Problem $1$:

Modular multiplication and division are fundamental operations in many public$-$key cryptosystems. These operations are, for the most part, time$-$consuming and are commonly regarded the computational bottleneck in these applications. As a result, system designers nowadays turn to VLSI implementation. Your task, as a novice system designer, is to design the $2-$bit mod$-5$ multiplication module using only basic logic gates. To guide you in this design problem are the following guidelines:

i$.$ In modular multiplication, the number $0$ is often ignored. Thus, in your design, the following $2-$bit number representation is used: ${00}_{2}1,{01}_{2}2,1023$ and $1124.$

ii$.$ Some examples of mod$-5$ multiplication:

$42-=3(mod5),33-=4(mod5).$

iii. The block diagram of the $2-$bit mod$-5$ multiplication module appear as shown in Figure $1.$

Tasks:

Determine the truth tables and logic circuits $($AND$-$OR implementation$)$ of the $2-$bit mod$-5$ multiplication module. Simplify the logic circuits as much as possible.

Figure $1$