Generalized bent functions - sufficient conditions and related constructions

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by  S. Hodžić

Generalized bent functions - sufficient
conditions and related constructions
S. Hodˇzi´c E. Pasalic
∗ †
6
1
0
2
Abstract

n
a The necessary and sufficient conditions for a class of functions f : Zn2 Zq, where
→
J q 2 is an even positive integer, have been recently identified for q = 4 and q = 8. In
≥
9 this article we give an alternative characterization of the generalized Walsh-Hadamard
2 transformin terms ofthe Walshspectra ofthe componentBooleanfunctions of f, which

then allows us to derive sufficient conditions that f is generalized bent for any even q.
]
O The case when q is not a power of two, which has not been addressed previously, is
treated separately and a suitable representation in terms of the component functions is
C
employed. Consequently, the derivedresults leadto genericconstructionmethods ofthis
.
h class of functions. The main remaining task, which is not answered in this article, is
t
a whether the sufficient conditions are alsonecessary. There are some indications that this
m mightbe truewhichisalsoformallyconfirmedforgeneralizedbentfunctions thatbelong
[ to the class of generalized Maiorana-McFarland functions (GMMF), but still we were
unable to completely specify (in terms of necessity) gbent conditions.
1
v
Keywords: Generalized bent functions, (generalized) Walsh-Hadamard transform,
4
8 (generalized) Marioana-McFarlandclass.
0
8
0 1 Introduction
.
1
0 A generalization of Boolean functions was introduced in [2] for considering a much larger
6 class of mappings from Zn to Z . Nevertheless, due to a more natural connection to cyclic
1 q q
v: codes over rings, functions from Zn2 to Zq, where q ≥ 2 is a positive integer, have drawn even
more attention [3]. In [3], Schmidt studied the relations between generalized bent functions,
i
X constant amplitude codes and Z -linear codes (q = 4). The latter class of mappings is called
4
r
a generalized bent(gbent)functionsthroughoutthisarticle. Therearealsoothergeneralizations
of bent functions such as bent functions over finite Abelian groups for instance [9]. A nice
survey on different generalizations of bent functions can be found in [15].
Thereareseveralreasonsforstudyinggeneralizedbentfunctions. Inthefirstplacethereis
acloseconnectionoftheseobjectstostandardbentfunctionsandforinstancetherelationship
between the bent conditions imposed on the component functions of gbent functions (using
a suitable decomposition) for the quaternary and octal case were investigated in [8] and
∗University of Primorska, FAMNIT, Koper, Slovenia, e-mail: [email protected]
†University of Primorska, FAMNIT & IAM, Koper, Slovenia, e-mail: [email protected]
1
[11], respectively. Also, in many other recent works [6, 7, 10] the authors mainly consider
the bentness of the component functions for a given prescribed form of a gbent functions.
In particular, it was shown in [10] that some standard classes of bent functions such as
Mariaona-McFarland class and Dillon’s class naturally induce gbent functions. A particular
class of the functions represented as f(x) = c a(x)+c b(x) were thoroughly investigated in
1 2
terms of the imposed conditions on the coefficients c Z and the choice of the Boolean
i q
∈
functions a and b, so that f is gbent [12]. A more interesting research challenge in this
context is to propose some direct construction methods of functions from Zn to Z , which
2 q
for suitable q may give a nontrivial decomposition into standard bent functions that possibly
do not belong to the known classes of bent functions. The second reason for the interest in
these objects is a close relationship between certain objects used in the design of orthogonal
frequency-division multiplexing (OFDM) modulation technique which in certain cases suffers
from relatively high peak-to-mean envelope power ratio (PMEPR). To overcome these issues,
the q-ary sequences lying in complementary pairs [1] (also called Golay sequences) having a
low PMEPR can be easily determined from the generalized Boolean function associated with
this sequence, see [4] and the references therein.
