Final answer:
Alice's RSA private key cannot be calculated accurately with the information provided, since the public exponent given as 35 is not relatively prime to the Euler's totient function of the product of the primes. The correct public exponent is necessary to find the private exponent using the Extended Euclidean Algorithm.
Step-by-step explanation:
In an RSA cryptosystem, the private key is derived from the public key and the two prime numbers used to generate the keys. Given that Alice has chosen two prime numbers, 13 and 17, we can calculate the modulus n by multiplying them, which gives us 221. The public key consists of the modulus n and the public exponent e. Alice's public key is given as 35, but this seems to be an error because the public exponent, which is usually denoted by e, should be relatively prime to φ(n), the Euler's totient function of n, and should be less than φ(n). For the primes 13 and 17, φ(n) would be (13-1)*(17-1) = 192.
Assuming that the provided '35' is a typographical error and is actually the public exponent e, and that e was meant to be a commonly used value such as 5 or 17 and not 35, we would then use the Extended Euclidean Algorithm to find the private exponent d, which satisfies the congruence ed ≡ 1 (mod φ(n)). We cannot solve for d using 35 since it's not coprime to 192.
If we assume e is a valid exponent, we can find d such that ed ≡ 1 (mod 192). For example, if e was 5, then d would be 77 since 5 * 77 = 385, which is 1 more than a multiple of 192.
Without the correct public exponent, it is not possible to calculate the accurate private key for Alice. Therefore, we cannot provide the private key unless the correct public exponent is given. Once we have the correct value for e, the private key can then be calculated.