Question $3$

a$.$ How does a zero$-$knowledge proof actually work? What information is being proven, and what information remains hidden?

b$.$ Give three $(3)$ real$-$world applications of zero$-$knowledge proofs.

c$.$ What are lattices, and how are they used in cryptography?

d$.$ State three $(3)$ advantages of lattice$-$based cryptography.

e$.$ State three $(3)$ challenges or limitations associated with lattice$-$based cryptography?

f$.$ How does lattice$-$based cryptography provide security against quantum attacks, and what makes it a promising candidate for post$-$quantum cryptography?