Final answer:
Without the prime factors of N in the RSA encryption algorithm, one cannot calculate the private key required to decrypt a message. In this scenario, factoring N = 667 gives prime factors 23 and 29, but the private key needed to decrypt the message "100" cannot be determined with the given information.
Step-by-step explanation:
The student has intercepted an encrypted message "100" using the RSA encryption system. Given the public key pair (e = 3, N = 667), to find the unencrypted message, one would usually calculate the private key component d such that d is the multiplicative inverse of e modulo ϕ(N), where ϕ is the Euler's totient function. However, the value of ϕ(N) cannot be determined without the prime factors of N.
Therefore, to proceed, we must factor N to find these prime factors. Upon factoring, we find that 667 = 23 × 29. Using these factors, ϕ(667) = (23 - 1)(29 - 1) = 22 × 28 = 616. Now we can find d such that it satisfies the congruence ed ≡ 1 mod 616.
To decrypt the message c = "100", the student will use the equation m = c^d mod N, where m is the original unencrypted message. Unfortunately, without the private key value d, we cannot complete the decryption process in this case. Thus, we are unable to determine the unencrypted message without additional information.