This talk delves into the rich mathematical framework of functions and polynomials over finite fields, emphasising their fundamental role in coding theory and symmetric cryptography. We will explore key properties, particularly those related to exponential sums, equation solvability, and structural characteristics essential for assessing cryptographic strength and optimising code design. A central focus will be on advanced algebraic techniques for functions over finite fields, with applications in cryptographic function theory. Our goal is to refine their algebraic properties, enhance their effectiveness in function-based coding schemes, and deepen mathematical insights into their behaviour. Additionally, we will discuss recent results in solving fundamental algebraic equations over finite fields, shed light on emerging methodologies and theoretical advancements, and present perspectives on their potential impact in both symmetric cryptography and modern coding theory.