Question 1. Let H: {0,1}* → {0, 1}" be a hash function that achieves the one-way and collision resistance security properties. (b). Show that the new function H' achieves collision resistance. Suppose that we want to use H for the deployment of a password-based user authentication mechanism that is a variant of the standard password hashing mechanism as follows: when loading a new password P selected by the user with ID U, instead of choosing a random salt, the password P is hashed. Then, the n-bit digest H(P) and password P serve as inputs to produce a fixed-length hash code. Namely, the information that is stored in the password file is the triple (U, H(P), H(H(P)||P)). During password verification for user ID U, the system receives a password P* and (i) computes H(P*), (ii) computes H(H(P*)||P*), and (iii) checks if H(H(P*)||P*) matches the value H(H(P)||P) that is stored for U. 1 Does the aforementioned mechanism provide protection against offline dictionary attacks? Justify your answer. Recall that in such an at- tack, the attacker obtains the system password file and compares the password hashes against precomputed hashes of commonly used pass- words.

Question 1. Let H: {0,1}* → {0, 1}" be a hash function that achieves the one-way and collision resistance security properties. (b). Show that the new function H' achieves collision resistance. Suppose that we want to use H for the deployment of a password-based user authentication mechanism that is a variant of the standard password hashing mechanism as follows: when loading a new password P selected by the user with ID U, instead of choosing a random salt, the password P is hashed. Then, the n-bit digest H(P) and password P serve as inputs to produce a fixed-length hash code. Namely, the information that is stored in the password file is the triple (U, H(P), H(H(P)||P)). During password verification for user ID U, the system receives a password P* and (i) computes H(P), (ii) computes H(H(P)||P), and (iii) checks if H(H(P)||P*) matches the value H(H(P)||P) that is stored for U. 1 Does the aforementioned mechanism provide protection against offline dictionary attacks? Justify your answer. Recall that in such an at- tack, the attacker obtains the system password file and compares the password hashes against precomputed hashes of commonly used pass- words.