An existence proof was developed before to the discovery of any specific public-key schemes, such as RSA, in order to demonstrate that public-key encryption is theoretically conceivable. The objective of this demonstration was to demonstrate that public-key encryption is possible. Consider the functions f1(x1) = z1, f2(x2, y2) = z2, and f3(x3, y3) = z3, where all of the values are integers and 1 is less than or equal to xi, yi, and zi is more than N. A vector M1 with length N, in which the kth item is the value of the function f1, can be used to represent the function f1 (k). Similar to f1, f2, and f3, the NxN matrices M2 and M3 can be used to represent these functions. The encryption and decryption process is going to be modeled as a series of table lookups, and the values of N in those tables are going to be extremely high. Such tables would be enormous, making their use impractical, yet they are not impossible to build. The plan is carried out in the following manner: Construct M1 by performing a random permutation on all of the integers ranging from 1 to N; that is, each integer only appears once in M1 overall. Construct M2 so that each row contains a different, randomly chosen permutation of the first N numbers. In conclusion, complete the M3 form in order to fulfill the condition: