On ternary rotation symmetric bent functions

Claudio Moraga; Milena Stanković; Radomir S. Stanković

doi:10.1515/itit-2025-0026

Article Open Access

On ternary rotation symmetric bent functions

Claudio Moraga
Claudio Moraga received his B.Sc. in E.E. from the Catholic University of Valparaiso (UCV), Chile, in 1961 and M.Sc. in E.E., Massachusetts Institute of Technology (MIT), USA, in 1962, He obtained his Ph.D. in E.E. (Summa cum Laude) from the Technical University “Federico Santa Maria” (UTFSM), Valparaiso, Chile, in January 1972. In November 1974 Claudio Moraga became an Alexander von Humboldt Research Fellow at the Dept. Computer Science, University of Dortmund, FRG. In April 1985 to September 1986, he was a Professor of Computer Science, in the area of Computer Architecture, at the Dept. Mathematics and Computer Science, University of Bremen, FRG. From October 1986 to February 2002, he was a Professor of Computer Science, in the area Theory of Automata at the Dept. Computer Science, University of Dortmund, FRG. From March 2006 through December 2015, Claudio Moraga was a Researcher Emeritus at the European Centre for Soft Computing, Mieres, Asturias, Spain. Claudio Moraga has served as a member of the IEEE Technical Committee on Multiple-valued Logic, and holds several awards, including the “Long Service Award for outstanding contributions to Multiple-valued Logic since 1971” from the IEEE TC on Multiple-valued Logic, received in May 2004 and the best paper award in Multiple-valued Logic at the ISMVL 2010. In 2005, Claudio Moraga was awarded the title of the Doctor honoris causa of the University of Niš, Serbia. He is the author of a few books, three of them with Radomir S. Stanković and Jaakko Astola published by Wiley/IEEE Press, Claypool & Morgan, and Springer, respectively.
, Milena Stanković
Milena Stanković received the B.Sc. degree in electronic engineering from the Faculty of Electronics, University of Niš, Serbia, in 1976, and M. Sc., and Ph. D. degrees in Computing by the same Faculty in 1982 and 1988, respectively. She was a Professor at the Department of Computer Science, Faculty of Electronics, University of Niš, Serbia, until 2018, when she is retired. Milena Stanković served as the Head of the Department of Computing at the Faculty of Electronic Engineering and Head of CIITLab (Computational Inteligence and Information Technologies Laboratory). Her research interests include switching theory, multiple-valued logic, spectral techniques and data mining.
and Radomir S. Stanković
Radomir S. Stanković received the B.Sc. degree in electronic engineering from the Faculty of Electronics, University of Niš, Serbia, in 1976, and M.Sc. and Ph.D. degrees in applied mathematics from the Faculty of Electrical Engineering, University of Belgrade, Serbia, in 1984 and 1986, respectively. He was a Professor at the Department of Computer Science, Faculty of Electronics, University of Niš, Serbia, until 2017, when he moved at the Mathematical Institute of SASA, Belgrade, Serbia, working there until 2019, when he is retired. In 1997, he was awarded by the Kyushu Institute of technology Fellowship and worked as a visiting researcher at the Department of Computer Science and Electronics, Kyushu Institute of Technology, Iizuka, Fukuoka, Japan. In 2000 he was awarded by the Nokia Professorship by Nokia, Finland. From 1999 until 2017, he worked in part at the Tampere International Center for Signal Processing, Department of Signal Processing, Faculty of Computing and Electrical Engineering, Tampere University of Technology, Tampere, Finland, first as a visiting professor and from 2009 until 2017 as an adjunct professor. His research interests include switching theory, multiple-valued logic, spectral techniques, and signal processing. He is an author of a few books, three of them, with Jaakko Astola, published by Springer, and another two with Jaakko Astola and Claudio Moraga, published by Wiley/IEEE Press, and Claypool & Morgan.

Published/Copyright: March 10, 2026

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Abstract

This paper is concerned with the existence, generation and counting of ternary rotation symmetric bent functions. The concept of rank of functions is introduced and it is shown that rotation symmetric bent ternary functions do exist and may be given a compact value vector representation with entries ordered by rank. A basic set of 3-place ternary rotation symmetric functions is extracted from a database of 3-place ternary bent functions. With the help of spectral invariant operations and some new operations a first set of 624 ternary rotation symmetric bent functions with n = 3 was obtained.

