Large class of many-to-one mappings over quadratic extension of finite fields

Many-to-one mappings (including $1$-to-$1$ mappings) over finite fields play important roles in cryptography and coding theory. This paper provides a unified framework for studying the many-to-one property of polynomials of the form $f(x) = h(a x^q + b x + c) + u x^q + v x$, where $h(x) \in \mathbb{