Proof of exchange and use of key pairs in RSA Algorithms

[Preface] A few notes in RSA Algorithm Research
RSA algorithm (Class C style description ):
Set P and Q to two prime numbers;
N = P * Q;
T = (P-1) * (Q-1 );
Find a certain number of E to meet the requirements of E and T (the number of M-grams between E and T is only 1 );
Find a number D to satisfy (E * D) % T = 1;
Then: (D, N) and (E, N) are interchangeable encryption (decryption) key pairs.
Set the encrypted plaintext to M and the encrypted ciphertext to C. The following process assumes that (E, N) is the public key for encryption and (D, N) is the private key for decryption.
Encryption: C = (E Power of M) % N;
Decryption: M = (E Power of C) % N;
It proves that (D, N) and (E, N) are interchangeable encryption (decryption) key pairs:
If (D, N) and (E, N) are interchangeable, the following two conditions must be met:
1. (D * E) % T = 1;
Proof: (E * D) % T) = (D * E) % T );
2. D and T need to be satisfied
Proof: false certificates D and T are not mutually exclusive, which can be described as follows:
D = a1 * X;
T = a2 * X;
By (D * E) % T = 1;
==> (A1 * X * E) % (a2 * X) = 1;
==> A1 * X * E = B * a2 * X + 1;
==> (A1 * E-B * a2) * E = 1;
=> E can only = 1, and a1 * E-B * a2 = 1, for example, E = 1, that is, the anti-evidence fails.
Proof complete

Author: Zhang Yu (data recovery )"