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Question (a) If an intruder happens to see a one-time password, H j ( k ) H^j(k)  for some 1 ≤ j < n 1 \leq j < n , what should be the computational task for the intruder to get the next password? To determine the computational task for the intruder to get the next password, H j − 1 ( k ) H^{j-1}(k) , the intruder would need to compute the hash function in reverse. However, hash functions like SHA-3 are designed to be one-way functions, meaning they are computationally infeasible to invert. Therefore, the intruder cannot feasibly compute H j − 1 ( k ) H^{j-1}(k)  from H j ( k ) H^j(k)  directly due to the one-way property of cryptographic hash functions. The task is computationally infeasible, implying it is practically impossible to obtain the next password H j − 1 ( k ) H^{j-1}(k)  if the hash function is secure. Question (b) What is the security problem if the passwords are dispensed in the order that H ( k ) H(k)  comes first, then H 2 ( k ) H^2(k) , ⋯ \cdots ⋯ , to H n ( k ) H^n(k) ? If the passwords are dispensed in the order H ( k ) H(k) , H 2 ( k ) H^2(k) , ⋯ \cdots ⋯ , to H n ( k ) H^n(k) , the security problem is that once an intruder captures a password, H i ( k ) H^i(k) , they can easily compute subsequent passwords H i + 1 ( k ) H^{i+1}(k) , H i + 2 ( k ) H^{i+2}(k) , etc. by simply applying the hash function repeatedly. For example, if the intruder captures H ( k ) H(k) , they can compute H 2 ( k ) H^2(k) , H 3 ( k ) H^3(k) , and so on, by applying the hash function H H  iteratively. This defeats the purpose of a one-time password system because the intruder can predict future passwords and gain unauthorized access. Final (a) The computational task for the intruder to get the next password from H j ( k ) H^j(k)  to H j − 1 ( k ) H^{j-1}(k)  is infeasible due to the one-way nature of the hash function. (b) The security problem with dispensing passwords in the order H ( k ) H(k) , H 2 ( k ) H^2(k) , ⋯ \cdots ⋯ , to H n ( k ) H^n(k)  is that it allows an intruder who captures one password to easily compute all subsequent passwords, compromising the system's security.

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