Keywords: ternary functions; rotation symmetric functions; bent functions

1 Introduction

Bent functions were introduced in the Boolean domain, by Oscar Rothaus [1] in 1976, as the most non-linear functions, with the highest distance (2ⁿ⁻¹ − 2^(n/2)−1) to all affine functions on an even number of variables. The earliest extensions to the multiple-valued domain were presented by Kumar et al. [2] and by Luis et al. [3]. In the Boolean domain, due to their non-linearity, Boolean bent functions attracted the interest of people working mainly in Cryptography and in Coding Theory. Works on quaternary bent functions have been reported e.g., to study Generalized Boolean functions [4], [5]. Possibly the most mentioned contribution on an application of quaternary bent functions is the work of Schmidt [6] who considered quaternary bent functions in the reduction of the peak-to-power ratio to the lowest possible value in multi-code code-division.multiple.access.systems.(MC-CDMA). In the prime-based multiple-valued domain, bent functions are considered as combinatoric mathematical objects with heavy challenges, like characterization, recognition, classification, generation, and counting multiple-valued bent functions.

Rotation Symmetric Boolean functions, are defined as an extension of symmetric functions under a condition that the cyclic shift of the value assignment to variables is the main characterizing feature. They were introduced by Pieprzyk and Qu [7] in 1999 in the context of hashing, (although they have been considered earlier in [8] as idempotents.) Properties of these functions may be found e.g., in Chapter 5 of [9], where the short denomination RotS was used and will also be used in the present paper. Moreover, in [9] it is pointed out, that there are Boolean bent functions which are RotS. This remark was one motivation for the present paper, focusing on ternary bent functions.

RotS functions may be seen as an extended variation of symmetric functions. Symmetric functions may be traced back to the early works of Shannon [10] and Komamiya [11]. Works on ternary symmetric functions started to appear in 1965 [12], followed by [13], [14], [15]. To the best knowledge of the authors, a first study of ternary symmetric bent functions was disclosed in [16] and a first study on ternary RotS functions was considered in [17]. The present paper follows this line of research and is concerned with ternary RotS bent functions. In what follows, the abbreviation RotSB will be used to denote rotation symmetric bent functions.

2 Formalisms

In this study, work will focus on ternary functions defined as mappings f: (ℤ ₃)ⁿ → ℤ ₃, where ℤ ₃ denotes the ring of non-negative integers smaller than 3. Therefore, all operations will be done modulo 3.

Definition 1:

Given an n-place ternary function, with value vector F, its composition C = (c ₀, c ₁, c ₂) identifies the number of times that the values 0, 1, and 2 appear, respectively, in F. Obviously c ₀ + c ₁ + c ₂ = 3ⁿ. If 1 F denotes a vector of the same length as F with all entries equal to 1, and F′ = F ⊕ 1 F then the composition of F′ equals C′, a one step cyclic shifted composition C. For a given n, a distribution D = (d ₀, d ₁, d ₂) states the repetition of values, but without identifying which values are repeated d-times. Therefore, a distribution covers a composition and all its possible permutations. [4], [18], [19].

Definition 2:

A multiple-valued function f is called symmetric, if its value does not change with any permutation of its arguments. Symmetric functions have obviously at least two arguments.

Definition 3:

A 3-place ternary function f is called rotation symmetric (RotS) if its value does not change with any cyclic shift of the value of its arguments. If the value of the arguments, seen as a vector, is a constant, the cyclic shift makes no changes and the function may have any value (from the value set). If a value assignment is not a constant vector, a cyclic shift will produce a cycle of length 3. Cycles based on the value assignments (012) and (021) are disjoint since they do not share components. A 3-place ternary RotS function has different values at places with value assignments belonging to disjoint cycles, respectively. (See Table 1).

Table 1:

Value assignments to the arguments and rank of 3-place ternary functions.

