TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.4, April 201
4, pp. 3010 ~ 3
0
1
4
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i4.4773
3010
Re
cei
v
ed Se
ptem
ber 10, 2013; Revi
se
d No
vem
ber
18, 2013; Accepted Decem
ber 1, 201
3
On the Algebraic Immunity of Boolean Function
Cao Hao*, Wang Huige
Coll
eg
e of Scie
nce, Anhu
i Sci
ence a
nd T
e
chnol
og
y Un
ivers
i
t
y
, F
eng
ya
ng
233
10
0, Chin
a
*Corres
p
o
ndi
n
g
author, em
ail
:
caohao
20
00
8
54@
163.com
A
b
st
r
a
ct
In view
of the constructio
n
requ
ire
m
e
n
ts o
f
Boole
an fun
c
tions w
i
th ma
ny go
od crypt
ogra
p
h
y
prop
erties, thro
ugh
the
an
alysi
s of the
rel
a
tio
n
shi
p
b
e
tw
een
the functi
on
val
ues
on th
e v
e
c
t
ors w
i
th w
e
igh
t
not
mor
e
th
an
d
an
d th
e
alg
ebra
i
c i
mmunit
y
, a
met
hod
to
deter
mine
the
hi
gher
or
der
a
l
ge
braic
i
m
mu
nit
y
function
is giv
en. Mea
n
w
h
ile
, a meth
od th
at appr
opri
a
te
chan
ge i
n
the
functi
on v
a
lu
e
w
i
thout reduc
i
n
g
alg
ebra
i
c i
mmunity
is pr
od
u
c
ed, a
n
d
usi
n
g it, a
n
exa
m
ple
to c
onstru
c
t Bool
ea
n fu
nction
w
i
th o
p
t
ima
l
prop
erties in th
e alg
ebra
i
c i
m
mu
nity, non
lin
e
a
rity, bala
n
ce a
nd corre
latio
n
i
m
mu
nity etc is prese
n
ted.
Ke
y
w
ords
: Bo
ole
an functi
on, alg
ebra
i
c i
m
mu
nity (AI), s
uppo
rt set, corelatio
n
i
mmunity (CI)
, nonli
n
e
a
rity
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
As an i
m
po
rtant tool in th
e de
signi
ng
and a
nal
ysi
s
of crypto
syst
em, Boolea
n
function
has be
en a
resea
r
ch fo
cus i
n
crypto
grap
hy. To
resi
st varia
b
le
kn
own atta
cks,
a vari
ety o
f
cryptog
r
a
phi
c prope
rties h
a
ve be
en
put
forward,
such a
s
co
rrel
a
tion imm
unity, bala
n
cedn
ess,
nonlin
earity, etc.
In 2003,
a
ne
w cleve
r
attack
on
st
re
am ciphe
rs,
the so call
ed algeb
rai
c
atta
ck
[1], which i
s
based
on th
e
solvin
g ove
r
determi
ned
n
online
a
r
multi
v
ariable
eq
u
a
tions bet
we
en
the initial key and the outpu
ts of Key Stre
am Gene
rato
r (KSG), bri
n
gs a complet
e
ly new criteri
o
n
f
o
r
t
he de
sig
n
of
se
cur
e
st
rea
m
ciph
e
r
sy
st
em
s,
known a
s
al
g
ebrai
c i
mmu
nity (AI) [2, 3
]. To
resi
st alg
ebra
i
c attack, alg
ebrai
c imm
u
n
i
ty of
Boolean function
ca
nnot be t
oo l
o
w. He
nce, it is
very mea
n
in
gful to
con
s
t
r
uct
Boole
a
n
functio
n
s wi
th high
AI. Being b
a
sed
on th
e
stud
y of
algeb
rai
c
attacks,
schola
r
s have
alre
a
d
y pre
s
ente
d
many const
r
uction
s of Bo
olean fu
nctio
n
s
with high AI by using different
approaches [3-15]. Howe
ver, it is still a difficult problem
to
meeting vari
o
u
s othe
r goo
d cryptog
r
a
p
h
ic prope
rt
ie
s when
con
s
tructing Bool
ea
n function
s with
high AI.
