Let q be a power of a prime p, and let Fq be the finite field with q elements. For any rational function h(x) ∈ Fq(x), its value set is defined as

Vh = { h(α) | α ∈ Fq ∪ {∞}} ⊆ Fq ∪ {∞}

If h(x) = f(x)/g(x) ∈ Fq(x) is a rational function of degree d, that is, f(x),g(x) ∈ Fq[x] are such that 

GCD(q,f)=1 and d = max{deg(f),deg(g)}, 

then one has the trivial bound 

Ceiling((q+1)/d) <= #Vh <= q+1.

A rational function for which the lower bound above is achieved will be called minimal value set rational function, abbreviated as m.v.s.r.f. The previous definition is analogous to the well-known notion of minimal value set polynomials. Polynomials with small value sets have been investigated by many authors over the past decades. The subject, important in its own right, is known to be relevant in other branches of mathematics. For instance, a close connection between minimal value set polynomials and Frobenius nonclassical curves was recently established.

In [2021/1] a connection between small value sets functions and Galois theory is exploited, proving that h(x) having a small value set is equivalent to the field extension Fq(x)/Fq(h(x)) being Galois.

In [2022/2] two-to-one mappings are investigated. Such functions have received a lot of attention due to their applications to cryptography and to the construction of classes of cryptographic functions. In this paper, we propose a new method to obtain m-to-1 functions based on Galois groups of rational functions.