Inthisarticle,weaddressanimportantproblemofspecifyingtheconditionsthatf :Zn
2 →
Z is gbent. In difference to the previous work [10, 11], where the sufficient and necessary
q
conditions when q = 4 and q = 8 were derived, we consider the general case of q being even
and subsequently derive some sufficient conditions for f to be gbent. We emphasize the fact
that the sufficient and necessary conditions for q = 8 were derived in a nontrivial manner
employing so-called Jacobi sums and the same technique could not be applied for larger q
of the form 2h. Nevertheless, our sufficient conditions completely coincide in this case and
thereforethey arealso necessary as well. Thatoursufficientconditions may alsobenecessary
at the same time is further supported by the fact that the GMMF class of gbent functions
essentially satisfies these conditions, see Section 4.2. The major difficulty in proving that
the sufficiency is at the same necessity as well lies is the hardness of dealing with certain
character sums.
The whole approach and the sufficient conditions derived here is based on an alternative
characterizationandcomputationofthegeneralizedWalsh-Hadamardspectralvaluesthrough
usingthestandardWalsh spectraofthecomponentBoolean functionsa whenf :Zn Z is
i 2 → q
(uniquely)representedasf(x)= a (x)+2a (x)+ +2h 1a (x). Whilethisrepresentation
0 1 − h 1
allows for a relatively easy treatment of the ca·s·e· q = 2h,−it turns out that it is not so
efficient when considering even q in the range 2h 1 < q < 2h. Even though given the input
−
and output values this representation is still unique for even 2h 1 < q < 2h, to give some
−
sufficient conditions for the gbent property in this case we were forced to consider a different
form of f which necessarily contains the coefficient q/2 in its representation. Thus, in this
case (again to avoid some difficult character sums) the function f is rather represented as
f(x)= qa(x)+a (x)+2a (x)+ +2h 2a (x) which then simplify the analysis of their
2 0 1 ··· − h−2
properties. Using these representations we derive a compact and simple formula to compute
the generalized Walsh-Hadamard spectra in terms of the spectra of the component functions
of f. Based on this formula some sufficient conditions for the gbent property are derived
which in turn gives us the possibility to specify certain generic classes of gbent functions.
2
The rest of this article is organized as follows. In Section 2, some basic definitions con-
cerning (generalized) bent functions are given. A new convenient formula for computing the
generalized Walsh-Hadamard spectra of f : Zn Z in terms of the spectral values of its
2 → q
component functions is derived in Section 3. In Section 4, a set of sufficient conditions on
the Walsh spectra of the Boolean component functions of f, ensuring that f is gbent, are
specified. It turns out that in some particular cases these conditions are also necessary, but
whether these sufficient conditions are also necessary, in general, is left as an open problem.
The problem of designing gbent functions, satisfying the set of sufficient conditions intro-
duced previously, is addressed in Section 5 one trivial method for this purpose are given.
The task of selecting the component functions, that satisfy the set of sufficient conditions,
in a non-trivial way appears to be rather difficult. Some concluding remarks are given in
Section 6.
2 Preliminaries
Wedenotethesetofintegers, realnumbersandcomplexnumbersbyZ,RandC,respectively,
and the ring of integers modulo r is denoted by Z . The vector space Zn is the space of all
r 2
n-tuples x = (x ,...,x ), where x Z with the standard operations. For x = (x ,...,x )
1 n i 2 1 n
∈
and y = (y ,...,y ) in Zn, the scalar (or inner) product over Z is defined as x y =
1 n 2 2 ·
x y x y . The same inner product of two vectors x,y Z , when defined modulo
1 1 n n q
⊕···⊕ ∈
q, will be denoted by “ ”, thus x y = x y + +x y (mod q). The addition over Z, R
1 1 n n
⊙ ⊙ ···
and C is denoted by “+”, but also the addition modulo q and it should be understood from
the context when reduction modulo q is performed. The binary addition over Z is denoted
2
by in a few cases when we use this addition. The cardinality of the set S is denoted by
⊕
S . If z = u+vi C, then z = √u2+v2 denotes the absolute value of z, and z = u vi
| | ∈ | | −
denotes the complex conjugate of z, where i2 = 1, and u,v R. We also denote u= Re(z)
− ∈
and v = Im(z).
