1. (a) (b) : Let h1 {0,1} {0, 1} and h2 {0,1} {0,1} be hash functions...

 

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1. (a) (b) : Let h1 {0,1} {0, 1}" and h2 {0,1} {0,1}" be hash functions with output length of n bits. Construct a hash function h {0, 1}* {0, 1}2n by concatenating the outputs of h and h, that is, h(x) = h(x)||h2(x) where || is string concatenation. Suppose h is collision resistant but h is not. Can we say that his collision resistant? Explain/Prove your claim. Let m and n be positive integers such that m>n. Let A be an n x m matrix, i.e., n rows and m columns, over the binary field F2. Define a hash function h: {0,1}{0, 1}" by h(x) = Ax. Here, Ar is the matrix-vector product, the input binary string r is considered as a binary vector of length m. The output of the hash is a binary vector of length n. Note that all operations are done mod 2. Is h preimage resistant? Is it collision resistant? Explain. 1. (a) (b) : Let h1 {0,1} {0, 1}" and h2 {0,1} {0,1}" be hash functions with output length of n bits. Construct a hash function h {0, 1}* {0, 1}2n by concatenating the outputs of h and h, that is, h(x) = h(x)||h2(x) where || is string concatenation. Suppose h is collision resistant but h is not. Can we say that his collision resistant? Explain/Prove your claim. Let m and n be positive integers such that m>n. Let A be an n x m matrix, i.e., n rows and m columns, over the binary field F2. Define a hash function h: {0,1}{0, 1}" by h(x) = Ax. Here, Ar is the matrix-vector product, the input binary string r is considered as a binary vector of length m. The output of the hash is a binary vector of length n. Note that all operations are done mod 2. Is h preimage resistant? Is it collision resistant? Explain.