Refining the search space of Alltop monomials

Alltop functions have applications to code-division multiple access (CDMA) systems and mutually unbiased bases (MUBs). Alltop functions construct MUBs and CDMA signal sets, and this motivates the continued search for further Alltop functions. The discovery of further Alltop functions is hindered by the computational complexity of verification. This paper narrows the search space through the introduction of a characteristic cubic polynomial and provides a verification process for Alltop monomials with a reduced computational complexity. Computational results lead to the conjecture that there are no further Alltop monomials.


Introduction
In the 1970s, Alltop [1] began research into wireless signal sets with low correlation for use as signals in CDMA systems. In 1980, Alltop successfully constructed periodic sequences which came close to the known bounds for use in these systems, using a cubic polynomial over the field F p where p was a prime greater than 3.
This construction was extended to all prime power fields F p r for p > 3, r ≥ 1 and used to construct mutually unbiased bases (MUBs) [9]. MUBs are important for quantum key distribution algorithms and CDMA signal sets [3].
Identifying a MUB is a computationally expensive process which is being explored numerically [5]. As such, exploring Alltop functions potentially adds to the space of MUBs that could be used for quantum key distribution algorithms. There are two known families of Alltop functions [1,7] which are both suitable for MUBs and CDMA signal set constructions. These applications motivate further searches for Alltop functions. There has been recent computational work in this direction [8].
In 2013, a family of Alltop functions was discovered which is inequivalent to the cubic polynomial introduced by Alltop in 1980 [7]. We continue from this family, exploring polynomials in order to search for additional families of Alltop functions. The computational complexity of naively verifying an Alltop functions is not practical. For this reason, our focus in this paper is on refining the search space of Alltop functions and reducing the computations required to verify an Alltop function.
Throughout this paper, we cover the following. In Sect. 2, we describe the known literature surrounding planar functions and Alltop functions. In Sect. 3, the existing families of Alltop functions are discussed and the form of Alltop functions is further explored. Section 4 introduces the characteristic cubic (CC) polynomial. The naive (exhaustive) computational requirements are also discussed here before a theoretical approach is taken in Sect. 5-7 which significantly reduces the computational requirements for verifying Alltop monomials. An improved process is set out in Sect. 8, followed by some computational results and related conjectures in Sect. 9. Finally, the results are summarized in Sect. 10. functions do not exist when p = 3 [6]. Therefore, when discussing Alltop functions, we note that p ≥ 5. The property of a function being planar is preserved when the function is composed with an additive (or linearized) polynomial [2]. [2] A function L(x) : F p r → F p r is called additive or linearized if L(x) + L(y) = L(x + y) for all x, y ∈ F p r . All linearized functions on F p r have the shape

Definition 4 (Linearized polynomial)
where a k ∈ F p r .
All known planar functions on fields of characteristic p ≥ 5 have the shape of a Dembowski-Ostrom (DO) polynomial [4].
Equivalence is important to consider in this case as we are dealing with a finite number of polynomials and want to know how many are unique given a specific field. Definition 6 [7] Two functions, f 1 , f 2 , are extended affine (EA) equivalent if there exist affine functions l 1 , l 2 , l 3 such that We apply this definition to planar and Alltop functions.
Lemma 1 [2] Let l 1 , l 2 , l 3 be affine functions with l 1 (x) and l 2 (x) permutations. If (x) is a planar function on F p r and then (x) is also a planar function and is EA-equivalent to (x).
Any function which is EA-equivalent to a planar function is itself a planar function [2], therefore, all planar functions can be partitioned into EA-equivalence classes. This also holds true for Alltop functions. Lemma 2 [7] Let l 1 , l 2 , l 3 be affine functions with l 1 (x) and l 2

(x) permutations, and D(x) a DO polynomial. If A(x) is an Alltop polynomial on F p r and
is also an Alltop polynomial and is EA-equivalent to A(x).