The set of all Boolean functions in n variables, that is the mappings from Zn to Z is
2 2
denotedby . Especially, thesetofaffinefunctionsinnvariableswedefineas = g(x) =
n n
B A {
a x b a Zn, b 0,1 . The Walsh-Hadamard transform (WHT) of f at any
· ⊕ | ∈ 2 ∈ { }} ∈ Bn
point ω Zn is defined by
∈ 2
Wf(ω)= 2−n2 ( 1)f(x)⊕ω·x.
−
x Zn
X∈ 2
A function f , where n is even, is called bent if and only if W (ω) = 1 for all ω Zn,
∈ Bn | f | ∈ 2
andthese functions only exist forn even. Ifn is odd,a functionf is said tobesemibent
n
∈ B
if W (ω) 0, √2 , for every ω Z . We call a function from Zn to Z (q 2 a positive
f ∈ { ± } ∈ 2 2 q ≥
integer) a generalized Boolean function in n variables [8]. We denote the set of such functions
by n and for q = 2 the classical Boolean functions in n variables are obtained.
GBq
Let ζ = e2πi/q be a complex q-primitive root of unity. The generalized Walsh-Hadamard
transform (GWHT) of f q at any point ω Zn is the complex valued function
∈GBn ∈ 2
f(ω)= 2−n2 ζf(x)( 1)ω·x.
H −
x Zn
X∈ 2
3
A function f q is called generalized bent (gbent) function if (ω) = 1, for all ω Zn.
∈ GBn |Hf | ∈ 2
If q = 2, we obtain the (normalized) Walsh transform W of f . Two n-variable Boolean
f n
∈B
functions f,g are said to be disjoint spectra functions if W (ω)W (ω) = 0, for every
n f g
∈ B
ω Zn [13].
∈ 2
A (1, 1)-matrix H of order p is called a Hadamard matrix if HHT = pI , where HT is
p
−
the transpose of H, and I is the p pidentity matrix. A special kind of Hadamard matrix is
p
×
the Sylvester-Hadamard or Walsh-Hadamard matrix, denoted by H , which is constructed
2k
using Kronecker product H = H H , where
2k 2⊗ 2k−1
1 1 H H
H = (1); H = ; H = 2k−1 2k−1 .
1 2 1 1 2k H H
(cid:18) − (cid:19) (cid:18) 2k−1 − 2k−1 (cid:19)
For afunctiong onZn,the(1, 1)-sequence definedby(( 1)g(v0),( 1)g(v1),...,( 1)g(v2n−1))
2 − − − −
is called thesequence of g, wherev = (v ,...,v ),i = 0,1,...,2n 1, denotes thevector
i i,0 i,n 1
in Zn2 whose integer representation is i, that is, i =− jn=−01vi,j2j. We t−ake that Zn2 is ordered
as
P
(0,0,...,0),(1,0,...,0),,(0,1,...,0),...,(1,1,...,1) ,
{ }
and the vector v = (v ,...,v ) Zn is uniquely identified by i 0,1,...,2n 1 .
i i,0 i,n−1 ∈ 2 ∈ { − }
3 Motivation and Conjecture on GWHT
In this section, we recall some results related to quaternary and octal gbent functions [8, 11]
in terms of GWHT. The necessary and sufficient conditions for gbent property derived in
[8, 11] for q = 4 and q = 8 motivates us to conjecture that similar sufficient conditions
are valid for arbitrary even q, which is then proved in Section 5. Notice that proving the
necessity of these conditions turns out to be hard though there are certain indications that
the sufficient conditions given in Theorem 4.1 are also necessary.
