Question

Alice and Bob agreed to use RSA algorithm for the secret communication. Alice securely

choose two primes, p=5 and q=11 and a secret key d=7. Bob uses this public key and sends
a cipher text 18 to Alice.
Find the corresponding public key

Answer 1

The RSA public key consists of a modulus and a public exponent. The modulus is obtained by multiplying the two prime numbers p=5 and q=11, resulting in n=55. The public exponent e is found such that ed ≡ 1 (mod ϕ(n)), and for d=7 and ϕ(n)=40, the public exponent e is 23.

Step-by-step explanation:

The question involves the RSA encryption algorithm, which is a part of computer security and encryption methods in computers and technology. Alice and Bob are using the RSA algorithm for secure communication. Alice has chosen two primes p=5 and q=11, and a secret exponent d=7. With these values, to calculate the RSA public key, we first find n by multiplying p and q:

• n = p × q = 5 × 11 = 55

Next, we calculate e, which is the public exponent. First, we need to find Euler's totient function ϕ(n), which for RSA is calculated as:

• ϕ(n) = (p - 1)(q - 1) = (5 - 1)(11 - 1) = 4 × 10 = 40

Given d and ϕ(n), we need to find an e such that ed ≡ 1 (mod ϕ(n)). Since d=7, we want to find an e that satisfies the congruence 7e ≡ 1 (mod 40). The value of e that satisfies this congruence is 23 since 7 × 23 = 161, which is 1 more than a multiple of 40 (160).

Therefore, the public key components are n = 55 and e = 23