Value assignments			Rank
Representative
000			0
001	010	100	1
002	020	200	2
011	110	101	3
012	120	201	4
021	210	102	5
022	220	202	6
111			7
112	121	211	8
122	221	212	9
222			10

Definition 4:

The lexicographic first value assignment in a cycle will be its “representative”. (If a value assignment is constant, it is its own representative.) If the representatives are ordered lexicographically, their position in the list (starting with 0), will be called their rank. (See Table 1).

Definition 5:

Let

(1) κ p , n = n + ( p − 1 ) ( p − 1 ) = ( n + p − 1 ) ! n ! ( p − 1 ) !

In the case of ternary functions, p = 3 and if the number of variables is chosen to be 3, then n = 3, and κ _3,3 = 5 ! 3 ! 2 ! = 54 2 = 10 .

The number of 3-place ternary symmetric functions is 3 κ 3,3 = 3¹⁰ = 59,049 [17], [20].

If p = 3 and n = 4, κ _3,4 = 6 ! 2 ! ⋅ 4 ! = 6.5 2 = 15.

The number of 4-place ternary symmetric functions is 3 κ 3,4 = 3¹⁵ = 14,348,907.

Definition 6:

Let

(2) N p n 1 n ∑ d | n φ n d p d ,

where φ denotes the Euler’s Totient function [21] and d|n as usual means that d divides n.

Then if p = 3 and n = 3,

N 3 3 = 1 3 ∑ d | 3 φ 3 d 3 d = 1 3 φ 3 3 + φ 1 3 3 = 1 3 2 × 3 + 1 × 27 = 11 ⋅

The number of 3-place ternary RotS functions is obtained with 3 N 3 ( 3 ) = 3¹¹ = 177,147 [17]. Notice that N ₃(3) = 11 equals the number of ranks of 3-place RotS ternary functions, as shown in Table 1.

If p = 3, n = 4,

N 3 4 = 1 4 ∑ d | 4 φ 4 d 3 d = 1 4 φ 4 3 + φ 2 3 2 + φ 1 3 4 = 1 4 2 × 3 + 1 × 3 2 + 1 × 3 4 = 1 4 6 + 9 + 81 = 24 ,

from which the number of 4-place ternary RotS functions is

3 N 3 4 = 3 24 = 282 , 429 , 536 , 481.

Definition 7:

Let ε denote a principal cubic root of unity. Then, with i = − 1 , ε = exp(2πi/3) = (−1 + i 3 )/2. Notice that ε ² = ε*, the complex conjugate of ε.

It holds ε + ε ² = −1, from where 1 + ε + ε ² = 0.

If V = [v ₁, v ₂, …] is a ternary vector, ε ^V: = ε v 1 , ε v 2 , … .

Definition 8:

[4] The Vilenkin–Chrestenson transform matrix in the ternary domain is given by:

(3) VC n = VC 1 ⊗ n , where VC 1 = 1 1 1 1 ε ε 2 1 ε 2 ε

The exponent “⊗n” denotes here the n-fold Kronecker product of VC(1) with itself.

Accordingly, VC(2) = VC(1) ⊗ VC(1).

(4) VC 2 = 1 1 1 1 ε ε 2 1 ε 2 ε 1 1 1 1 ε ε 2 1 ε 2 ε 1 1 1 1 ε ε 2 1 ε 2 ε 1 1 1 1 ε ε 2 1 ε 2 ε ε ε ε ε ε 2 1 ε 1 ε 2 ε 2 ε 2 ε 2 ε 2 1 ε ε 2 ε 1 1 1 1 1 ε ε 2 1 ε 2 ε ε 2 ε 2 ε 2 ε 2 1 ε ε 2 ε 1 ε ε ε ε ε 2 1 ε 1 ε 2

Lemma 1:

[4], [5], [22], [23]. An n-place ternary function f with value vector F is bent iff the absolute value of all the spectral coefficients of the Vilenkin–Chrestenson spectrum S_VC of ε ^F equals 3^n/2.

(5) S V C ε F = VC ∗ n ⋅ ε F

Example 1:

Let f be a ternary 2-place function with F = [0 0 0 0 1 2 0 2 1]^T.