If 2h 1 < q 2h, to any generalized function f : Zn Z , we may associate a unique
− ≤ 2 → q
sequence of Boolean functions a (i = 0,1,...,h 1) such that
i n
∈ B −
f(x)= a (x)+2a (x)+22a (x)+...+2h 1a (x), x Zn. (1)
0 1 2 − h−1 ∀ ∈ 2
The functions a (x), i = 0,1,...,h 1, are called the component functions of the function
i
−
f(x). When q = 4 it was shown that the function f(x) = a (x) + 2a (x), a ,a , is
0 1 0 1 n
∈ B
gbent if and only if a (x) and a (x) a (x) are bent Boolean functions [8]. Note that the
1 0 1
⊕
last condition implies that a (x) is not necessarily bent (it can be affine for instance), and
0
consequently only a (x) needs to be bent. In addition, the GWHT of the function f in this
1
case is expressed in terms of the WHT transforms of the functions a (x) and a (x) a (x),
1 0 1
⊕
i.e., we have
1
(u) = [(W (u)+W (u))+i(W (u) W (u))], u Zn.
Hf 2 a1 a0⊕a1 a1 − a0⊕a1 ∀ ∈ 2
However, we may rewrite this equality so that we view as a linear combination of W
Hf a1
and W , where the coefficients are complex numbers, that is,
a0 a1
⊕
1 1
(u) = +i W (u)+ i W (u). (2)
Hf 2 a1 2 − a0⊕a1
(cid:18) (cid:19) (cid:18) (cid:19)
4
In the case when q = 8, for f 8 given by
∈ GBn
f(x)= a (x)+2a (x)+22a (x), (3)
0 1 2
the GWHT of f is given by the following lemma.
Lemma 3.1 [10, 11] Let f 8 as in (3). Then,
∈ GBn
4 (u) = α W (u)+α W (u)+α W (u)+α W (u), (4)
Hf 0 a2 1 a0⊕a2 2 a1⊕a2 3 a0⊕a1⊕a2
where α = 1+(1+√2)i, α = 1+(1 √2)i, α = 1+√2 i, α = 1 √2 i.
0 1 2 3
− − − −
Remark 3.1 A special case of selecting a (x) = 0 appears to be interesting. In the first
0
place, the condition relating the Walsh coefficients becomes simpler, that is,
4 (u) = 2(1+i)W (u)+2(1 i)W (u), u Zn.
Hf a2 − a1⊕a2 ∀ ∈ 2
Then, assuming further that a (x) = 0 would actually give 4 (u) = 4W (u), meaning
1 Hf a2
that we only have one bent function and that the function f(x) = 4a (x) is gbent though
2
its codomain only takes the values from the set 0,4 . In general, any function defined as
{ }
f(x)= qa(x) is gbent if and only if a(x) is a bent function.
2
Remark 3.2 Apart form the trivial case discussed in Remark 3.1, we may also consider
other suitable choices for the component functions a ,a and a . Fixing a to be bent we may
0 1 2 2
consider a ,a to be suitably chosen affine functions so that the above conditions are
0 1 n
∈ A
satisfied. Indeed, since a being bent implies that the addition of any affine function to it does
2
not affect the bent property we can assume that a for i= 0,1. It is well-known that for
i n
∈ A
a (x) = a +a x +...+a x , if the Walsh transform of f(x) at point u is W (u) then
i i,0 i,1 1 i,n n f
the transform of f(x)+ai(x) at point u is (−1)ai,0Wf(u+a(i)), where a(i) ∈ Zn2 is given as
a(i) = (a ,...,a ). Hence, (4) can be rewritten as,
i,1 i,n
4 (u) =α W (u)+α ( 1)a0,0W (u+a(0))+α W (u)+α ( 1)a0,0W (u+a(0)).
Hf 0 a2 1 − a2 2 a1⊕a2 3 − a1⊕a2
Notice that in (4) is again a linear combination of the WHTs of the functions a (x),
f 2
H
a (x) a (x), a (x) a (x), a (x) a (x) a (x). Moreover, the following theorem imposes
0 2 1 2 0 1 2
⊕ ⊕ ⊕ ⊕
the conditions for the function f 8 to be a gbent function.
