CRYPTOGRAPHY math foundations

1. Let () = ⁶ + ³ + ² + + 1 and () = ⁴ + ³ + 1 in (2)[]. Find the quotient of () () and the remainder.

2. Solve the system of modular equations: x =2 mod 3, x = 3 mod 7 and x = 2 mod 11.

3.Let Galois Field GF(2⁴) with irreducible polynomial + + 1, f(x) = ³ + 1, and g(x) = + 1.

4.find c(x) = f(x)g(x) in GF(2⁴) (remember to do remainder operation).

5.Express the multiplication c(x) = f(x)g(x) as bit string multiplication using the equivalence ax³ + bx² + cx + d == (abcd). For example, ³ + 1 == (1001).

(optional students need to look for the fast exponentiation algorithm an do it accordingly) Use the fast exponentiation algorithm to calculate 23⁶⁹ mod 71