Then ɛ ^F = 1 1 1 1 ε ε 2 1 ε 2 ε ^T and its VC spectrum is given by:

S VC = VC ∗ 2 ⋅ ε F = 1 1 1 1 ε 2 ε 1 ε ε 2 1 1 1 1 ε 2 ε 1 ε ε 2 1 1 1 1 ε 2 ε 1 ε ε 2 1 1 1 1 ε 2 ε 1 ε ε 2 ε 2 ε 2 ε 2 ε 2 ε 1 ε 2 1 ε ε ε ε ε 1 ε 2 ε ε 2 1 1 1 1 1 ε 2 ε 1 ε ε 2 ε ε ε ε 1 ε 2 ε ε 2 1 ε 2 ε 2 ε 2 ε 2 ε 1 ε 2 1 ε ⋅ 1 1 1 1 ε ε 2 1 ε 2 ε = 3 3 3 3 3 ε 2 3 ε 3 3 ε 3 ε 2

It becomes apparent that all spectral coefficients have magnitude 3, which corresponds to 3^n/2 since n = 2 variables. Therefore the given function is bent.

Notice that (5) represents the product of a matrix and a vector. A calculation with Mat Lab or a similar calculating system is straightforward. Moreover it allows working directly with VC* ( 1 ) ⊗ n without preliminarily calculating the matrix VC*(n) as done in the previous example.

Lemma 2:

If n = 3, necessary conditions for a ternary function to be bent are: the sum mod 3 of all its entries ≡ 0. and its distribution is (6, 9, 12) [4]. Notice that this distribution implies that all three values are present in the entries of the value vector of the function and that the function is not balanced.

Lemma 3:

(Based on [10], [11]) If a ternary function is symmetric it may be given a compact value vector representation of length κ _3,n according to the ranks of the permuted assignments to the arguments.

Corollary 3.1:

If a ternary function is RotS it may be given a compact value vector representation of length N ₃(n) according to the ranks of the cyclic permuted assignments to the arguments. See Figures 1 and 2.

Figure 1:

Ranks of a 3-place ternary function.

Figure 2:

Top: Karnaugh-type map of the ternary RotSB F ₀. Bottom: Compact representation of F ₀ according to ranks.

Definition 9:

The places of a ternary RotS function, such that the value assignment to the arguments (x ₁, x ₂, x ₃), are (0, 0, 0), (1, 1, 1) and (2, 2, 2), will be called “neutral”.

3 Ternary RotS bent functions

Some elementary basic questions will be considered in this section:

Are there ternary RotSB functions?
If yes, are there any ways to generate or find them?
If yes, how many such functions are there?

Notice that these questions comprise existence, generation, and counting RotSB functions. This indicates that the present work is a mathematical contribution. Answers to these questions are needed before stating questions about possible practical applications.

For the first question, it seems reasonable to consider the simplest case, i.e., n = 3 and to recall the necessary conditions for a ternary function to be bent: the function must exhibit all three values, the sum mod 3 of its entries ≡ 0 and its distribution must be (6, 9, 12) [4]. The necessary conditions for a 3-place ternary function to be RotS, after Definition 3 (and Table 1) are that entries at places with the same rank must have the same value and entries at places with ranks 4 and 5 must be different. Recall Corollary 3.1: ternary RotS functions may be represented by a compact value vector of length N ₃(n) whose entries are ordered according to their ranks.

To answer with “yes” the first question, a random search was done to find N ₃(3) = 11 entries under the necessary conditions of both bent and RotS functions. The entries should have a (6, 9, 12) distribution and as shown in [16] the entries corresponding to the cycle (0, 1, 2) must be different from the entries corresponding to the cycle (0, 2, 1). Furthermore, to reduce the search complexity (from 11 to 8 unknowns), the class of RotS functions with the value 0 in its neutral places was considered. It took about 10 min to manually find the following 3-place RotSB function shown in Figure 2. A basic map with the ranks corresponding to the places when n = 3 is provided in Figure 1.