∈GBn
Theorem 3.1 [10] Let f 8 as in (3). Then:
∈GBn
1) If n is even, then f is generalized bent if and only if a , a a , a a , a a a are
2 0 2 1 2 0 1 2
⊕ ⊕ ⊕ ⊕
all bent, and
( ) W (u)W (u) = W (u)W (u), for all u Zn;
∗ a0⊕a2 a1⊕a2 a2 a0⊕a1⊕a2 ∈ 2
2) If n is odd, then f is generalized bent if and only if a , a a , a a , a a a are
2 0 2 1 2 0 1 2
⊕ ⊕ ⊕ ⊕
semi-bent satisfying
( ) : W (u) = W (u) = 0 W (u) = W (u) = √2; or
∗∗ a0⊕a2 a2 ∧ | a1⊕a2 | | a0⊕a1⊕a2 |
W (u) = W (u) = 0 W (u) = W (u) = √2,
a1⊕a2 a0⊕a1⊕a2 ∧ | a0⊕a2 | | a2 |
for all u Zn.
∈ 2
5
In general, a formula which gives the GWHT of the function f given by (1) is given by the
following theorem.
Theorem 3.2 [10, 11] The Walsh-Hadamard transform of f : Zn Z , 2h 1 < q 2h,
2 → q − ≤
where f(x)= hi=−01ai(x)2i, ai ∈ Bn is given by
HfP(u) = 2−h ζPi∈I2i (−1)|J|WPt∈J∪Kat(x)(u). (5)
I⊆{0X,...,h−1} J⊆XI,K⊆I
This implicit expression does not reveal the fact that of a function f represented as in
f
H
(1) can be given explicitly as a linear combination (with complex coefficients that can be
efficiently computed) of the WHTs of some linear combinations of its component functions
a (x), i= 0,1,...,h 1. Therefore, for an arbitrary generalized Boolean function f given by
i
−
(1), it is of great importance to develop a more useful formula for its GWHT which will be
given in the next section.
Before we state our conjecture regarding the GWHT and the conditions ( )-( ) in gen-
∗ ∗∗
eral, we first formalize our observations. Let Θ (x) be the function defined as
i
Θi(x)= ( 1)zi,0a0(x)⊕zi,1a1(x)⊕...⊕zi,h−1ah−1(x), (6)
−
wherez = (z ,z ,...,z ) Zhandidenotesitsintegerrepresentation,i = 0,...,2h 1.
i i,0 i,1 i,h−1 ∈ 2 −
Remark 3.3 Note that the function Θ (x) actually gives ( 1) powered to all possible linear
i
−
combinations of the component functions a (x),a (x),...,a (x). In addition, we always
0 1 h 1
have ζ2qah−1(x) = ( 1)ah−1(x) for q = 2h. −
−
Forq = 8 = 23,thush = 3,letusconsiderf :Zn Z givenby(3). Sinceζ4a2(x) = ( 1)a2(x),
2 → 8 −
the GWHT is given as:
(u) = ζf(x)( 1)ux = ζa0(x)+2a1(x)( 1)a2(x) ux. (7)
f · ⊕ ·
H − −
x Zn x Zn
X∈ 2 X∈ 2
Hence, for q = 8 we have z = (z ,z ) Z2, Θ (x) = ( 1)z0a0(x) z1a1(x), where
0 1 ∈ 2 z − ⊕
Θ (x) = Θ (x) = 1,
0 (0,0)
Θ (x) = Θ (x) = ( 1)a0(x),
1 (1,0)
−
Θ (x) = Θ (x) = ( 1)a1(x), (8)
2 (0,1)
−
Θ (x) = Θ (x) = ( 1)a0 a1(x),
3 (1,1) ⊕
−
and
ζa0(x)+2a1(x) = 2 2(α Θ (x)+α Θ (x)+α Θ (x)+α Θ (x)),
− 0 0 1 1 2 2 3 3
where α are given in Lemma 3.1. Regarding the GWHT of the function f given by (1), we
i
propose the following conjecture.
6
Conjecture 1 Let f q and Θ (x) be given by (1) and (6), respectively. Then ζf(x)
∈ GBn i
can be represented as a linear combination of functions Θ (x), i = 0,1,...,2h 1, where the
i
−
coefficients α are complex numbers, i.e.,
i
2h 1
ζf(x) = ζPih=−01ai(x)2i = − αiΘi(x). (9)
i=0
X
Furthermore, for a given f q the coefficients α can be computed efficiently.
