How Multiplication Table is generated for GF(2^2) field I was unable to solve the multiplication table given in the book for $\mathrm{GF}(2^2)$.However, I have managed to solve the addition table.

Acoording to the Book multiplication is the AND operation, but when I applied this I did not get the answer given in the book.

This is the Table given in the book:


  


I couldn't replicate the same answers for 10x10 or 11x11.
$\mathrm{GF}(2^2)$ is the finite field of 4 elements, and has minimal polynomial $x^2+x+1$. Throughout this question I will use $ab$ to denote $ax+b$ (ie $10=1*x+0$) - this is standard notation when considering finite fields over $\mathbb F_2$ since it aligns with how we consider bits in bytes.

As you have already seen, addition is done by bitwise xor:
$ab+cd := ab\oplus cd$

For multiplication, we do standard polynomial multiplication, but then must reduce by the minimal polynomial. That is, we use the identity that $x^2=x+1$. So:

$$\begin{aligned}
 ab\cdot cd
   &amp;= (a*x+b)(c*x+d)
\\ &amp;= (a * c)x^2+(a* d + b * c)x + b*d
\\ &amp;= (a* d + b * c +  a * c) x + [b*d +  a * c]
\end{aligned}$$
Now, as discussed above + is xor (denoted $\oplus$), and the hint given by your book is that multiplication of the coefficients is the same as an AND operation (which I'll denote with $\&amp;$):

$$\begin{aligned}
 ab\cdot cd
   &amp;= (a\&amp;d \oplus b\&amp;c \oplus  a\&amp;c) x \oplus [b\&amp;d \oplus  a\&amp;c]
\\ &amp;= [a\&amp;d \oplus b\&amp;c \oplus  a\&amp;c][b\&amp;d \oplus  a\&amp;c]
\end{aligned}$$