Question 1 (5 marks): (Cryptographic Hash Functions) Consider the following proposal for a hash function h. Let p be a large prime and let g be a generator modp. Represent messages as sequences x=x1,x2,xn where the xi are numbers modp. We define the hash of such a message x by h(x)= gx1+x2++xnmodp 1. (2 marks) Show that this hash function is good for use as a message digest for error correction purposes, in the sense that it has a high probability of detecting errors if messages x are transmitted in the form (x,h(x)), and a message (x,y) that is received is treated as correct if h(x)=y. In particular, for concreteness, suppose that the messages Alice will be sending are all of length two (x=x1,x2), and are generated uniformly at random. Suppose also that we know that communications channel that we are using contains occasional bursts of random noise (e.g., from sunspots on a radio channel) that may potentially damage every bit of the message (x,h(x), including the hash. When Alice sends a message (x,h(x)), with probability p, the channel delivers the message to Bob exactly as transmitted. With probability 1p, the channel delivers to Bob a message (y,z) selected uniformly at random, where y=y1,y2 and y1,y2,z are all numbers mod p. (The channel never delivers a message to Bob if Alice did not send a message.) Bob accepts a message (y,z) that he receives as a correct transmission of a message y from Alice if h(y)=z, and treats it as corrupted otherwise