I've been studying a cryptographic scheme that uses the Chinese Remainder Theorem (CRT) for decryption. Here's a brief overview of the encryption and decryption process:
Setup:
Alice chooses two distinct prime numbers, p and q She ensures
gcd(p, q) = 1, gcd(p,q-1)=1, gcd(p−1,q)=1
Alice computes N=pq and publishes it.
Alice calculates
q’ such that q.q’≡1(mod p−1)
p’ such that p.p’≡1(mod q−1 )
Encryption:
Bob encrypts a message M as C=M^N mod N
Decryption:
Alice decrypts C by finding M′:
M′ ≡ C^q’ (mod p )
M′ ≡ C^p’ (mod q )
By Using the Chinese Remainder Alice Obtain that M = M′
My Question:
Explain Why does the decrypted message M′ exist and why is it equal to the original message M?
If possible write mathematical proof
I'm a computer science student 3rd year . I have studied the Chinese Remainder Theorem and Fermat's Little Theorem, which seem to be key to understanding this proof. However, I'm struggling to remember how to solve this problem after hours of failed attempted, So i could use the help
