So I fully understand the how the RSA algorithm works, but now I am trying to reason with the formula. I want to know:

why the public key e and the private key d in the RSA encryption have to satisfy the equation ed = 1 mod (p − 1)(q − 1)?

Is it because of the standard modular arithmetic rule where 1 mod anything is 1, or is there a more to this answer?

  • $\begingroup$ What research have you done? This is answered in standard textbook explanations of RSA, which are numerous. There is no value to the world in having us re-type out what is already found in standard textbooks and in many online resources. $\endgroup$ – D.W. May 23 '14 at 21:59
  • $\begingroup$ That quote is incorrect; the congruence only needs to hold mod $\:\operatorname{L}\hspace{-0.03 in}\operatorname{cm}\hspace{.02 in}(\hspace{.04 in}p\hspace{-0.04 in}-\hspace{-0.05 in}1,q\hspace{-0.04 in}-\hspace{-0.05 in}1)\;$. $\hspace{1.34 in}$ $\endgroup$ – user12859 May 24 '14 at 9:26
  • $\begingroup$ hey @D.W. if you don't have an answer, don't comment. Ive tried many web resources and found nothing. If its all there why not link them. $\endgroup$ – joker May 24 '14 at 11:01
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    $\begingroup$ @joker For example: en.wikipedia.org/wiki/RSA_(cryptosystem)#Proofs_of_correctness. $\endgroup$ – Yuval Filmus May 24 '14 at 14:08

The relation of $e$ and $d$ is set so that encryption and decryption are inverses of each other, that is, if you encrypt a message then the ciphertext decrypts to the original message. Encryption and decryption are $$ c = m^e \pmod{n}, \quad p = c^d \pmod{n}, $$ where $n = pq$ is part of the public key, $m$ is the message, $c$ the ciphertext, and $p$ the decrypted plaintext. We want $p = m$, i.e., $$ m = m^{ed} \pmod{n}. $$ This equation holds for all $m$ iff $ed \equiv 1 \pmod{\lambda(n)}$, where $\lambda(n)$ is Carmichael's function. For $n = pq$, $\lambda(n) = \mathrm{lcm}(p-1,q-1)$. In practice, $\lambda(n)$ is replaced by Euler's function $\varphi(n) = (p-1)(q-1)$.

  • $\begingroup$ $\langle n\hspace{.02 in},\hspace{-0.03 in}e,\hspace{-0.03 in}d\hspace{.03 in}\rangle = \langle 143,\hspace{-0.04 in}7\hspace{-0.02 in},\hspace{-0.03 in}43\rangle \;$ is a counter-example to your iff sentence. $\;\;\;\;$ $\endgroup$ – user12859 May 24 '14 at 9:23

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