∈GBn i
Note that Conjecture 1 covers all the values of even q in the range q (2h 1,2h]. Clearly,
−
∈
in the case when q = 8 = 2h (similarly when q = 4), we had that ζa0(x)+2a1(x)+4a2(x) =
( 1)a2(x)ζa0(x)+2a1(x), and consequently we represent only ζa0(x)+2a1(x) as a linear combina-
−
tion of functions Θ (x), i = 0,1,2,3. This representation is proved useful later for deriving
i
sufficient conditions of gbent property and for generalizing Theorem 3.1 but covering all
values of q, where q is even.
3.1 New GWHT formula
In this section, we prove Conjecture 1 and consequently a new GWHT formula for any
generalized function f q, which computes by using the Walsh spectral values of the
∈ GBn Hf
component functions and the coefficients α , is derived.
i
Let f :Zn Z , 2h 1 < q 2h, where again f(x)= a (x)+2a (x)+...+2h 1a (x),
a (x) . 2Fo→r coqnven−ience, w≤e introduce the coefficients0c = 2i,1for i = 0,...,h− h1−,1thus
i n i
writin∈g Bf(x) = hi=−01ciai(x). Notice that whatever formal representation of f is u−sed (see
also Example 3.1), once the function f has been specified in terms of its input and output
P
values,thedecompositionintotheBooleancomponentfunctiona (x)asgivenaboveisunique
i
and any other representation can be transformed into this form.
Assume now that the function f can be represented as a linear combination of the func-
tions Θ (x) as in (9), that is,
i
2h 1
ζf(x) = ζPih=−01ciai(x) = − αiΘi(x), (10)
i=0
X
for some complex numbers αi C and Θi(x) = ( 1)zi,0a0(x)⊕···⊕zi,h−1ah−1(x), as given by (6).
∈ −
The main task is to find the coefficients α such that (10) holds for every x Zn.
i ∈ 2
Consider an arbitrary but fixed x Zn such that (a (x),...,a (x)) = z Zh, where
′ ∈ 2 0 ′ h−1 ′ k ∈ 2
k is the integer representation of a binary vector z . To relate the functions Θ to the rows
k i
(columns) of the Hadamard matrix we need the following useful identification. It is well-
known that the rows of the Hadamard matrix H of size 2h 2h are the evaluations of
2h ×
all linear functions in , that is, the k-th row of H (alternatively the k-th column since
Bh 2h
H = HT ) can be expressed as H(k) = ( 1)zk y y Zh , where z is fixed. Therefore,
2h 2h 2h { − · | ∈ 2} k
(k)
(Θ (x),Θ (x),...,Θ (x)) = H .
0 ′ 1 ′ 2h 1 ′ 2h
−
7
Indeed, for a fixed x Zn the value of a binary vector (a (x),...,a (x)) = z is also
′ ∈ 2 0 ′ h−1 ′ k
fixed and it is easy to verify that,
(Θ0(x′),Θ1(x′),...,Θ2h−1(x′)) = ((−1)zk·z0,(−1)zk·z1,...,(−1)zk·z2h−1) = H2(kh),
where z ,z ,...,z are elements of the set Zn. Furthermore, for this particular (but arbi-
0 1 2h 1 2
trary) value x the f−act that (a (x),...,a (x)) = z implies that
′ 0 ′ h 1 ′ k
−
ζf(x′) = ζPih=−01ciai(x′) = ζzk⊙(c0,...,ch−1). (11)
Now, if we define the column matrix Λ = [α ]2h 1 to be a matrix of the coefficients α , the
i i=−0 i
previous discussion together with (10) implies that
α
0
α
H2(kh) ...1  = H2(kh)Λ= ζzk⊙(c0,...,ch−1).
