Prior to the discovery of any specific public-key schemes, such as RSA, an existence proof was developed whose purpose was to demonstrate that public-key encryption is possible in theory. Consider the functions f1(x1) = z1; f2(x2, y2) = z2; f3(x3, y3) = z3, where all values are integers with 1 xi, yi, zi N. Function f1 can be represented by a vector M1 of length N, in which the kth entry is the value of f1(k). Similarly, f2 and f3 can be represented by N * N matrices M2 and M3. The intent is to represent the encryption/decryption process by table look-ups for tables with very large values of N. Such tables would be impractically huge but could, in principle, be constructed. The scheme works as follows: Construct M1 with a random permutation of all integers between 1 and N; that is, each integer appears exactly once in M1. Construct M2 so that each row contains a random permutation of the first N integers. Finally, fill in M3 to satisfy the following condition: f3(f2(f1(k),p),k) = p for all k, p with 1 k, p N In words, 1. M1 takes an input k and produces an output x. 2. M2 takes inputs x and p giving output z. 3. M3 takes inputs z and k and produces p. The three tables, once constructed, are made public. a. It should be clear that it is possible to construct M3 to satisfy the preceding condition. As an example, fill in M3 for the following simple case: Convention: The ith element of M1 corresponds to k = i. The ith row of M2 corresponds to x = i; the jth column of M2 corresponds to p = j. The ith row of M3 corresponds to z = i; the jth column of M3 corresponds to k = j. We can look at this in another way. The ith row of M1 corresponds to the ith column of M3. The value of the entry in the ith row selects a row of M2. The entries in the selected M3 column are derived from the entries in the selected M2 row. The first entry in the M2 row dictates where the value 1 goes in the M3 column. The second entry in the M2 row dictates where the value 2 goes in the M3 column, and so on. b. Describe the use of this set of tables to perform encryption and decryption between two users. c. Argue that this is a secure scheme.