I have been given $N=2021$ and $E=5$. I am to encrypt the the word 'he' where h is 18 and e is 15. Then I am to find D, and k, and decipher the encrypted message.

My first question is whether i do h and e separately or if i consider the message $1815$?

So far, I started with my message being $1815$. I found the totient of $2021$ to be $1932$. Then: $$1815^5(\mod2021)=913$$ Does this mean that the encrypted message that I would send to the receiver is 913?

To find D, i used $ED \equiv 1 \mod1932$ which gave me $D=773$ but this seems very high so I think I may have gone wrong somewhere?

  1. You are correct so far encrypting the massage 'he'. You should view 'he' as a $1815$. We have $1815^5 $(mod $2021)=913$, where $913$ is the encrypted massage you should send to the receiver.

  2. Since we know totient of $2021$ is $1932$, then $D$ is the multiplicative inverse of $E$ mod $1932$. You are answer is correct, where $D=773$. Then, decrypting a massage $x$ is just simply solve the mod $x^{773}$ (mod $2021)$, and the result is the decrypted massage.

  • $\begingroup$ Doesn't this make x=2021? I thought we wanted it to be 1815? $\endgroup$ – user319635 Mar 16 '16 at 15:04
  • $\begingroup$ The encrypted message is $913$ so you compute $913^d = 913^{773}$ modulo $2021$ (modulo $pq= N$!), and this results in $1815$ again, so "h e" as expected. $\endgroup$ – Henno Brandsma Mar 19 '16 at 14:14

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