Answer :
Oscar cannot break Alice's private key in the RSA public key cryptosystem based on the given information of [tex]\( n = 221 \) and \( e = 13 \)[/tex].
Breaking the RSA private key requires factoring the modulus [tex]\( n \)[/tex] into its prime factors, which is computationally difficult. The RSA public key cryptosystem is based on the difficulty of factoring large composite numbers into their prime factors. In this case, Oscar knows the public key components [tex]\( n \) and \( e \)[/tex], but breaking the private key requires finding the prime factors of [tex]\( n \)[/tex]. If Oscar could factorize [tex]\( n = 221 \)[/tex] into its prime factors, they could calculate Euler's totient function [tex]\(\phi(n)\)[/tex] which would allow them to compute the private key exponent [tex]\( d \)[/tex]. However, factoring [tex]\( n \)[/tex] is a computationally difficult problem, especially for large prime numbers.
In this scenario, the modulus [tex]\( n = 221 \)[/tex] is a small composite number, and it can be easily factored into [tex]\( n = 13 \times 17 \)[/tex]. But this factorization is not useful for breaking the private key because it doesn't provide any information about the private key exponent [tex]\( d \)[/tex]. To break the private key in a real-world scenario, the modulus [tex]\( n \)[/tex] would need to be a much larger number, typically the product of two large prime numbers. The security of RSA relies on the difficulty of factoring such large numbers, making it computationally infeasible to break the private key without additional information or advanced factorization techniques.
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