Analysis:

The map of Figure 2 shows that the entries of the function comprise all three values and by counting, the distribution (6, 9, 12) is obtained. Moreover, the sum mod 3 of all entries is congruent with 0. Furthermore, it is simple to see in the compact representation, that the function vector has the same entry at places with the same rank and has different entries for the ranks 4 and 5, which correspond to the disjoint cycles based on the value assignments (012) and (021), respectively. Finally, a direct calculation of the Vilenkin–Chrestenson spectrum of F ₀ (according to Eq. (5)) proved that the spectral coefficients had the absolute value 3 3 , which corresponds to 3^n/2, since n = 3. Therefore, F ₀ is the value vector of a 3-place ternary RotSB function. Its polynomial representation will be shown in Eq. (7):

Remark 1:

It is possible to calculate the distribution of F ₀ from the compact representation. Recall however, that the example belongs to the class of RotS functions whose 3 neutral places have the entry 0, (at ranks 0, 7 and 10, respectively), whereas each of the other 0-entries (of the compact representation) corresponds to three 0-entries of the full representation. Therefore, this leads to 3 + 3⋅(3) = 12 entries with value 0. The 1-entries lead to 3⋅(2) = 6 and the 2-entries lead to 3⋅(3) = 9. It follows that the distribution is (6, 9, 12), as required for bentness.

What we have done up to now answers the first question clearly with “yes”.

Now, to the second and third questions:

In the case of ternary bent functions there is not known method to generate bent functions “from scratch”. New bent functions are obtained based on known bent functions and applications of spectral invariant operations [4] or tensor sums [16]. For other possibilities related to Boolean bent functions, which might possibly be extended to the ternary domain, see e.g. [5], [22]. Similar (but not the same) constraints apply in the case of ternary RotSB functions.

Lemma 4:

[16] Let F denote the value vector of a ternary RotSB function. Then, with k ∊ {1, 2}:

(6) F ′ = F ⊕ k · 1 F , F ′ ′ = 2 · F , F ′ ′ ′ = 2 · F ⊕ k · 1 F

are also RotSB.

Proof:

The operations shown are spectral invariant [4] and they preserve bentness. Furthermore, they shift or scale the entries of F by a constant, which preserves the conditions for rotation symmetry.

Remark 2:

Other spectral invariant operations like adding a scaled or value shifted value vector of some arguments to a RotS function does not preserve the rotation symmetry, since this would affect in a different way, places with the same rank, but at different rows or columns of the Karnaugh map.

Remark 3:

2⋅F ⊕ 2⋅ 1 F represents the ternary complement of F since F ̄ = 2⋅ 1 F − F and −1 ≡ +2 mod 3.

Lemma 5:

Let F denote the value vector of a 3-place ternary RotS function. Swapping the entries with ranks 4 and 5 leads to another RotS.

Proof

follows from Definition 3.

If a database of n-place ternary bent functions is available, then it is possible to run a computer based search for RotS functions. In [24] a database of 858 3-place ternary bent functions was indicated. A search for RotS functions of the same class as F ₀, characterized by neutral places with entry 0 was done. The functions shown in Table 2 were found in less than 10 s using a Lenovo Think Pad laptop with 16 GB memory and 2.8 GHz clock-rate. These functions are independent of each other in the context of Lemma 4.

Table 2:

3-place ternary RotSB functions with 0-entries at the shaded neutral places.

Rank	1	2	3	4	5	6	7	8	9	10
F ₁	0	0	1	0	1	1	0	2	2	0
F ₂	0	1	0	0	2	2	0	1	2	0
F ₃	0	1	2	0	1	1	0	2	0	0
F ₄	0	2	0	1	0	1	0	2	1	0
F ₅	1	0	1	0	2	2	0	0	2	0
F ₆	1	0	2	1	0	0	0	2	1	0
F ₇	1	0	2	2	0	0	0	2	1	0
F ₈	1	1	0	1	0	0	0	2	2	0
F ₉	1	1	0	2	0	0	0	2	2	0
F ₁₀	1	1	2	0	2	2	0	0	0	0
F ₁₁	2	1	0	0	1	1	0	0	2	0
F ₁₂	2	1	2	1	0	0	0	1	0	0
F ₁₃	2	1	2	2	0	0	0	1	0	0

Remark 4:

Notice that F ₀ was not found when searching, however F ₀ and F ₅ “commute” when swapping the entries with ranks 4 and 5. Both are RotS and bent!

