The iteration of polynomials and rational functions over finite fields have recently become an active research topic. These dynamical systems have found applications in diverse areas, including cryptography, biology and physics. In cryptography, iterations of functions over finite fields were popularized by the Pollard rho algorithm for integer factorization ; its variant for computing discrete logarithms is considered the most efficient method against elliptic curve cryptography based on the discrete logarithm problem . Other cryptographical applications of iterations of functions include pseudorandom bit generators , and integer factorization and primality tests [8, 9].
When we iterate functions over finite structures, there is an underlying natural functional graph. For a function over a finite field , this graph has nodes and a directed edge from vertex to vertex if and only if . It is well known, combinatorially, that functional graphs are sets of connected components, components are directed cycles of nodes, and each of these nodes is the root of a directed tree from leaves to its root; see, for example, .
Some functions over finite fields when iterated present strong symmetry properties. These symmetries allow mathematical proofs for some dynamical properties such as period and preperiod of a generic element, (average) “rho length” (number of iterations until cycling back), number of connected components, cycle lengths, etc. In this paper we are interested on these kinds of properties for Chebyshev polynomials over finite fields, closely related to Dickson polynomials over finite fields. These polynomials, specially when they permute the elements of the field, have found applications in many areas including cryptography and coding theory. See  for a monograph on Dickson polynomials and their applications, including cryptography; for a more recent account on research in finite fields including Dickson polynomials, see .
Previous results for quadratic functions are in ; iterations of have been dealt in  and iterations of Rédei functions over non-binary finite fields appeared in [14, 15]. Related to this paper, iterations of Chebyshev polynomials over finite fields have been treated in . The graph and periodicity properties for Chebyshev polynomials over finite fields when the degree of the polynomial is a prime number are given in .
In this paper we study the action of Chebyshev functions of any
degree over finite fields. We give a structural theorem for the functional graph from which it is not hard to derive many periodicity properties of these iterations. In the literature there are two kinds of Chebyshev polynomials: normalized and not normalized. We use the latter ones, generally known as Dickson polynomials of the first kind. In odd characteristic both kinds of Chebyshev polynomials are conjugates of each other, and so their functional graphs are isomorphic. However, this is not the case in even characteristic. Using the normalized version trivializes since we getif is even, and if is odd, where is the th degree Chebyshev polynomial. As a consequence, we work with the non normalized version that is much richer in characteristic . Not much is known about Chebyshev polynomials over binary fields; see  for results over the -adic integers.
In Section II we introduce relevant concepts for this paper like -series and their associated trees. These trees play a central role in the description of the Chebyshev functional graph. Several results about a homomorphism of the Chebyshev functional graph, as well as a relevant covering notion, are given in Section III. A decomposition of the Chebyshev’s functional graph is given in Section IV. This decomposition leads naturally into three parts: the rational, the quadratic and the special component. Section V treats the rational and quadratic components. The special component is dealt in Section VI. The main result of this paper (Theorem 4), a structural theorem for Chebyshev polynomials, is given in Section VII. We provide several examples to show applications of our main theorem. As a consequence of our main structural theorem, in this section we also obtain exact results for the parameters and for Chebyshev polynomials, where is the number of cycles (that is, the number of connected components), is the number of cyclic (periodic) points, is the expected value of the period, is the expected value of the preperiod, and is the expected rho length.
We denote by a finite field with element, where is a prime power, and the ring of integers modulo . Let and denote the multiplicative group of inverse elements of and , respectively. Let denote the equivalence class of modulo . For with , we denote by and the multiplicative order of in and , respectively. It is easy to see that if in , then , otherwise . For we denote by the radical of which is defined as the product of the distinct primes divisors of . We can decompose where and which we refer as the -decomposition of . If is a function defined over a finite set , we denote by its functional graph.
The main object of study of this paper is the action of Chebyshev polynomials over finite fields . The Chebyshev polynomial of the first kind of degree is denoted by . This is the only monic, degree- polynomial with integer coefficients verifying for all . Table I gives the first Chebyshev polynomials.
A remarkable property of these polynomials is that for all . In particular, , where denotes the composition of with itself times. Describing the dynamics of the Chebyshev polynomial acting on the finite field is equivalent to describing the Chebyshev’s graph .