Families of Alltop functions
There are two existing families of Alltop functions. The first family is the cubic function A(x) = x 3 which exists over any field F p r for characteristic p ≥ 5 [1,9]. The second family was discovered in 2013 by Hall, Rao and Gagola [7].
Theorem 1 [7] Let p ≥ 5 be an odd prime and r an integer such that 3 does not divide The constraints of both field and characteristic in Theorem 1 suggest that further Alltop functions may exist and could also form an infinite family. We begin our exploration by assessing the result when two difference operators are applied to a polynomial. For a polynomial, f (x) = n i=0 a i x i , of degree n over the field F p r , the first difference function simplifies to This leads to the following result.
a i x i be a polynomial of degree n over a field F p r . Then, f ,α (x) is a polynomial of degree ≤ n − 1, with equality holding iff p n.
Lemma 3 can be extended to second difference functions.
a i x i be a polynomial of degree n. Then, f ,α ,β (x) is a polynomial of degree n − 2 iff p n and p (n − 1).
Thus, for p ≥ 5, applying any two difference operators to the cubic function, x 3 , yields a linear function. Polynomials of lower order similarly reduce to constant functions, which cannot be permutations. Turning to higher order polynomials, we next see that Alltop functions of degree 4 do not exist.

Lemma 4
A quadratic function is not a permutation over any field of characteristic p ≥ 3. Proof Let f (x) = ax 2 + bx + c be a quadratic function over a field, F, where a = 0.
Either way, there exist distinct x and y with f (x) = f (y), so f is not a permutation.

Lemma 5 There does not exist an Alltop function of degree 4 over any field of characteristic p ≥ 5.
Proof Result follows from Corollary 1 and Lemma 4.
Note that, Lemma 5 may not extend to all functions of even degree. An example of an even degree permutation polynomial is the function f (x) = x 4 ± 3x ∈ F 7 [10] which, by Corollary 1, allows the existence of even degree Alltop functions.

Alltop function search space
We are interested in reducing the search space of Alltop functions, as an improved search speed may enable the discovery of additional Alltop functions. We begin by arguing that it is reasonable to reduce the search space from all polynomials over F to a general form related to the requirement that the second difference function of an Alltop function must be a permutation. Even so, for a naive search, the number of operations required to identify whether a polynomial is an Alltop function is too large to be considered feasible (see Table 1, details explained later) and so we also consider ways to reduce this computational load.
In order to find a general form, we seek to identify polynomials f (x) for which f ,α ,β (x) is most likely to be a permutation function. Considering the infinite family of Alltop functions given by Hall et al. [7], as well as Alltop's cubic function [1], we note that these families share the form x p k + p j + p i .
Both Eqs. (6) and (7) match the following form that is given in [7] relating to function equivalence.

Lemma 6 [7] Let p be an odd prime and r a positive integer. Any function f over F p r of the form
This form is useful since, as we show in Lemmas 7 and 8, only functions of this form relate to DO polynomials.

Definition 7 (Characteristic Cubic polynomial)
Let p be an odd prime and r a positive integer. We denote polynomials over F p r of the form to be characteristic cubic (CC) polynomials.

Lemma 7 Let p be an odd prime, r a positive integer and f
We can then classify each of the components as follows: More generally, if f (x) is a polynomial with n > 1 terms, then f ,α (x) = D 1 (x)+ D 2 (x) + · · · + D n (x) in similar fashion. Since the sum of two DO polynomials is EAequivalent to a DO polynomial [2], f ,α (x) is EA-equivalent to a DO polynomial.
Further, we show that, over F p r , only polynomials EA-equivalent to CC polynomials have difference functions that are EA-equivalent to DO polynomials.