In this
pap
er,
having
deepl
y studie
d
o
n
t
he relation
s
b
e
twee
n the
AI and
the Su
p
port
set
of Boolean fu
nction
s, a suf
f
icient co
ndition that
modifying several value
s
of the Boolean fun
c
t
i
on
in so
me p
o
in
t does not d
e
c
re
ase AI is
pre
s
ente
d
. Using thi
s
m
e
thod, con
s
tru
c
tion of Boole
a
n
function
s with
goo
d crypt
ogra
phi
c
p
r
o
pertie
s
,
such
that alge
braic imm
unity, balan
ce
dne
ss,
nonlin
earity a
nd co
rrelation
immunity, is pre
s
ente
d
.
2. Preliminary
A function
f
:{0,1}
n
→
{0,1
} is call
ed n-va
riable Boole
a
n
function. We
denote
B
n
the set of
all n-va
riable
Boolean fu
n
c
tion
s from
{
0
,1}
n
to {0,1}.
Denote 0
f
= {x
{0,1}
n
| f (x)=
0} a
nd 1
f
=
{x
{0,1}
n
| f (x)= 1}, which
are called off set and
sup
p
o
rt set.
Any n-variab
le Boolean
function ha
s a unique repre
s
e
n
tation
as a multivariate
polynomial ov
er
GF
(2), call
ed alge
brai
c
norm
a
l form (ANF) :
n
n
j
i
j
i
n
j
i
i
i
n
i
n
x
x
x
a
x
x
a
x
a
a
x
x
x
f
2
1
,
,
2
,
1
,
1
1
0
2
1
)
,
,
,
(
(1)
Whe
r
e th
e
co
efficients
an
d
d
enote t
he
GF
(2
) a
d
d
i
tion. The
alg
ebrai
c
degree, deg
(
f
), is the num
ber of variabl
es in the hig
hes
t order te
rm with non
ze
ro co
efficient.
In
the ANF of f(x), it is
s
a
tis
f
ied that:
)
2
(
2
1
GF
a
k
i
i
i
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
On the Algeb
raic Im
m
unity of Boolean F
unctio
n
(Cao
Hao
)
3011
)
(
}
,
,
2
,
1
{
)
sup(
,
,
2
,
1
x
f
a
ij
i
i
x
ij
i
i
(
2
)
Whe
r
e
sup
(
x) denote
s
the seri
al num
bers of 1 in x=(x
1
,x
2
,…,x
n
), i.e., sup(x
)
={i|x
i
=1}.
An impo
rtant
tool to
stud
y the crypto
grap
hic prop
erties of Boo
l
ean fu
nction
s, called
Spectrum (d
e
noted by S
f
(w)), is defin
ed
as:
n
x
wx
x
f
n
f
w
S
}
1
,
0
{
)
(
)
1
(
2
)
(
,
(w
{0,1
}
n
)
(3)
Nonli
nea
rity of Boolean function is a
n
importa
nt Cry
p
togra
ph ind
e
x, which is d
e
fined as
the minimum
distan
ce bet
wee
n
the functi
on an
d all affine functio
n
s, den
oted by N
f
. It can be
depi
cted by the equ
ation a
s
follows:
N
f
=2
n-1
(1
-m
ax
{|S
f
(w)|,w
{0,1}
n
}
(4)
An n-vari
able
Boolean fu
n
c
tion
f
is
call
ed
m
-ord
er
correlation im
mune, if for a
n
y
w
F
2
with 1
wt
(
w
)
m
, we
have
S
f
(
w
)=0, wh
ere
wt
(
w
) d
e
notes the
Ha
mming
wei
g
h
t
of
w
. Furth
e
r
more, if S
f
(
0
)=0(i.e., f is balanced), f is call
ed
m
-o
rd
er
re
silient B
oolea
n fun
c
ti
on. The
relati
on
betwe
en the numbe
r of variable
s
, alge
b
r
aic d
egr
ee a
nd co
rrel
a
tion
immunity can be describ
ed
as
follows
:
m+n
d
(5)
If f is balan
ce
d, we have m
+
n
<
d.