The case when is a prime number was dealt by Gassert; see [7, Theorem 2.3]. In this paper we extend these results for any positive integer .
For the corresponding Chebyshev polynomial is given by . The graphs for and are shown in Fig. 1.
Next we review some concepts from . For and positive integers such that , the -series associated with is the finite sequence defined by the recurrence for and with if , and if .
We write to indicate that is the union of pairwise disjoint sets . If and is a rooted tree, denotes a graph with a unique directed cycle of length , where every node in this cycle is the root of a tree isomorphic to . We also consider the disjoint union of the graphs , denoted by , and for . If are rooted trees, is a rooted tree such that its root has exactly predecessors , and is the root of a tree isomorphic to for . If is a tree that consists of a single node we simply write . In particular, denotes a directed cycle with nodes. The empty graph, denoted by , is characterized by the properties: for all graphs , for all and .
We associate to each -series a rooted tree, denoted by , defined by the recurrence formula (see Fig. 2):
The tree has vertices and depth ; see Proposition 2.14 and Theorem 3.16 of .
The following theorem is a direct consequence of Corollary 3.8 and Theorem 3.16 of . As usual, denotes Euler’s totient function.
Let and be the -decomposition of . Denoting by the functional graph of the multiplication-by- map on the cyclic group , the following isomorphism holds:
A strategy to describe a functional graph of a function is decomposing the set in -invariant components. A subset is forward -invariant when . In this case the graph is a subgraph of . If , the set is backward -invariant. The set is -invariant if it is both forward and backward -invariant. In this case is not only a subgraph of but also a union of connected components and we can write , where . In this paper, we decompose the set in -invariant subsets such that each functional graph for is easier to describe than the general case and .
To describe a functional graph we need to describe not only the cyclic part but also the rooted trees attached to the periodic points. We introduce next some notation related to rooted trees (where the root is not necessarily a periodic point). Let , and be the set of its non-periodic points. We define the set of predecessors of by
We denote by the rooted tree with root , vertex set and directed edges for .
Iii Results on homomorphism of functional graphs
A directed graph is a pair where is the vertex set and is the edge set. A homomorphism between two directed graphs and , denoted by , is a function such that if then . In the particular case of functional graphs, a homomorphism is a function satisfying , or equivalently such that the following diagram commutes
It is easy to prove by induction that the relation implies for all , that is, is also a homomorphism for all . If in addition is bijective (as function from to ) then is an isomorphism of functional graphs. In this case the functional graphs are the same, up to the labelling of the vertices. The main result of this paper (Theorem 4) is an explicit description of , the functional graph of the Chebyshev polynomial over a finite field .
In the first part of this section we introduce the concept of -covering between two functional graphs and derive some properties. In the last part we apply these results to obtain some rooted tree isomorphism formulas which are used in the next sections.
In our case of study (functional graph of Chebyshev polynomials) we consider the set , where is the multiplicative subgroup of of order , and the following maps:
The inversion map given by .
The exponentiation map given by .
The map given by .
A useful relationship between these maps and the Chebyshev map are and . In other words we have the following commutative diagrams:
To describe the Chebyshev functional graph it is helpful to consider the homomorphism and to relate properties between these functional graphs. This homomorphism is not an isomorphism, but it has very nice properties that are captured in the next concept.
Let be a homomorphism of functional graphs and be a permutation (bijection) which commutes with (that is, ). Then is a -covering if for every there is such that (in other words, if the preimage of each point is a -orbit). The homomorphism is a covering if it is a -covering for some verifying the above properties.
We remark that a covering is necessarily onto and every isomorphism is a covering (with respect to the identity map , ). We note that the condition of being a -orbit for all implies that .
In  it is proved several properties of the map . Namely is surjective, , , and for , where and are the roots (in ) of which are distinct if . In particular, with our notation, we have that is a -covering between these functional graphs.
Next we prove some general properties for coverings of functional graphs that are used in the next section for the particular case of the covering . In the next propositions we denote by and the set of periodic and non-periodic points with respect to the map , respectively. We note that if is a homomorphism and then there is a such that . This implies , thus and we have . The next proposition shows that when is a covering this inclusion is in fact an equality.
Let be a permutation satisfying . If is a -covering then .
Let be the order of (i.e. ). It suffices to prove . If then there is a such that . Since we conclude that for some . Applying on both sides we obtain . In the same way, applying several times, we have by induction that for all . With we obtain , thus . ∎
The equation is equivalent to since .
