In the RSA public key cryptosystem, the private and public

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In the RSA public key cryptosystem, the private and public keys are (e, n) and (d, n) respectively, where n = p*q and p and q are large primes. Besides, n is public and p and q are private. Let M be an integer such that 0 < M < n and φ(n) = (p – 1) (q – 1). Now consider the following equations.
1. M’ = M^e mod n
M = (M’)^d mod n
2. ed ≡ 1 mod n
3. ed = 1 mod φ n
4. M’ = M^e mod φ n
M = (M’)^d mod φ n
Which of the above equations correctly represent RSA cryptosystem?

1. 1 and 2
2. 1 and 3
3. 2 and 4
4. 3 and 4

Correct Option: B

The RSA laboratories define RSA cryptosystems as the following
The RSA cryptosystem is a public-key cryptosystem that offers both encryption and digital signatures (authentication). Ronald Rivest, Adi Shamir and Leonard Adleman developed the RSA system in 1977; RSA stands for the first letter in each of its investor’s last names.
The RSA algorithm works as follows: take two large primes, p and q, and compute their product n = pq, n is called the modulus. Choose a number, e, less than n and relatively prime to (p – 1) (q – 1), which means e and (p – 1) (q – 1) have no common factors except 1. Find another number d such that (ed – 1) is divisible by (p – 1) (q – 1). The values e and d are called the public and private exponents, respectively. The public key is the pair (n, e); the private key is (n, d). The factors p and q may be destroyed or kept with the private key. It is currently difficult to obtain the private key d from the public key (n, e). However, if one could factor n into p and q, then one could obtain the private key d. Thus, the security of the RSA system is based on the assumption that factoring is difficult. The RSA cryptosystem in the problem is correctly represented by the equation 1 and 3.