Lemma 8 Let p be an odd prime, r is a positive integer and
where u k j ≥ 1. Consider the application of the difference operator upon f (x). From Lemma 7, each monomial x p k + p j +u k j gives the following difference function The leading term of (x + α) u k j gives a difference function f ,α (x) which contains the terms x p k +u k j and then the function f (x) must be EA-equivalent to a CC polynomial.
As noted above, all known planar functions on fields F p r for p ≥ 5 are DO polynomials [4]. The results of Lemmas 7 and 8 therefore reduce the search space of likely Alltop functions to CC polynomials. For simplicity, we will consider monomials of this form from now on.
Note that, not all CC polynomials are Alltop functions, though all known Alltop functions are EA-equivalent to CC polynomials. If an Alltop function g(x) is found that is not equivalent to a CC polynomial, the planar function g,α (x) must be nonequivalent to a DO polynomial. To quantify the reduced search space, the number of CC monomials over any finite field can be calculated; in fact, it is straightforward to show that the number of monomials of the form Understanding the number of CC monomials over a field allows us to begin computational exploration. In order to exhaustively verify the Alltop function requirements computationally, we are required to perform the following steps.
Exhaustive Process: Given a candidate function f (x) over F p r : Exhaustive computation therefore requires p r ×( p r −1) 2 evaluations of f ,α ,β (x) to successfully identify a monomial as an Alltop function, and therefore, 1/6 × r (r + 1)(r + 2) × p r × ( p r − 1) 2 evaluations of f ,α ,β (x) are needed to verify all CC monomials in a given field. Table 1 presents the number of f ,α ,β (x) calculations for some example fields.
Note that, the calculations presented are maximums, as the verification process can be terminated for a specific function f (x) as soon as we know that f ,α ,β (x) is not a permutation function. As the field size increases, the number of f ,α ,β (x) evaluations required in this exhaustive process becomes too large to feasibly compute. For Alltop functions, the exhaustive process will never terminate early.
There is structure within the field that can be exploited to achieve greater efficiency.

Subgroups and cosets
We explore the structure of CC polynomials over finite fields by assessing the application of the difference operator on a general CC monomial. From Lemma 7, the form In order to see the structure present in the range of f ,α ,β (x) for a non-Alltop function, we plot an example case. The evaluations of a second difference function of f (x) = x 7 over F 5 2 are plotted in Fig. 1, where ζ is used to represent a primitive element.
The graph in Fig. 1 suggests that a subgroup structure exists among non-Alltop functions. We demonstrate this structure by displaying the subgroup and the 1-coset for the example from Fig. 1 in Table 2. This result is explained by [10], Theorem 3.50: In particular, the solutions of f ,α ,β (x) = f ,α ,β (0) form a subgroup of the additive group of F p r . The function plotted is the second difference function of f (x) = x 7 over F 5 2 , with α = 1 and β = ζ + 3 = ζ 2 and shows clustering. The primitive polynomial used to generate the field is x 2 + 4x + 2 the same multiplicity, which is either 1 or a power of p, and the roots form a linear subspace of F p s , where F p s is regarded as a vector space over F p . This is an important result which shows that we no longer have to verify all outputs f ,α ,β (x) as permutations across all α, β for a given function. Theorem 2 also shows that if the subgroup, G, given by the solutions to f ,α ,β (x) = f ,α ,β (0) is nontrivial, then it has cardinality |G| = p v , for some positive v.

Theorem 2 [10] Let L(x) be a nonzero linearised polynomial over F p r and let the extension field F p s of F p r contain all the roots of L(x). Then, each root of L(x) has
If a polynomial, f (x), is an Alltop function, then f ,α ,β (x) is a permutation function and hence the roots of f ,α ,β (x) are trivial. Conversely, if the group of roots is non-trivial, then f (x) is not an Alltop function. The task of checking whether a given polynomial is an Alltop function has thus been reduced to a search for nontrivial roots of f ,α ,β (x). We next consider how the search space for these solutions can be reduced. Table 2 The subgroup and coset behaviour of the non-Alltop polynomial f (x) = x 7 . The second difference function was found by setting α = 1 and β = ζ + 3 = ζ 2 with the primitive polynomial x 2 + 4x + 2 over F 5 2 . These results are given in multiplicative field notation (M) and in vector space notation (VS) ζ 20 3ζ + 4 0 0 ζ 10 4ζ + 4 ζ 19 3ζ ζ 13 4ζ 0 0

Reducing the search space
The number of calculations of f ,α ,β (x) for a given f (x) = x p k + p j + p i can be reduced by removing the need to search over α; the variable used to define the first difference function. We show that it is sufficient to search for a nonzero solution for which α = 1. (x, α, β) for the equation f ,α ,β (x) = c, with x = 0. Then,  (x , α , β ) is also a solution, where α = 1 and x = 0.