 
 α 
 2h 1 2h 1
 −  ×
Notice that when z goes through Zh the value z (c ,...,c ) goes through Z , since
k 2 k ⊙ 0 h−1 q
the operation means cutting by modulo q. Therefore, it is convenient to define a column
matrix B as a⊙matrix of all corresponding powers of ζ, that is, B = [ζzi⊙(c0,...,ch−1)]2i=h−01 or
given in the matrix form as,
ζ0
ζc0
B =  . . (12)
.
.
 
 ζc0+···+c2h−1 
 
 
and obviously assuming (10) is valid the following system of equations must be satisfied
H Λ= B. (13)
2h
As mentioned previously, the function f q may be given in different forms, for
instance f(x)= d c b (x), where b bu∈t cGBnZ and in general c = 2i. Nevertheless,
i=0 i i i ∈Bn i ∈ q i 6
one can easily transform such a function into the form discussed above. Note that the
P
solution Λ of the system (13) implies that the equality (10) holds for any x Zn. The main
∈ 2
reason for this is the fact that the Hadamard matrix covers all possible values of the vector
(Θ (x),Θ (x),...,Θ (x)). Therefore, for any x Zn the evaluation of the component
0 1 2h−1 ∈ 2
functions (a (x),...,a (x)) implies that the corresponding Hadamard row multiplied with
0 h 1
−
Λ will always be equal to the corresponding power of ζ.
Since the determinant of the Sylvester-Hadamard matrix is given as det(H ) = 2h2h−1,
2h ±
using the fact that H 1 = 2 hHT (H is symmetric), we have that the unknown column
2−h − 2h 2h
matrix Λ= [α ]2h 1 is (uniquely) given by
i i=−0
Λ = H 1B = 2 hHT B = 2 hH B. (14)
2−h − 2h − 2h
In the following example, we illustrate a complete procedure of finding α with respect to
i
both discussed representations of the function f(x).
8
Example 3.1 Let us consider f(x) = 2b (x)+3b (x), for q = 6. Since 22 < q 23 then
0 1
≤
h =3, and f(x) can be rewritten in the form (1) as
f(x) = b (x)+2(b (x)+b (x))+4 0,
1 0 1
·
where we now identify a (x) = b (x), a (x) = b (x)+ b (x), and a (x) = 0. Considering
0 1 1 0 1 2
the system H Λ = B, where B = [ζk]23 1, we have that the matrix Λ = [α ]23 1 is given as
23 k=−0 i i=−0
Λ= 2 3H B, i.e.,
− 23
3 +i√3
α 2 2
0 1 i√3
α1  2 − 2 
 α  3 i3√3
2  2 − 2 
 α   3 i√3 
Λ=  α34  = 2−3 −32+−i3√23 
   −2 2 
 α5   3 +i√3 
 α6   92+i3√23 
 α7   23 i3√23 
   2 − 2 
 
In addition, from Θi(x) = (−1)zi,0a0(x)⊕zi,1a1(x)⊕zi,3·0, zi ∈ Z32 we have:
Θ (x) = Θ (x) = 1, Θ (x) = Θ (x)= ( 1)a0(x),
0 4 1 5
−
Θ (x) = Θ (x) = ( 1)a1(x), Θ (x) = Θ (x) = ( 1)a0(x) a1(x).
2 6 3 7 ⊕
− −
Hence, we have the following calculation:
23 1
−
ζf(x) = α Θ (x)= (α +α )Θ (x)+(α +α )Θ (x)+(α +α )Θ (x)+(α +α )Θ (x)
i i 0 4 0 1 5 1 2 6 2 3 7 2
i=0
X
= 2 3(2√3iΘ (x)+2Θ (x)+6Θ (x)+( 2√3i)Θ (x))
− 0 1 2 2
−
= 2 3(2√3i+2( 1)a0(x)+6( 1)a1(x) 2√3i( 1)a0(x) a1(x)). (15)
− ⊕
− − − −
Since a (x) = b (x) and a (x) = b (x) + b (x), for all values of the component functions
0 1 1 0 1
b (x) and b (x) we have that ζf(x) takes the following values:
0 1
ζf(x) = 2 3(2√3i+2( 1)a0(x)+6( 1)a1(x) 2√3i( 1)a0(x) a1(x))
− ⊕
− − − −
1, (b (x),b (x)) = (0,0)
0 1
ζ2 = 1 +i√3, (b (x),b (x)) = (1,0)
=  −2 2 0 1 .
