Proofs of two conjectures on ternary weakly regular bent functions

Abstract

We study ternary monomial functions of the form $f(x)=\Tr_n(ax^d)$, where $x\in \Ff_{3^n}$ and $\Tr_n: \Ff_{3^n}\to \Ff_3$ is the absolute trace function. Using a lemma of Hou \cite{hou}, Stickelberger's theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary monomial functions arising from \cite{hk1} are weakly regular bent, settling a conjecture of Helleseth and Kholosha \cite{hk1}. We also prove that the Coulter-Matthews bent functions are weakly regular.

Full text

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arXiv:0803.2878v1 [math.CO] 19 Mar 2008
PROOFS OF TWO CONJECTURES ON TERNARY WEAKLY REGULAR
BENT FUNCTIONS
TOR HELLESETH1, HENK D. L. HOLLMANN, ALEXANDER KHOLOSHA1,
ZEYING WANG, AND QING XIANG2
Abstract. We study ternary monomial functions of the form f(x) = Trn(axd), where x ∈ F3n
and Trn : F3n → F3 is the absolute trace function. Using a lemma of Hou [17], Stickelberger’s
theorem on Gauss sums, and certain ternary weight inequalities, we show that certain ternary
monomial functions arising from [12] are weakly regular bent, settling a conjecture of Helleseth
and Kholosha [12]. We also prove that the Coulter-Matthews bent functions are weakly regular.
1. Introduction and Summary of results
Let p be a prime, n ≥ 1 be an integer. We will use Fpn to denote the finite field of size pn,
and F∗
pn to denote the set of nonzero elements of Fpn. Let f : Fpn → Fpbe a function. The
Walsh (or Fourier) coefficient of f at b ∈ Fpn is defined by
Sf(b)=
?
x∈Fpn
ωf(x)−Trn(bx)
where Trn: Fpn → Fpis the absolute trace function, ω = e
of unity, and elements of Fpare considered as integers modulo p. In the sequel, Sa(b) is also
used to denote the Walsh transform coefficient of a function that depends on parameter a when
it is clear from the context which function we mean. The function f is said to be a p-ary bent
function (or a generalized bent function) if all its Walsh coefficients satisfy
2πi
p is a primitive complex pth root
|Sf(b)|2= pn.
A p-ary bent function f is said to be regular if for every b ∈ Fpn the normalized Walsh coefficient
p−n
f∗: Fpn → Fp. A bent function f is said to be weakly regular if there exists a complex number
u with |u| = 1 such that up−n
In such a situation, the function f∗is also a weakly regular bent function and it is called the
dual of f.
Binary bent functions are usually called Boolean bent functions, or simply bent functions.
These functions were first introduced by Rothaus [24] in 1976. Later Kumar, Scholtz, and
Welch [20] generalized the notion of a Boolean bent function to that of a p-ary bent function.
All known p-ary bent functions but one are weakly regular. The only known example of bent
but not weakly regular bent function was constructed by Helleseth and Kholosha [12].
Bent functions, and in general, p-ary bent functions are closely related to other combinatorial
and algebraic objects such as Hadamard difference sets in (F2n,+) [8], relative difference sets
[23], planar functions, and commutative semifields [6, 4, 26].
2Sf(b) is equal to a complex pth root of unity, i.e., p−n
2Sf(b) = ωf∗(b)for some function
2Sf(b) = ωf∗(b)for all b ∈ Fpn, where f∗: Fpn → Fpis a function.
For future use, we explicitly
Key words and phrases. Bent function, Gauss sum, perfect nonlinear function, planar function, Walsh trans-
form, weakly regular bent functions.
1Research supported by the Norwegian Research Council.
2Research supported in part by NSF Grant DMS 0701049.
1