The polynomial expressions corresponding to F ₀ and F ₅ are:

(7) f 0 x 1 , x 2 , x 3 = 2 x 3 ⊕ 2 x 3 2 ⊕ 2 x 2 ⊕ x 2 x 3 ⊕ 2 x 2 x 3 2 ⊕ x 2 2 ⊕ x 2 2 x 3 ⊕ x 2 2 x 3 2 ⊕ 2 x 1 ⊕ x 1 x 3 ⊕ x 1 x 3 2 ⊕ x 1 x 2 ⊕ x 1 x 2 x 3 2 ⊕ 2 x 1 x 2 2 ⊕ x 1 x 2 2 x 3 ⊕ 2 x 1 2 ⊕ 2 x 1 2 x 3 ⊕ x 1 2 x 3 2 ⊕ x 1 2 x 2 ⊕ x 1 2 x 2 x 3 ⊕ x 1 2 x 2 2 f 5 x 1 , x 2 , x 3 = 2 x 3 ⊕ 2 x 3 2 ⊕ 2 x 2 ⊕ x 2 x 3 ⊕ x 2 x 3 2 ⊕ 2 x 2 2 ⊕ 2 x 2 2 x 3 ⊕ x 2 2 x 3 2 ⊕ 2 x 1 ⊕ x 1 x 3 ⊕ 2 x 1 x 3 2 ⊕ x 1 x 2 ⊕ x 1 x 2 x 3 2 ⊕ x 1 x 2 2 ⊕ x 1 x 2 2 x 3 ⊕ 2 x 1 2 ⊕ x 1 2 x 3 ⊕ x 1 2 x 3 2 ⊕ 2 x 1 2 x 2 ⊕ x 1 2 x 2 x 3 ⊕ x 1 2 x 2 2

It is simple to understand that swapping the entries with ranks 4 and 5 in all other functions obtained from the database will preserve their RotS character. (Recall Definition 3). This poses however, an important question: does this swapping also preserve bentness? A direct check of the “swapped” 13 ternary RotS functions obtained from the database showed that they were also bent. The absolute value of the spectral coefficients of their Vilenkin–Chrestenson spectra was 3 3 = 3 ^1.5 as required for 3-place ternary bent functions.

Therefore we now have 26 ternary RotSB functions with n = 3. All these functions have 0-entries at the neutral places.

Is it possible to increase the number of 3-place ternary RotSB found without moving to search in a next larger database of ternary bent functions?

Definition 10:

Let U be a 3-place two-valued ternary function such that U (0, 0, 0) = U (1, 1, 1) = U (2, 2, 2) = 1 and otherwise U (x ₁, x ₂, x ₃) = 0.

Recall that ternary RotS functions may take any ternary value at the neutral places and that the search was restricted to RotS functions with value 0 at the neutral places. Therefore, e.g., F ₂ + U, will also be RotS, but it will not be bent since its distribution becomes (9, 9, 9), i.e., the function is balanced and it does not preserve the distribution (6, 9, 12) required for bentness. On the other hand, the composition of F ₂ ⊕ 2U becomes (9, 6, 12), and the arithmetic sum of its entries is 6 + 24 = 30, which is congruent with 0 in modulo 3. Therefore F ₂ + 2U satisfies the necessary conditions for bentness. A direct calculation shows that the coefficients of its Vilenkin–Chrestenson spectrum have the absolute value 3 3 , therefore, besides being RotS, the function is bent.

Definition 11:

(8) Let Alpha = F 1 , F 3 , F 4 , F 6 , F 8 , F 11 , F 12 and Beta = F 2 , F 5 , F 7 , F 9 , F 10 , F 13 .

All Alpha-functions have the composition (12, 9, 6) and these functions + U have the composition (9, 12, 6), whereas if 2U is added, their composition becomes (9, 9, 9). Therefore, The Alpha-functions + U satisfy the distribution (6, 9, 12) required for bentness. A check of their Vilenkin–Chrestenson spectra confirmed their bentness. Moreover, Alpha functions + U preserve their RotS character. It is simple to check that if for all Alpha functions the swap of entries with ranks 4 and 5 is done, the new functions also have the former property. We may speak of “extended Alpha functions”.

