 # Suppose an RSA communication between two parties, of Alice and Bob....

## Question

Suppose an RSA communication between two parties, of Alice and Bob.
The Bob's public key is (187, 73).
The Alice encrypts a message m and sends to Bob in the following encrypted message (ciphertext) c = 42.
However, an eavesdropper, the Eve, monitors the communication channel and manages to read the encrypted message c.
Show how Eve can decrypt c and to acquire knowledge of the original message m.

## Solution Preview

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It is first useful to give an overview of the RSA algorithm and then apply for the provided numerical example.
n must decomposed in two prime factors p and q, thus n=p*q
It is computed φ(n)=(p-1)*(q-1)
We have e – the public exponent; it must be lower than φ(n) and GCD(e, φ(n))=1 (to be co-prime numbers).
It is computed the private exponent d, such way that to have d*e=1 mod φ(n)
The encryption process assumes that ciphertext c = m^e mod n, where m is the message in plaintext.
The decryption operation takes place like: m=c^d mod n...

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