Let
f
(
x
),
g
(
x
)
∈
B
n
,
g
(
x
)
i
s
called an anni
hilator of
f
if
f
(
x
)·
g
(
x
)=
0
for all
x
∈
{0,1}
n
, d
enoting
An(
f
) a
s
the set of all annih
ilators of
f
. The algeb
rai
c
immunity of
f
is define
d
as f
o
llows:
AI(
f
)=min
{
g
∈
B
n
|
g
∈
Ann(
f
)
Ann(1
+
f
) a
nd
g
0 }
It is kno
w
n th
at for arbitrary
n
-variabl
e Boolean fun
c
tion
f
, we have AI(
f
)
n/2
[3]
.
To re
sist
algeb
rai
c
attack, com
b
ina
t
ion function
with high AI
sho
u
ld be
se
lected in th
e
desi
gn of KSG.
Therefore,
co
nstru
c
ting Bo
olean fun
c
tio
n
s with o
p
timal AI is nece
s
sary.
3. Judging the AI of Bo
o
l
ean Func
tio
n
Den
o
te W
<d
={x
{0,1}n|wt(x)<
d
}, W
>d
={
x
{0,1}n|
w
t(x
)
>d
}, W
=d
={x
{0,1}n|wt(x)=
d
}. For
any f
∈
B
n
, de
note
W
<
d
0
f
= {
1
,
2
,…,
s
}, W
=
d
0
f
= {
1
,
2
,…,
m
}, W
>
d
1
f
= {
1
,
2
,…,
t
}, W
=
d
1
f
=
{
1
,
2
,…,
k
}.
Cons
truc
t two matrix A=
(a
ij
)
m×s
and
B=(b
ij
)
k×t
, where a
ij
=1if and onl
y if
sup
(
j
)
su
p(
i
) , and
b
ij
=1 if and only if sup(
j
)
sup(
i
). On the AI of Boolean fun
c
tion, we hav
e
th
e
fo
llo
w
i
ng
c
o
nc
lus
i
on
：
Theorem 1
:
A and B are a
ll column full rank mat
r
ix
AI(f)
d.
Proof:
Firstly, we
only sho
w
that f
doe
s not exi
s
t no
n-zero a
nnihil
a
tor
with
de
gree
no
more than d-1.
For any g(x)
An(f) with deg(g)
d
-
1, we sh
ow tha
t
g(x)=0. Fro
m
Equation (2), it is
kno
w
n that if
x
∈
W
<
d
, then
g
(
x
)=0. It is obviou
s
that whe
n
x
∈
W
<
d
1
f
, we have:
g
(
x
)=
0
(6)
No
w, we sho
w
that the equation g(x
)
=0
still holds if
x
∈
W
<
d
0
f
.
O
w
ing
to
de
g(
g
)
d-
1
,
we
ha
ve
0
)
(
)
sup(
)
sup(
x
g
i
x
for any
i
∈
W
=
d
0
f
.
Fro
m
the Equ
a
tion (6)
and
the
definitio
n of
matrix
A, we have
0
)
(
1
j
j
i
s
j
g
a
.
Therefo
r
e, we
can
get equ
a
t
ions co
ntaini
ng
m homo
gen
e
ous li
nea
r e
q
uation
s
on va
riable
s
g(
1
),
g(
2
),…,g(
s
), and the
co
e
fficient matrix
o
f
the equ
ation
s
A is
col
u
m
n
full ran
k
.
