finding d. https://onedrive.live.com/redir?resid=12493230D3D9B40F!108735&authkey=!AB5KxOyUIlJP6EE&v=3&ithint=photo%2cpng – Jamie Mar 20 '16 at 07:27
@Jamie take exponent equal to $-1$, modulus to $\phi(n)$. – Henno Brandsma Mar 20 '16 at 09:23
@henno-bandsma I did try that before, but I thought it was wrong because I seem to be getting this error. Any idea what could be wrong? https://onedrive.live.com/redir?resid=12493230D3D9B40F!108736&authkey=!AMixJYWiCTwEn8I&v=3&ithint=photo%2cpng – Jamie Mar 20 '16 at 14:09
@jamie the API might not accept negative numbers, in which case you might need the extended Euclidean algorithm instead – Henno Brandsma Mar 20 '16 at 16:10
@henno-bandsma Thank you so much! Using the EEA worked perfect. Everything's working fine. – Jamie Mar 21 '16 at 00:24

Finding d given everything else in RSA

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