Lemma 9 Let f ,α ,β (x) be the second difference function of a CC monomial. Suppose there is a solution
Dividing by α p k + p j + p i and setting x = x α = 0, β = β α and α = α α = 1 gives That is, x , α = 1 and β give a solution of the equation, as claimed.
As a result of Lemma 9, it suffices to search for nonzero solution pairs (x, β) of the equation Thus, rather than exhaustively verifying whether f ,α ,β (x) is a permutation function for all x ∈ F p r and α, β ∈ F * p r , we need only show that there exist no nonzero solutions to Eq. (12). This is achieved by attempting to find at least one nonzero subgroup element.

Finding subgroup elements
In searching for non-trivial solutions to Eq. (12), it is convenient to consider three cases relating to the values of i, j and k. In all cases, we consider f (x) = x p k + p j + p i .

All exponents equal
When i = j = k, Eq. (12) reduces to Since β = 0, there are no nonzero solutions x of this equation; every function of the form x 3 p i is an Alltop function. This result is not surprising, since f (x) = x 3 p i is EA-equivalent to f 2 (x) = x 3 (see Definition 6).

Two exponents equal
Since β = 0 and we are seeking non-trivial solutions for x, this reduces further to or, equivalently, to The following lemma shows that we need not check every combination of x, β ∈ F * p r to determine whether Eq. (13) has a nonzero solution.

Lemma 10 Let f (x) = x p k +2 p ϕ be a polynomial over a field F p r . To verify that f (x) is an Alltop function it is sufficient to check
Denote a generator of F p r as ζ . Then, as β cycles through all the elements of F * p r , the powers β p k − p ϕ will cycle through all the powers of ζ γ . Likewise, x p k − p ϕ must also be a power of ζ γ for any nonzero value of x.
It follows that it is sufficient to check at most |F * p r |/|F * p θ | = ( p r − 1)/( p θ − 1) distinct values of β in seeking possible solutions to Eq. (13). Furthermore, it is not necessary to explicitly solve for x, as it is straightforward to check whether the resulting RHS value of Eq. (13) is a power of ζ γ .
For any given field F p r , Lemma 10 also allows us to reduce the number of index sets k = j = i = ϕ that must be included in our search for Alltop functions of the form x p k +2 p ϕ .
Note that, the same value of γ applies for any pairs of values of k and ϕ that differ by a constant amount k − ϕ, so we are seeking solutions for x and β among the same set of powers (powers of ζ γ ) in all these cases. Based on this observation, it suffices to search over pairs (k, ϕ) for which ϕ = 0, of which there are only r − 1 (k = 1, 2, . . . , r − 1). Furthermore, the search space can be further reduced by noting that it is not k − ϕ itself but gcd(k − ϕ, r ) which determines γ ; as a consequence, it is sufficient to search only over values of k − ϕ that are divisors of r . We have therefore demonstrated the following lemma.

Lemma 11
Consider the set of CC monomials of the form x p k +2 p ϕ over a field F p r . To assess whether any of these monomials are Alltop functions, it is sufficient to search for nonzero solutions of Eq. (13) with ϕ = 0 and k|r.