 ζ3 = 1, (b (x),b (x)) = (0,1)
 0 1
 ζ5 = 1 −i√3, (b (x),b (x)) = (1,1)
2 − 2 0 1


Hence, from (14) we have α = 2 hH(i)B, for i = 0,...,2h 1, and together with (10) we
i − 2h −
have that the GWHT is given as
2h 1 2h 1
− −
(u) = ζf(x)( 1)ux = ( 1)ux α Θ (x) = α W (u), u Zn, (16)
Hf − · − · i i i i ∀ ∈ 2
xX∈Zn2 xX∈Zn2 (cid:16) Xi=0 (cid:17) Xi=0
9
where
Wi(u) = Θi(x)( 1)u·x = ( 1)zi,0a0(x)⊕···⊕zi,h−1ah−1(x)⊕u·x, (17)
− −
x Zn x Zn
X∈ 2 X∈ 2
i.e., W (u) is the WHT of the function z a (x) z a (x) at point u Zn, where
z = (zi ,...,z ) Zh, i= 0,...,2hi,010. Now⊕·w·e·⊕stati,eh−t1hehm−1ain result of thi∈s se2ction.
i i,0 i,h−1 ∈ 2 −
Theorem 3.3 Let f : Zn Z , 2h 1 < q 2h, where f(x) is given by (1). Let the
2 → q − ≤
function Θ (x) be defined by (6), and let W (u) denote the WHT of the Boolean function
i i
z a (x) z a (x) at point u Zn as in (17), for i= 0,...,2h 1. Then:
i,0 0 ⊕···⊕ i,h−1 h−1 ∈ 2 −
1. ζf(x) can be represented as a linear combination of the functions Θ (x),
i
2h 1
ζf(x) = ζPih=−01ciai(x) = − αiΘi(x),
i=0
X
where α are given by
i
α = 2 hH(i)B,
i − 2h
and the matrix B is given by (12).
2. Consequently, (u) can be represented as a linear combination of W (u), i.e.,
f i
H
2h 1
−
(u) = α W (u), u Zn. (18)
Hf i i ∀ ∈ 2
i=0
X
For instance, Lemma3.1 is an easy corollary of theabove resultas illustrated in thefollowing
example.
Example 3.2 Let q = 8 = 2h, thus h = 3, and consider an arbitrary function f q given
∈ GBn
by f(x) = a (x)+2a (x)+4a (x), f : Zn Z . Then, the GWHT of f at some arbitrary
0 1 2 2 → 2
point u Zn is given by
∈ 2
f(u) = ( 1)ah−1(x)⊕u·xζPih=−02ai(x)2i = ( 1)a2(x)⊕u·xζa0(x)+2a1(x).
H − −
x Zn x Zn
X∈ 2 X∈ 2
Now we would like to represent ζa0(x)+2a1(x) as a linear combination of functions Θ (x) = 1,
0
Θ (x) = ( 1)a0(x), Θ (x) = ( 1)a1(x) and Θ (x) = ( 1)a0(x)+a1(x), i.e.,
1 2 3
− − −
ζa0(x)+2a1(x) = α Θ (x)+α Θ (x)+α Θ (x)+α Θ (x),
0 0 1 1 2 2 3 3
where the coefficients α C, i = 0,1,2,3. For such coefficients, all of the following equalities
i
∈
must be true:
1 =α +α +α +α , if (a (x),a (x)) = (0,0)
0 1 2 3 0 ′ 1 ′
ζ1 = α α +α +α , if (a (x),a (x)) = (1,0)
ζa0(x)+2a1(x) =  0− 1 2 3 0 ′ 1 ′ ,
ζ2 = α +α α +α , if (a (x),a (x)) = (0,1)
 ζ3 = α0 α1−α2+α3, if (a0(x′),a1(x′)) = (1,1)
0 1 2 3 0 ′ 1 ′
− −



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