All Beta-functions have the composition (12, 6, 9) and these functions + 2U have the composition (9, 6, 12), whereas if only U is added, their composition becomes (9, 9, 9). Therefore, the Beta-functions + 2U satisfy the distribution (6, 9, 12) required for bentness. A check of their Vilenkin–Chrestenson spectra confirmed their bentness. Moreover, Beta functions + 2U preserve their RotS character. It is simple to check that if for all Beta functions the swap of entries with ranks 4 and 5 is done, the new functions also have the former property. We may speak of “extended Beta functions”.

It is simple to conclude that with the properties of the extended Alpha- and extended Beta-functions, the number of ternary RotS bent functions found by searching (with value 0 at the neutral places), and swapping entries with ranks 4 and 5, will be duplicated, leading to 52 RotSB 3-place ternary functions.

Is it possible to increase the number of ternary RotSB functions beyond the number found/obtained up to now?

Recall that based on the 13 seeds from a database of 858 ternary bent functions in 3 variables under the constraint that neutral places have the entry 0, swapping entries with ranks 4 and 5, and properly increasing the entries at the neutral places, a preliminary total of 52 RotSB functions had been obtained. If we apply the 5 operations of Lemma 4, we obtain 5 × 52 = 260 new RotSB functions leading to 312 ternary RotSB functions.

In [25] it was reported that there are 303,264 ternary bent functions for n = 3 and degree 3, and in [4] it is reported that there are 12,623,364 ternary bent functions of degree up to 4. The degree of a function is given by the largest sum of the exponents of the variables in the product terms of its Positive Polarity Reed–Muller polynomial expression. The total number of ternary bent functions is still not known.

Is it still possible to increase the number of ternary RotSB found/obtained up to now without searching in a larger database of 3-place ternary Bent functions?

There is one operation that does not “officially” belong to the spectral invariant operations, which however preserves bentness:

Given a ternary bent function f (x ₁, x ₂, x ₃) then

(9) μ f x 1 , x 2 , x 3 = f x ̄ 1 , x ̄ 2 , x ̄ 3

is also a bent function [4]. Moreover, the value vector of μ f equals the value vector of f, however read bottom up. Therefore, if f (x ₁, x ₂, x ₃) is RotS then μ f will also be RotS, since the conditions of Definition 3 remain preserved.

The above means that the μ function applied to the already found ternary RotSB functions will duplicate their number. This leads to 624 ternary RotSB functions finally obtained from the 13 seed functions extracted from a moderate sized database of ternary bent functions.

4 Conclusions and further work

Three basic questions were studied and some possible answers were given: ternary RotSB functions do exist. No direct generation method is known, as is also the case for ternary bent functions, but operations to generate new RotSB functions from known RotSB functions were disclosed. Since the number of ternary bent functions is unknown this also applies to RotSB functions, however, a quite reasonable number of such functions for n = 3 and degree 4 was obtained, based on a set of only 13 “seed” RotSB functions obtained by a restricted search on a database of ternary bent functions. Two options are in the agenda for further research: one rather obvious is to search for equivalent classes of seeds in a larger database of ternary bent functions. The other one would consider work with “high level” spectral invariant operations. Recall the mirror function μ f x 1 , x 2 , x 3 = f x ̄ 1 , x ̄ 2 , x ̄ 3 . This function preserves bentness since it applies the complementation of an argument, which is a basic spectral invariant operation, to each argument. μ f x 1 , x 2 , x 3 is such a “high level” spectral invariant operation. For further study of RotSB functions, “high level” spectral invariant operations preserving RotS could be used. (Recall Lemma 4). In [4] an efficient formalism based on permutation matrices is introduced to work with this class of “high level” spectral invariant operations.

Further open questions include the case of p = 3 but n > 3 and the case of p > 3. These questions bring together the curse of dimensionality. Recall that for p = 3 and n = 4 there are 3²⁴ = 282,429,536,481 RotS functions (out of 3⁸¹ 4-place ternary functions) and to the best of our knowledge there is no database of ternary bent functions in 4 variables. We only know that 4-place ternary bent functions based on the sum of products of disjoint pairs of variables have composition (33, 24, 24) [4]. If p > 3 the situation is even more severe. Examples of bent functions (of the Maiorana–McFarland class) for p = 5 and p = 7 have been shown in [26], but they are neither symmetric nor RotS.