Obviou
sly, the equ
ation
s
only ha
s a
zero
solution,
i.e.,
g
(
x
)=0
al
so h
o
lds wh
en
x
∈
W
<
d
0
f
. Hen
c
e, f doe
s n
o
t exist non
-zero anni
hilator
with de
gre
e
n
o
more than d-1.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 3010 – 3
014
3012
Nex
t, we sh
o
w
that 1
f does n
o
t exist non-ze
ro a
n
n
i
hilator with
degree no m
o
re tha
n
d-1. For
any
g(x)
An(1
f) with
de
g(g)
d-1. Denote
g'(x
)
=g(1
x)
，
it nee
ds
only p
r
ove th
at
g'(x)=0,
we
will get g(x)=0. Similar to
the prev
ious
proof m
e
thod, we will
get
g'(x)=0, hence
,
g(x)=
0
, that is to s
a
y, 1
f d
oes n
o
t exist non-ze
ro a
nni
hilator with d
egre
e
no mo
re than d-1.
Therefore,
Neither f no
r 1
f exist no
n-zero
annihil
a
tor
with
deg
re
e no
mo
re t
h
an d
-
1,
namely AI(f)
d.
4. Cons
truc
ting Boolea
n Function
s
w
i
th high AI
Selec
t
T
W
<
d
1
f
，
U
W
=
d
0
f
，
S
W
>
d
0
f
，
V
W
=
d
1
f
，
de
note (W
<
d
0
f
)
T =
{
t
1
,
t
2
,…,
tl
1
}, U={
u
1
,
u
2
,…,
ul
2
}, (W
>
d
1
f
)
S=
{
s
1
,
s
2
,…,
sl
3
}, V= {
v
1
,
v
2
,…,
vl
4
}, where
l
1
l
2
，
l
3
l
4
. De
fine two mat
r
ix M and N
a
s
follo
ws: M
=
(
m
ij
)
l
2×
l
1
and N=(
n
ij
)
l
4×
l
3
, where
m
ij
=1if
a
n
d
only if sup(
tj
)
sup
(
ui
) and
n
ij
=1if and o
n
l
y
if sup(
vj
)
sup(
si
). We have the follow c
o
nc
lus
i
on:
Theorem 2
:
Let
f
∈
B
n
with
AI(
f
)=
d. Defi
ne
h
(
x
):
otherwise
x
f
T
x
S
x
x
h
)
(
V
0
U
1
)
(
If M and N are all colum
n
full ran
k
matri
x
, Then AI(h)
d.
Proof:
Firstly, we
sh
ow th
at h d
o
e
s
not
exist n
on-ze
ro an
nihilato
r
with d
egree
n
o
mo
re
than d-1.
For a
n
y g(x
)
An(
h
) wit
h
deg
(g
)
d
-
1
,
we sho
w
th
at g(x)=0. From Equatio
n
(2), it is
kno
w
n that if
x
∈
W
<
d
, then
g
(
x
)=0. It is obviou
s
that whe
n
x
∈
W
<
d
1
h
, we have:
g
(
x
)=
0
(7)
Now, we
onl
y show that
the
equation g(x)=0
still holds if
x
∈
W
<
d
0
h
= (W
<
d
0
f
)
T
.
Owin
g to deg
(
g
)
d-
1
,
w
e
ha
ve
0
)
(
)
sup(
)
sup(
x
g
ui
x
for any
ui
∈
U
.
From th
e Equation (7
) and the
definition of
matrix M, we
have
0
)
(
1
1
tj
j
i
l
j
g
m
. There
f
ore, we
ca
n
get equ
ation
s
contai
ning
l
2
homog
ene
ou
s lin
ear eq
ua
tions
on va
ri
able
s
g(
t
1
),
g(
t
2
),…
,
g
(
tl
1
)
, and
the
co
e
fficient matrix
of
the equatio
n
s
M is colum
n
full ran
k
. Obviously, the equatio
ns o
n
l
y
has a ze
ro solutio
n
, i.e.,
g
(
x
)=0 al
so h
o
lds
whe
n
x
∈
W
<
d
0
f
. Hen
c
e, h do
es n
o
t exist non-ze
ro an
nihilato
r with deg
re
e n
o
more than d-1.