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2 HELLESETH, HOLLMANN, KHOLOSHA, WANG, AND XIANG
state the relationship between planar functions and p-ary bent functions here.
F : Fpn → Fpn is said to be planar if the function from Fpn to Fpn induced by the polynomial
F(X + a) − F(X) − F(a) is bijective for every nonzero a ∈ Fpn. The following lemma gives the
relationship between planar functions and p-ary bent functions.
A function
Lemma 1.1. Let F : Fpn → Fpn be a function. Then F is planar if and only if Trn(aF(x)) is
p-ary bent for all a ∈ F∗
The proof of the lemma is fairly straightforward, see [5]. Almost all known planar functions
F : Fpn → Fpn are of Dembowski-Ostrom type, namely the corresponding polynomials F(X)
have the form F(X) =?
as follows. Let n,k ≥ 1 be integers such that gcd(k,n) = 1. Then the function F : F3n → F3n
defined by
3k+1
2 , ∀x ∈ F3n
is planar. Thus by Lemma 1.1, Trn(ax
bent functions are usually called the Coulter-Matthews bent functions. It is conjectured that
the Coulter-Matthews bent functions are weakly regular [12], [19]. (Strictly speaking, it was
only stated as an open problem in [12] to decide whether the Coulter-Matthews bent functions
are weakly regular or not. But most people believed that these functions are weakly regular
bent.) In a recent paper [19], it was proved that the Coulter-Matthews bent functions are weakly
regular in two special cases. We confirm the conjecture in this paper. Therefore our first result
in this paper is
pn.
i,jaijXpi+pj∈ Fpn[X]. The Coulter-Matthews planar functions are
special since they are not of Dembowski-Ostrom type. These planar functions can be defined
F(x) = x
3k+1
2 ) is 3-ary bent for every nonzero a ∈ F3n. These
Theorem 1.2. Let n,k ≥ 1 be integers such that gcd(k,n) = 1.
Trn(ax
3n, is weakly regular bent.
Then the bent function
3k+1
2 ), a ∈ F∗
Helleseth and Kholosha [12] surveyed all proven and conjectured classes of p-ary monomial
bent functions f : Fpn → Fpof the form f(x) = Trn(axd), where a ∈ F∗
is odd. (See Table 1 in [12].) In that paper, besides mentioning that it is an open problem to
decide whether the Coulter-Matthews bent functions are weakly regular, the authors also made
the following conjecture.
pn, d is an integer, and p
Conjecture 1.3. ([12]) Let n = 2k with k odd. Then the ternary function f mapping F3n to
F3and given by
f(x) = Trn
?
is a weakly regular bent function if a = ξ
4
and ξ is a primitive element of F3n. Moreover, for
b ∈ F3n the corresponding Walsh transform coefficient of f(x) is equal to
ax
3n−1
4
+3k+1?
3k+1
Sf(b) = −3kω
±Trk
„
b3k+1
a(I+1)
«
where I is a primitive fourth root of unity in F3n.
We will show that the ternary functions in the above conjecture are indeed weakly regular
bent. It still remains to prove the second part of the conjecture. We state our second result in
this paper as
Theorem 1.4. Let k be an odd positive integer, and let n = 2k. Then the ternary function
f : F3n → F3defined by
f(x) = Trn(ax
3n−1
4
+3k+1), ∀x ∈ F3n,