Let be a homomorphism satisfying and . We have .
Let , (in particular ). By definition, there is an integer such that . This implies . Since and we have , thus . ∎
If then since is surjective.
Let be a permutation satisfying , and be a -covering. The equality holds.
The inclusion follows from Propositions 1 and 2 (see also Remark 2). To prove the other inclusion we consider with (in particular ) and such that (this is possible because is surjective). We have to prove that there is a point such that . By definition there is an integer such that and we have . Since is a -covering, from we have that for some integer and define . Using that and commute we obtain and (because ). To conclude the proof we have to show that and it suffices to prove that . Since we have by Proposition 1 (see also Remark 1). ∎
With the same notation and hypothesis of Proposition 3, if we denote by and we have that the restricted function is onto. We want to find conditions to guarantee that is a bijection. We recall that the order of a permutation is the smallest positive integer such that . This implies that the cardinality of the -orbit of a point , given by , is a divisor of .
Let be a permutation of order . A point is -maximal, if the sequence of iterates: are pairwise distinct (that is, if the -orbit of has exactly elements).
An important particular case is when is the identity map. In this case every point is -maximal.
Let be a permutation satisfying , be a -maximal point of and be a -covering. We denote by and . Then the restricted map is a bijection.
By Proposition 3 we have that is onto. To prove that is -to- we consider such that . Then there is an integer such that . If the order of the permutation is , we can suppose that and we also have . We consider the smallest integers such that for (they exist because ). We want to prove that . Consider the smallest integer such that . We have that is an isomorphism of a functional graph (since is bijective and ), thus, by Proposition 1, . We have that (in particular because and is a predecessor of ). We have that and by the minimality of we conclude that . In a similar way we prove the other inequality obtaining ; let us denote by . We have with . Using that is -maximal we conclude that and as desired. ∎
Iii-B Rooted tree isomorphism formulas
Let be a homomorphism of functional graph. We consider a point and the sets and . When and the restricted map is a bijection, this map determines an isomorphism between the rooted trees and (i.e. a bijection between the vertices preserving directed edges). In this case we say that is a rooted tree isomorphism and the trees and are isomorphic which is denoted by . Sometimes, when the context is clear, we abuse notation and write when these trees are isomorphic.
The first result is about the trees attached to the map . Since and are closed under multiplication we have and .
Let and be the -decomposition of and , respectively. Let and be two -periodic points. Then and .
The sets and are multiplicative cyclic groups of order and , respectively. In general, if is a multiplicative cyclic group of order with , , and is the map given by we prove that . Indeed, if is a generator of and is the map given by , then (where denotes the multiplication-by- map). This implies that is an isomorphism of functional graphs. Since all the trees attached to periodic points in are isomorphic to (Theorem 1) the same occurs for the trees attached to periodic points in . ∎
If is an odd integer and , then and are isomorphic.
Consider the map given by . Since is an odd integer, the Chebyshev polynomial is an odd function and we have . Thus is an isomorphism of functional graphs and the results follows from Proposition 4. ∎
Let . Then, and are isomorphic.
We consider the isomorphism of functional graphs given by (it is an isomorphism because is bijective and ). The results follows from Proposition 4. ∎
Let with and . Then, and are isomorphic.
We consider the homomorphism (it is a homomorphism because ). This homomorphism is in fact a -covering because where is a root of . We note that is not -maximal if and only if since is a permutation of order ; this is equivalent to . If , then is -maximal and the result follows from Proposition 4. ∎
Iv Splitting the functional graph into uniform components
The most simple case of functional graph is when the trees attached to the periodic points are isomorphic. In this case describing the functional graph is equivalent to describing the cycle decomposition of the periodic points and the rooted tree attached to any periodic point. We start with a definition.
A functional graph is uniform if for every pair of periodic points the trees and are isomorphic.
In this section we decompose the set in three -invariant sets: (the rational component), (the quadratic component) and (the special component), obtaining a decomposition of the Chebyshev functional graph
Moreover, we prove that the functional graphs of the right hand side are uniform (Proposition 10). We describe each component separately.
We have is -invariant if and only if is -invariant.
() Let . We have and (because is forward -invariant). Therefore and then . This proves that