Nex
t, we
sho
w
that 1
f do
es n
o
t exist n
on-ze
ro an
nih
ilator with
de
gree
no mo
re
than d-
1. For any
g(x)
An
(
1
f) with
deg(g)
d
-
1. Den
o
te
g'
(x)=g(1
x)
，
it need
s
only prove t
hat
g'(x)=0,
we
will get g(x)=0. Similar to
the prev
ious
proof m
e
thod, we will
get
g'(x)=0, hence
,
g(x)=
0
, that is to s
a
y, 1
f d
oes n
o
t exist non-ze
ro a
nni
hilator with d
egre
e
no mo
re than d-1.
Therefore,
Neither f no
r 1
f exist no
n-zero
annihil
a
tor
with
deg
re
e no
mo
re t
h
an d
-
1,
namely AI(f)
d.
Similar to
th
e p
r
eviou
s
p
r
oof meth
od,
we
will
get
that 1
h
doe
s n
o
t exist
n
on-ze
ro
annihilato
r wi
th degre
e
no
more tha
n
d-1.
Therefore,
Neither h n
o
r 1
h exist non
-zero an
nihilat
o
r with de
gree no mo
re t
han d
-
1,
namely AI(h)
d.
Usi
ng
Th
eor
e
m 2
, we can
con
s
tru
c
t a
class of Boole
an functio
n
s
with AI no less than d
from a given
Boolean function
f
with
AI(f)=d. Fo
r example, let
f
(
x
) b
e
a n-v
a
riabl
e Majo
rity
Boolean
fun
c
tion, where n
i
s
eve
n
, we
ca
n con
s
tru
c
t B
oolea
n fun
c
ti
ons with
go
o
d
cryptog
r
ap
h
i
c
prop
ertie
s
.
5. Example of Con
s
tr
ucting Boolean
Function
s
w
i
th Good
Cr
y
p
togr
aphic Properti
e
s
In this section, we will
present a example of
constructing a 4-variable Boolean f
unctions
with goo
d cry
p
togra
phi
c propertie
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
On the Algeb
raic Im
m
unity of Boolean F
unctio
n
(Cao
Hao
)
3013
Example:
Le
t n=4
and
d
=
2 in
Th
eore
m 2
, and
sel
e
ct a
majo
rity functio
n
f(x)
rand
omly.
The fun
c
tion values a
nd spectrum of f(x)
ca
n be de
scribe
d by the followin
g
table:
Table 1. The
Functio
n
Valu
es an
d Spect
r
um of f(x)
w
0000
0001
0010
0011
0100
0101
0110
0111
f
(
w
)
1
1
1
011
00
S
f
(w
)
0
-1/4
-1/2
1/4 -1/2
-1/4
0
1/4
w 1000
1001
1010
1011
1100
1101
1110
1111
f
(
w
)
1
1
1
000
00
S
f
(
w
)
-1/4
0 -1/4
0
1
/4
0
1
/4
0
From th
e tabl
e, it coul
d e
a
s
ily get that
N
f
=4. The
Th
e N
f
i
s
hi
gh, and
i
s
clo
s
e to
Bent
function
(the
Nonli
nea
rity of 4-va
riabl
e
Bent fun
c
tion
is
6).
Acco
rding to
calcul
ations,
we
ge
t the
ANF of f(x) as
follows
:
f
(
x
)=
x
1
x
2
x
4
+
x
1
x
3
x
4
+
x
1
x
2
+
x
2
x
3
+
x
3
x
4
+1
Select
T
=
{
0
0
00,000
1}, U={001
1,110
0}, S={0
1
11,10
1
1
,1101
} a
nd
V={01
01,10
1
0
,1001
},
then we
will g
e
t the matrix M and N which ar
e ind
u
ced
by T, U, S an
d V as follows:
0
1
1
1
M
,
1
1
0
0
1
0
1
0
1
N
Becau
s
e
the
matrix M an
d
N a
r
e all
col
u
mn full ran
k
matrix, so th
e AI of the Boolea
n
function h
(
x)
deci
ded by
Theorem 2
is
optimal.
Next, we discu
ss any oth
e
r pro
p
e
r
ties of h(
x). Accordin
g to cal
c
ulatio
ns, we
get the
table whi
c
h
can pre
s
e
n
t the function val
ues a
nd sp
e
c
trum of h(x).