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PROOFS OF TWO CONJECTURES ON TERNARY WEAKLY REGULAR BENT FUNCTIONS3
is a weakly regular bent function if a = ξ
3k+1
4
and ξ is a primitive element of F3n.
Our proofs of Theorem 1.2 and 1.4 rely on a lemma of Hou [17]. The idea is of a p-adic nature,
and it has been used successfully a few times in the literature (see for example, [15], [9]): Given
a function f : Fpn → Fp, it is usually difficult to compute the Walsh coefficients Sf(b) explicitly;
sometimes, even computing the absolute values of Sf(b) is difficult. However, such difficulties
can sometimes be bypassed by divisibility considerations. To this end, we first introduce Gauss
sums, Stickelberger’s theorem on Gauss sums, and Hou’s lemma.
2. The Teichm¨ uller character, Gauss sums, Stickelberger’s Theorem, and Hou’s
lemma
Let p be a prime, q = pn, and n ≥ 1. Let ω = e
and let Trnbe the trace from Fqto Z/pZ. Define
2πi
p be a primitive complex pth root of unity
ψ : Fq→ C∗,ψ(x) = ωTrn(x),
which is easily seen to be a nontrivial character of the additive group of Fq. Let
χ : F∗
q→ C∗
be a character of F∗
q(the cyclic multiplicative group of Fq). We define the Gauss sum by
g(χ) =
?
a∈F∗
q
χ(a)ψ(a).
Note that if χ0is the trivial multiplicative character of Fq, then g(χ0) = −1. Gauss sums can be
viewed as the Fourier coefficients in the Fourier expansion of ψ|F∗
characters of Fq. That is, for every c ∈ F∗
1
q − 1
χ∈X
where X denotes the character group of F∗
q.
One of the elementary properties of Gauss sums is [3, Theorem 1.1.4]
qin terms of the multiplicative
q,
ψ(c) =
?
g(χ)χ−1(c), (1)
g(χ)g(χ) = q, if χ ?= χ0. (2)
A deeper result on Gauss sums is Stickelberger’s theorem (Theorem 2.1 below) on the prime
ideal factorization of Gauss sums. We first introduce some notation. Let a be any integer not
divisible by q − 1. Then there are unique integers a0,...,an−1with 0 ≤ ai≤ p − 1 for all i,
0 ≤ i ≤ n − 1 such that
a ≡ a0+ a1p + ··· + an−1pn−1(modq − 1).
We define the (p-ary) weight of a (mod q − 1), denoted by w(a), as
w(a) = a0+ a1+ ··· + an−1.
For integers a divisible by q − 1, we define w(a) = 0.
Next let ξq−1be a complex primitive (q −1)th root of unity. Fix any prime ideal p in Z[ξq−1]
lying over p. Then Z[ξq−1]/p is a finite field of order q, which we identify with Fq. Let ωpbe
the Teichm¨ uller character on Fq, i.e., an isomorphism
ωp: F∗
q→ {1,ξq−1,ξ2
q−1,...,ξq−2
q−1}
satisfying
ωp(α)(mod p) = α, (3)

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4HELLESETH, HOLLMANN, KHOLOSHA, WANG, AND XIANG
for all α in F∗
characters of Fq.
Let P be the prime ideal of Z[ξq−1,ξp] lying above p. For an integer a, let νP(g(ω−a
the P-adic valuation of g(ω−a
p ). The following classical theorem is due to Stickelberger (see [21,
p. 7], [3, p. 344]).
q. The Teichm¨ uller character ωphas order q−1; hence it generates all multiplicative
p )) denote
Theorem 2.1. Let p be a prime, and q = pn. Let a be any integer not divisible by q − 1. Then
νP(g(ω−a
p)) = w(a).
Next we state Hou’s lemma using the notation developed in this paper.
Lemma 2.2. ([17]) Let f : F3n → F3be a function. We have
(i) f is a ternary bent function if and only if ν3(Sf(b)) =n
(ii) f is a weakly regular bent function if and only if ν3(Sf(0)) =n
for all b ∈ F∗
2for all b ∈ F3n.
2and ν3(Sf(b)−Sf(0)) >n
2
3n.
3. Proofs of the Main Results
We will first prove Theorem 1.2. The proof is relatively easy since most of the work has been
done in [11].
2πi
Let F : Fpn → Fpn be a function, and ω = e
[11], the following notation was introduced:
p be a primitive complex pth root of unity. In
SF(a,b) =
?
x∈Fq
ωTrn(aF(x)+bx)
and
K = Q(ω), W+
K= {ωi|0 ≤ i ≤ p − 1}, W−
Kis the group of roots of unity in K∗. We quote the following theorem
K= {−ωi|0 ≤ i ≤ p − 1}.
Note that WK= W+
from [11].
K∪W−
Theorem 3.1. ([11]) Let q be an odd prime power. Let F be a planar function on Fq with
F(0) = 0 and F(−x) = F(x) for all x ∈ Fq. Then we have
i)
?
?
a∈F∗
q
SF(a,0)=0
a,b∈Fq
SF(a,b)=
?
a,b∈Fq
SF(a,b)2= q2
ii) For all a ∈ F∗
qand b ∈ Fq
SF(a,b) = εa,b(?p∗)n,εa,b∈ WK,
where p∗= (−1)
planar function, then
p−1
2 p. Moreover, if F is of Dembowski-Ostrom type or F is the Coulter-Matthews
εa,0∈ {±1} andεa,b· εa,0∈ W+
K.
We are ready to give the proof of Theorem 1.2.

