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 f$1($x$1)=$ z$1$; f$2($x$2,$ y$2)=$ z$2$; f$3($x$3,$ y$3)=$ z$3,$ where all values are integers with $1...$ xi$,$ yi$,$ zi $...$ N$.$ Function f$1$ can be represented by a vector M$1$ of length N$,$ in which the kth entry is the value of f$1($k$).$ Similarly,