Corresponding author: Claudio Moraga, Technical University Dortmund, Otto-Hahn-Str. 12, Dortmund 44227, Germany, E-mail: claudio.moraga@udo.edu

About the authors

Claudio Moraga

Claudio Moraga received his B.Sc. in E.E. from the Catholic University of Valparaiso (UCV), Chile, in 1961 and M.Sc. in E.E., Massachusetts Institute of Technology (MIT), USA, in 1962, He obtained his Ph.D. in E.E. (Summa cum Laude) from the Technical University “Federico Santa Maria” (UTFSM), Valparaiso, Chile, in January 1972. In November 1974 Claudio Moraga became an Alexander von Humboldt Research Fellow at the Dept. Computer Science, University of Dortmund, FRG. In April 1985 to September 1986, he was a Professor of Computer Science, in the area of Computer Architecture, at the Dept. Mathematics and Computer Science, University of Bremen, FRG. From October 1986 to February 2002, he was a Professor of Computer Science, in the area Theory of Automata at the Dept. Computer Science, University of Dortmund, FRG. From March 2006 through December 2015, Claudio Moraga was a Researcher Emeritus at the European Centre for Soft Computing, Mieres, Asturias, Spain. Claudio Moraga has served as a member of the IEEE Technical Committee on Multiple-valued Logic, and holds several awards, including the “Long Service Award for outstanding contributions to Multiple-valued Logic since 1971” from the IEEE TC on Multiple-valued Logic, received in May 2004 and the best paper award in Multiple-valued Logic at the ISMVL 2010. In 2005, Claudio Moraga was awarded the title of the Doctor honoris causa of the University of Niš, Serbia. He is the author of a few books, three of them with Radomir S. Stanković and Jaakko Astola published by Wiley/IEEE Press, Claypool & Morgan, and Springer, respectively.

Milena Stanković

Milena Stanković received the B.Sc. degree in electronic engineering from the Faculty of Electronics, University of Niš, Serbia, in 1976, and M. Sc., and Ph. D. degrees in Computing by the same Faculty in 1982 and 1988, respectively. She was a Professor at the Department of Computer Science, Faculty of Electronics, University of Niš, Serbia, until 2018, when she is retired. Milena Stanković served as the Head of the Department of Computing at the Faculty of Electronic Engineering and Head of CIITLab (Computational Inteligence and Information Technologies Laboratory). Her research interests include switching theory, multiple-valued logic, spectral techniques and data mining.

Radomir S. Stanković

Radomir S. Stanković received the B.Sc. degree in electronic engineering from the Faculty of Electronics, University of Niš, Serbia, in 1976, and M.Sc. and Ph.D. degrees in applied mathematics from the Faculty of Electrical Engineering, University of Belgrade, Serbia, in 1984 and 1986, respectively. He was a Professor at the Department of Computer Science, Faculty of Electronics, University of Niš, Serbia, until 2017, when he moved at the Mathematical Institute of SASA, Belgrade, Serbia, working there until 2019, when he is retired. In 1997, he was awarded by the Kyushu Institute of technology Fellowship and worked as a visiting researcher at the Department of Computer Science and Electronics, Kyushu Institute of Technology, Iizuka, Fukuoka, Japan. In 2000 he was awarded by the Nokia Professorship by Nokia, Finland. From 1999 until 2017, he worked in part at the Tampere International Center for Signal Processing, Department of Signal Processing, Faculty of Computing and Electrical Engineering, Tampere University of Technology, Tampere, Finland, first as a visiting professor and from 2009 until 2017 as an adjunct professor. His research interests include switching theory, multiple-valued logic, spectral techniques, and signal processing. He is an author of a few books, three of them, with Jaakko Astola, published by Springer, and another two with Jaakko Astola and Claudio Moraga, published by Wiley/IEEE Press, and Claypool & Morgan.

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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
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Received: 2025-08-28

Accepted: 2026-01-23

Published Online: 2026-03-10

This work is licensed under the Creative Commons Attribution 4.0 International License.

https://doi.org/10.1515/itit-2025-0026

Keywords for this article

ternary functions; rotation symmetric functions; bent functions

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