Prior to the discovery of any specific public-key schemes, such as RSA, an existenceproof was developed whose purpose was to demonstrate that public-key encryption ispossible in theory. Consider the functions f₀(x₁) =z₁; f₂(x₂, y₂) = z₂; f₃(x₃, y₃) = z₃,where all values are integers with 1 ≤x_i, y_i, z_i≤N. Function f1 can be represented by avector M1 of length N, in which the kth entry is the value of f₁(k). Similarly, f₂ and f₃can be represented by N xN matrices M2 and M3. The intent is to represent theencryption/decryption process by table lookups for tables with very large values of N.Such tables would be impractically huge but could be constructed in principle. Thescheme works as follows: Construct M1 with a random permutation of all integersbetween 1 and N; that is, each integer appears exactly once in M1. Construct M2 sothat each row contains a random permutation of the first N integers. Finally, fill in M3to satisfy the following condition:

f₃(f₂(f₁(k), p), k) = p

for all k, p with 1 ≤k, p ≤ N

To summarize,

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 tok = j.

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.