I How Does the Equation D x E mod Φ(N) = 1 Relate to RSA Encryption?

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RSA encryption relies on the relationship between the encoding key (E), decoding key (D), and the totient function (Φ(N)). The equation D x E mod Φ(N) = 1 indicates that D and E are multiplicative inverses modulo Φ(N), which is crucial for the encryption and decryption process. Understanding this relationship helps in determining D once E is chosen, ensuring that the decryption process correctly retrieves the original plaintext message (M) from the encrypted message (C). The discussion highlights the importance of the totient function in RSA and suggests that additional resources, like Wikipedia and online calculators, can aid in grasping these concepts. Mastery of these equations is essential for a complete understanding of RSA encryption.
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TL;DR Summary
Understanding the RSA mathematics
Please help me understand the RSA encryption.
I understand part of it, then I don't understand.

Definitions:

M=plain text message
C=encrypted message
N=public key
E=encoding key
D=decoding key
P,Q are prime factors on N

I get that
P x Q = N (where P and Q are primes).

I get that

Φ(N) = (P-1) x (Q-1)

I get that

MEmodN = C

I get that

CDmodN = M

But after this I don't understand the relevance of this equation to solve D and E

D x E mod Φ(N) = 1

Seems to be some step or something obvious I'm missing?
 
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