1. **Problem Statement:** Create and explore the Galois Field GF($2^4$) in binary and polynomial form, construct it using an irreducible polynomial, and implement an affine mapping $f(X_i) = [AX_i + B] \bmod 2^4$ with given matrix $A$ and vector $B$ derived from digits of a NUTECH ID. 2. **Galois Field GF($2^4$) Construction:** - GF($2^4$) is a finite field with $16$ elements. - Elements can be represented as 4-bit binary vectors or polynomials of degree less than 4 over GF(2). - Choose an irreducible polynomial of degree 4 over GF(2), for example: $$p(x) = x^4 + x + 1$$ - Field elements are polynomials modulo $p(x)$. 3. **Field Arithmetic Rules:** - Addition is bitwise XOR. - Multiplication is polynomial multiplication modulo $p(x)$. 4. **Affine Mapping Definition:** - Given digits $a,b,c,d$ from NUTECH ID, convert each to 4-bit binary. - Construct matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. - Compute determinant $\det(A) = ad - bc$ in GF($2^4$). - If $\det(A) = 0$, replace $a$ by $b$ and recompute. - Vector $B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$ where $b_1,b_2$ are last two digits of NUTECH ID. - For each $X_i = \begin{bmatrix} x_i \\ x_{i+1} \end{bmatrix}$ with $i=0,...,15$, compute: $$f(X_i) = (A X_i + B) \bmod 2^4$$ 5. **Lookup Boxes Construction:** - Use first coordinate of $f(X_i)$ to fill Box-1 (4x4 matrix). - Use second coordinate of $f(X_i)$ to fill Box-2 (4x4 matrix). 6. **Summary:** - This process combines Boolean algebra (bitwise operations), set theory (field elements), number theory (irreducible polynomials), and cryptography (affine mappings). 7. **Programming Implementation:** - Implement GF($2^4$) arithmetic with chosen irreducible polynomial. - Define matrix and vector from NUTECH ID digits. - Compute affine mapping for all $X_i$. - Construct and visualize lookup boxes. **Final Note:** This is a conceptual framework; actual numeric results depend on your specific NUTECH ID digits.