Table 2. The
Functio
n
Valu
es an
d Spect
r
um of h(x)
w
0000
0001
0010
0011
0100
0101
0110
0111
h
(
w
)
0
0
1
1
1001
S
h
(
w
)
0
0
0
-
1/2
0
0
1
/2
0
w
1000
1001
1010
1011
1100
1101
1110
1111
h
(
w
)
1
0
0
1
1100
S
h
(
w
)
0
0
1/2
0
0
0
0
1/2
It is easily to get that N
h
=4. From the t
able, we
can
see the
spe
c
trum of h(x) on the
vectors with weig
ht
not morn
th
an
1
are all
0,
so
h(x) i
s
a
1-re
silient Bool
ea
n functio
n
. From
Equation (5),
It is easy to get that the re
silie
n
c
y of h(x) is
opt
imal amon
g
all the nonli
n
ear
function
s. T
herefo
r
e
h(x
)
a
c
hieve
s
optimal
in
many
crypto
grap
hic p
r
op
erties,
such
as
balan
ce
dne
ss, algeb
rai
c
immunity, non
linearity, and
correlatio
n immunity.
6. Conclusio
n
We p
r
o
pose
d
a sufficien
t conditio
n
that modifyin
g seve
ral va
lues
of the
Boolean
function in some point d
oes not de
crease
AI, and pre
s
ente
d
a const
r
u
c
tion of 4-vari
able
Boolean
fun
c
tion
with
good
crypto
grap
hic p
r
o
pertie
s
,
su
ch that
alge
brai
c im
mun
i
ty,
balan
ce
dne
ss, nonline
a
rit
y
and correl
ation immuni
ty. However,
how to effectively select
the
point (a
nd th
en modify the
values
of the
s
e p
o
ints) i
s
a difficult p
r
o
b
lem. If this
probl
em
coul
d be
effectively sol
v
ed, it will b
e
meani
ngful t
o
the
co
n
s
tru
c
tion
of cryptosyste
m
with
high
se
cu
rit
y
,
and it is also the furthe
r re
search.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 3010 – 3
014
3014
Ackn
o
w
l
e
dg
ements
The p
ape
r i
s
supp
orte
d
by
NC
FS
(605
730
26); Anhui
Province Natu
ral Scien
c
e
Re
sea
r
ch Proje
c
t (KJ2
0
10B059
); Anhui Provin
ce Natural
Science Re
sea
r
ch Proje
c
t
(KJ2
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8
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)
; Anhui Provi
n
cial
Natural Scien
c
e Fo
un
dation (1
208
0
85QF1
19
).
Referen
ces
[1]
Courto
is N,
Meier W
.
Al
g
ebra
i
c attacks
on stre
am c
i
ph
ers
w
ith
li
ne
ar fee
dback.
Advanc
es i
n
Cryptol
ogy-Eur
ocrypt
. Berlin.
200
3; 265
6: 34
5 - 359.
[2]
Armkneckt F
.
Improvin
g fast alg
ebra
i
c attac
ks.
Fast Software Encryption.
Delhi. 2
0
0
4
; 3017: 65 - 8
2
.
[3]
Courtois N. Fast algebraic
attacks
on stream ciphers
w
i
th linear feedbac
k,
Advances
in
Cryptol
ogy-
Crypto
. Califor
nia. 20
03; 27
2
9
: 176-1
94.
[4]
Dala
i DK, M
a
i
t
ra S, Sarkar
S. Basic th
eor
y
in c
onstructi
on
of Boo
l
e
a
n
functio
n
s
w
i
t
h
ma
xim
u
m
possi
ble a
n
n
i
hi
lator immu
nit
y
.
Desig
n
s Co
des
and Crypto
gra
phy
. 200
6; 40(
1): 41-58.
[5]
Z
epen
g Z
,
Ji
nfeng
C, Gu
ozh
en
X,
Ha
o C.
