Question

Using your younger brother’s RSA Public Key (N=133,e=7), one of his friends sends him the message "5" (the number 5 is the complete message). Decrypt the message to your brother (and you should advise him to choose a key that is not so easy to break). (Provide an integer value).

Answer 1

To decrypt the message, your brother needs to use his private key, which is derived from the RSA Public Key. The private key consists of two values, d and N, where d is the modular multiplicative inverse of e modulo φ(N). In this case, N=133 and e=7. To find d, we need to calculate the value of φ(N), also known as Euler's totient function of N. For a prime number, φ(N) = N-1, so in this case, φ(133) = 132. Using the Extended Euclidean Algorithm, we can find that d ≡ 19 (mod 132), so your brother's private key is (N=133, d=19). To decrypt the message, we use the formula for RSA decryption: M ≡ C^d (mod N), where M represents the decrypted message, C represents the ciphertext (the number 5 in this case), d is the private key, and N is the modulus. Substituting the values, we find that 5^19 ≡ 84 (mod 133). Therefore, the decrypted message for your brother is 84.

Step-by-step explanation:

To decrypt the message, your brother needs to use his private key, which is derived from the RSA Public Key. The private key consists of two values, d and N, where d is the modular multiplicative inverse of e modulo φ(N). In this case, N=133 and e=7.

To find d, we need to calculate the value of φ(N), also known as Euler's totient function of N. For a prime number, φ(N) = N-1, so in this case, φ(133) = 132.

Next, we can use the Extended Euclidean Algorithm to find the modular multiplicative inverse of e modulo φ(N). The algorithm states that if a and b are coprime positive integers, then there exist integers x and y such that ax + by = 1. In this case, we need to find the value of d such that 7d + 132y = 1.

Solving this equation, we find that d ≡ 19 (mod 132). So, your brother's private key is (N=133, d=19).

To decrypt the message, we use the formula for RSA decryption: M ≡ C^d (mod N), where M represents the decrypted message, C represents the ciphertext (the number 5 in this case), d is the private key, and N is the modulus.

Substituting the values, we get M ≡ 5^19 (mod 133). Using modular exponentiation, we can calculate that 5^19 ≡ 84 (mod 133). Therefore, the decrypted message for your brother is 84.