Distinct exponents
When k = j = i, Eq. (12) can again be reduced. As β = 0 and we seek nonzero solutions for x, Eq. (12) reduces to The following lemma shows that, in the same way as Lemma 10, it is not necessary to check every combination of x, β ∈ F * p r to determine whether this equation has a nonzero solution.
Lemma 12 Let f (x) = x p k + p j + p i be a polynomial over a field F p r , where k = j = i.
To verify that f (x) is an Alltop function, it is sufficient to check τ 2 +τ 2 pairs of nonzero values (x, β) from the field F p r , where τ = p r −1 Denote the generator of F p r as ζ ; then, as β cycles through all the elements of F * p r , the powers β p j − p i and β p k − p i will cycle through various powers of ζ γ . Likewise, x p k − p i and x p j − p i must also be powers of ζ γ for any nonzero value of x.
It follows that it is sufficient to check at most |F * p r |/|F * It is possible to reduce the number of CC monomials of the form f (x) = x p k + p j + p i , where k = j = i, that must be checked to find Alltop functions for any given field F p r . Similarly to the discussion supporting Lemma 11, we are interested in the pairs of values of j and i and pairs of values of k and i which differ by constant amounts j − i and k − i, respectively. For any given value of these differences, we are seeking solutions for x and β among the same set of powers of ζ γ . This gives the following lemma.
Lemma 13 Consider the set of CC monomials of the form x p k + p j + p i over a field F p r . To assess whether any of these monomials are Alltop functions, it is sufficient to search for nonzero solutions of Eq. (14) over tuples (k, j, i) for which i = 0.
Note that, all CC monomials with distinct exponents for which i = 0 are EAinequivalent (see Definition 6). This means there are r −1 2 = (r −1)(r −2) 2 unique CC monomials with distinct exponents in a given field. A further reduction similar to the second reduction argument in Lemma 11 may be possible but due to the additional term in the gcd of Eq. (15), any such reduction is not expected to have a significant impact on the overall verification process. As a result, this optimization is omitted from this paper.
In these last two sections, we have reduced the number of functions to be tested by restricting the search space to the set of CC monomials; we have also reduced the number of operations required to determine if a CC monomial is an Alltop function and quantified this reduced operation count. The full improved process is given in the following section.

Optimization results
Throughout this paper, we have reduced the number of monomials needed to verify whether any of these is an Alltop function over a given field, F p r . We have removed Table 3 Upper bounds for the total number of times different field elements need be substituted into Eqs. (13) and (14), with θ = 1. This table replicates the fields given in Table 1 to demonstrate the large efficiency gain over the exhaustive process in Sect. 4

F Two Exponents Equal
Distinct Exponents  by running the relevant calculations on the Queensland University of Technology's high-performance computer cluster using an Intel Xenon E5-2680v2 processor and requiring a maximum of 4GB of memory. These computational results indicate that the tests of steps 2)c)ii) and 3)b)ii) routinely take far fewer than the maximum number of steps before determining that a candidate monomial is not an Alltop function. For run-times displayed in Table 4, the exhaustive process for F 7 4 ran for a week without terminating. As a result, we did not attempt the exhaustive process for larger fields.
The improved process takes far less time to compute than the exhaustive process. Not only are there fewer functions and less field elements to test, but also the process of each check is greatly reduced. Due to this additional efficiency, the run-time saving is even greater than the difference that Tables 1 and 3 imply.
The improved process, coupled with Tables 3 and 4 combine to show, through an analysis of the subgroups in f ,α ,β (x) over F p r , that the time to verify Alltop functions over large fields can be greatly reduced.
Despite the improved process, only monomials which belong to the Alltop function families presented in [1] and [7] have so far been found over larger fields. All fields F p r with 5 ≤ p ≤ 97 and r ≤ 4, and fields F p r with 5 ≤ p ≤ 59 and r ≤ 20 have been tested. These results suggest the following two conjectures:

Conjecture 1 Let p be an odd prime and r a positive odd integer. A monomial f (x) = x a over F p r is an Alltop function only if it is EA-equivalent to the function A(x) = x 3 (Alltop's cubic function from [1]).
Conjecture 2 Let p be an odd prime and r any positive integer. Over the field F p r , there exist only two families of Alltop monomials: Alltop's cubic function [1] and the family given by Hall et al. [7].

Conclusion
After considering the link between Alltop functions and planar functions, the form of known Alltop functions, and the link between planar functions and DO polynomials, we have introduced the CC polynomial form and proved that only functions of this form have difference functions of the DO form. Thus, we have established it as a likely form for Alltop functions. We have also demonstrated a subgroup structure related to the second difference functions for CC monomials. By exploiting this structure, we have significantly reduced the number of calculations required to detect whether a given CC monomial is Alltop compared to the exhaustive search.
From our computational search, we conjecture that all Alltop monomials belong to one of two existing families [1,7].