Spec
tral
An
al
ysis of T
w
o
Bo
ole
an F
u
nctio
n
s
an
d T
hei
r
Derivativ
e
s.
Ch
ines
e Journ
a
l o
f
Electronics
. 2
011; 20(
4): 747
-749.
[6]
Meier W, Pasa
lic E, Carl
et C. Alge
braic
attacks and d
e
com
positi
on of Bo
o
l
ea
n functio
n
s.
Advanc
es i
n
Cryptol
ogy-Eur
ocrypt
. Berlin. 200
4; 302
7: 47
4-49
1.
[7]
W
e
igu
o
Z
,
Yong D, Ning D, Guozh
en
X. A Char
acter
i
zatio
n
of Algebra
i
c Immune Bo
ole
a
n
F
unctions
.
Journ
a
l of Beij
i
ng Un
iversity o
f
Posts and T
e
leco
mmu
n
icati
o
ns
. 2007; 3
0
(5)
:
55-57.
[8]
Qiang
M, Lu
Shen
g C, F
a
n
g
W
e
i F
.
C
o
n
s
truction
of Bo
ole
an F
u
nctio
n
s
w
i
th M
a
ximu
m Alge
bra
i
c
Immunit
y
.
Jo
ur
nal of Softw
are
. 2010; 21(
7): 1758-
176
7.
[9]
Xi
ao
w
e
n
X, Ai-
guo W
,
Z
h
i-ju
n Z
.
Construction of Rotatio
n
S
y
mmetric Bo
ole
an F
uncti
on
s
w
i
th Go
od
Cr
yptogr
aph
ic Properti
es.
Jou
r
nal of Electro
n
i
cs & Informati
on T
e
chn
o
l
ogy
. 2012; 34(
10): 235
8-23
62.
[10]
Yong-
jua
n
W
,
Shi-
w
u
Z
.
Con
s
truction of B
o
ole
an
fu
nctio
n
s
w
i
th optimum alg
ebra
i
c
imm
unit
y
.
Jo
urna
l
of Computer A
pplic
atio
ns
. 20
12; 32(1): 4
9
-5
1, 73.
[11]
Guang
pu G,
W
en-fen
L. T
he N
o
tes on the
Li
ne
ar
Str
u
ctures
of Ro
tation S
y
mm
etric Boo
l
e
a
n
F
unctions.
Jou
r
nal of Electro
n
i
cs & Informati
on T
e
chn
o
l
ogy
. 2012; 34(
9): 2273-
227
6.
[12]
Yongta
o
P,
W
enfeng
Q.
Cons
tructi
on
of Bal
anc
ed
Correl
a
tio
n
-Immune
F
uncti
o
n
s
w
i
t
h
H
i
g
h
e
s
t
Degr
ee.
Journ
a
l of Electron
ic
s & Informatio
n
T
e
chnol
ogy
. 2
006; 28(
12): 23
55-2
358.
[13]
Chu
and
a Q, Yi
ng-d
a
Y. Al
ge
b
r
aic D
egr
ee
of
a Cl
ass B
ool
e
an F
u
nction
A
nni
hil
a
tors.
Acta El
ectronic
a
Sinic
a
.
201
2; 40(6): 117
7-1
1
7
9
.
[14]
Hao
C, Sh
imin
W, Huige
W. C
onstructio
n
s of
od
d var
i
ab
les
s
y
mmetric
bo
ol
ean
functi
ons
w
i
t
h
sec
o
n
d
-
order corr
elati
o
n immunit
y
.
Co
mp
uter Eng
i
n
e
e
rin
g
and A
ppl
i
c
ations
. 20
12;
48(2): 83-
85.
[15]
Jiao M, Qiao
ya
n W
.
Constr
uctions of Boo
l
e
a
n
F
unctions
w
i
t
h
Optimal Alg
e
b
raic Immun
i
t
y
.
Journal o
f
Beiji
ng U
n
ivers
i
ty of Posts and T
e
leco
mmun
i
catio
n
s.
200
9; 32(4): 73-
76.
Evaluation Warning : The document was created with Spire.PDF for Python.