The general method for constructing the parameters of the RSA cryptosystem can be described as follows:

• Select two primes p and q
• Let  =  and determine Φ (N) = (p - 1)(q - 1)
• Randomly choose  in the range 1 < e <  Φ , such that gcd (e,N) = 1
• Determine  as the solution to ed ≡ 1 mod Φ ()
• Publish (e,N) as the public key

a. Show that a valid public key pair can still be constructed if we use only one prime , such that N =  and Φ (N) = (p - 1).
b. If we use this "one-prime" RSA construction and publish the public key (e, N),why is it easy to recover the secret key ?

c. Let RSA(M) denote the encryption of the message  using the pair (M₁, M₂). Show that the RSA encryption function has the following property for any two messages M₁ and M₂ 

RSA (M1 x M2) = RSA(M1) x RSA(M2)

That is, the encryption of a product is equal to the product of the encryptions.

 Tasks:
 
a.  Show that "one-prime" construction produces a valid public key 

b.  Show the steps to recover  

c.  Mathematical argument to show the property 

Referencing style: APA style