1. Consider a hash function that starts with an 32-bit initial value of c09db506 and repeatedly XORs it with 32-bit blocks of the input to produce a 32-bit output.

For example: the input de0fc223 would generate the following output: () = =

and the input cd1aa915 f08d7db7 would generate the following output: ( ) = =

(a) Approximately how many steps would be required to perform a brute force attack to find a preimage for a given output of this hash function? (2 pts) (b) Approximately how many steps would be required to perform a brute force birthday attack to find a collision for this hash function? (2 pts) (c) Use the properties of the XOR operation to find a 32-bit preimage for the output 666cdce9. (3 pts) (d) Find a 32-bit second preimage for the output fd0a61a4 calculated above. (3 pts) (e) Find a collision other than the one demonstrated by your answer to (d). (hint: pick a 32-bit input, then find a 64-bit input that generates the same output using the properties of XOR.) (3 pts)