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PROOFS OF TWO CONJECTURES ON TERNARY WEAKLY REGULAR BENT FUNCTIONS5
Proof of Theorem 1.2:
Let F : F3n → F3n be defined by F(x) = x
3k+1
2 , ∀x ∈ F3n. For any
2 ), ∀x ∈ F3n. By Lemma 2.2,
3n, ν3(Sf(b) − Sf(0)) >n
nonzero a ∈ F3n, let f : F3n → F3be defined by f(x) = Trn(ax
it suffices to show that ν3(Sf(0)) = n/2, and for every b ∈ F∗
As F is a planar function on F3n, by Theorem 3.1,
3k+1
2.
Sf(0) = SF(a,0) = εa,0(√−3)n.
Therefore ν3(Sf(0)) =n
For any b ∈ F∗
Sf(b) − Sf(0) =
2.
3n, we have
?
x∈Fq
ωTrn(aF(x)−bx)−
?
x∈Fq
ωTrn(aF(x))= SF(a,−b) − SF(a,0).
By Theorem 3.1, we have
SF(a,−b) = εa,−b(√−3)n, SF(a,0) = εa,0(√−3)n,
and
εa,0∈ {±1} andεa,−b· εa,0∈ W+
K.
Therefore,
Sf(b) − Sf(0)=(√−3)n(εa,−b− εa,0)
(√−3)nεa,0(ωj− 1),
=
where ω is a complex primitive cubic root of unity, and j ∈ {0,1,2}.
Fix any prime ideal p in Z[ξq−1] lying over 3. Let P be the prime ideal of Z[ξq−1,ω] lying
above p. Since νP(3) = 2, we see that
ν3(Sf(b) − Sf(0)) >n
2
Note that for j = 0, we have νP(ωj− 1) = ∞; and for j = 1 or 2, we have νP(ωj− 1) = 1. As
νP(√−3)n= n, we have
νP(Sf(b) − Sf(0)) = νP(ωj− 1) + νP(√−3)n> n.
Hence we have shown that ν3(Sf(b)−Sf(0)) >n
Remark 3.2. It was shown in [17] that if f : Fn
(p − 1)n ≥ 4, then
deg(f) ≤(p − 1)n
In [17], after the proof of the bound in (4), it was mentioned that when p and n are both odd
with n ≥ 3, it is not known if the bound in (4) is attainable. Let n ≥ 3 be an integer. Then the
Coulter-Matthews bent functions Trn(ax
) has degree n. Therefore by Theorem 1.2, these
functions provide examples of weakly regular bent functions of degrees attaining the bound in (4).
⇐⇒ νP(Sf(b) − Sf(0)) > n.
2. The proof of theorem is now complete.
p→ Fpis a weakly regular bent function and
2
. (4)
3n−1+1
2
We now make some preparation for the proof of Theorem 1.4. Let Ci(i = 0,1,2,3) denote
the cyclotomic classes of order four in the multiplicative group of Fpn, i.e., Ci= {ξ4t+i| t =
0,...,f − 1}, where ξ is a primitive element of Fpn and f =pn−1
expressions in the indices numbering the cyclotomic classes are taken modulo 4.
4
. Throughout this section all
Lemma 3.3. ([13]) Let p be an odd prime with p ≡ 3 (mod 4) and let n = 2k with k odd.
Raising elements of Cito the (pk+1)thpower results in apk+1
classes of order two in the multiplicative group of Fpk. Moreover, C0and C2map onto the squares
and C1and C3onto the non-squares in F∗
pk.
2
-to-1 mapping onto the cyclotomic