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Citation: Gao, Z.; Feng, J.-e. Research
Status of Nonlinear Feedback Shift
Register Based on Semi-Tensor
Product. Mathematics 2022,10, 3538.
https://doi.org/10.3390/
math10193538
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Accepted: 26 September 2022
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mathematics
Review
Research Status of Nonlinear Feedback Shift Register Based on
Semi-Tensor Product
Zhe Gao †and Jun-e Feng ∗,†
School of Mathematics, Shandong University, Jinan 250100, China
*Correspondence: fengjune@sdu.edu.cn
† These authors contributed equally to this work.
Abstract:
Nonlinear feedback shift registers (NFSRs) are the main components of stream ciphers
and convolutional decoders. Recent years have seen an increase in the requirement for information
security, which has sparked NFSR research. However, the NFSR study is very imperfect as a result
of the lack of appropriate mathematical tools. Many scholars have discovered in recent years that
the introduction of semi-tensor products (STP) of matrices can overcome this issue because STP can
convert the NFSR into a quasi-linear form. As a result of STP, new NFSR research has emerged from
a different angle. In view of this, in order to generalize the latest achievements of NFSRs based on
STP and provide some directions for future development, the research results are summarized and
sorted out, broadly including the modeling of NFSRs, the analysis of the structure of NFSRs, and the
study of the properties of NFSRs.
Keywords:
semi-tensor product; nonlinear feedback shift register; equivalence; nonsingularity;
stability
MSC: 93C10
1. Introduction
As the information age has progressed, the nation, society, and people have become
more concerned with information security, and cryptography has emerged as a key tool for
doing so. Symmetric and asymmetric cryptosystems are the two basic types of cryptography
used today. Block ciphers and stream ciphers are examples of symmetric cryptosystems.
Due to its benefits for quick encryption and decryption, easy hardware implementation,
minimal error propagation, and simple application protocol, stream cipher is frequently
employed in vital departments and various mobile communication systems for confidential
communication. The feedback shift register (FSR) is a primary structural component of
stream cipher.
FSR, as a finite state automaton that can be represented by difference equations of a
set of Boolean functions, in addition to its application in stream ciphers, is also widely used
in communication systems, digital circuits and other fields [
1
–
3
]. The research of FSRs has
a history of more than 50 years. In 1967, the famous scholar Golomb [
4
] introduced the
fundamental characteristics of FSRs and the generation mechanism of the shift sequences,
and these two concepts serve as the foundation for FSR study today. Different from linear
FSR (LFSR), nonlinear FSR (NFSR) has received more attention due to its higher level of
security. In stream ciphers, NFSR, as a generator of pseudo-random sequences, has been
researched by numerous scholars. Especially after the European eSTREAM project, many
NFSR-based stream cipher algorithms emerged, such as the hardware finalists Grain [
5
],
Trivium [
6
], and Mickey [
7
]. In convolutional codes, NFSR is the main building block of
decoding algorithms such as threshold decoding [
8
,
9
]. In the threshold decoding algorithm,
the syndrome sequence formed by the received information symbols and the received
Mathematics 2022,10, 3538. https://doi.org/10.3390/math10193538 https://www.mdpi.com/journal/mathematics
Mathematics 2022,10, 3538 2 of 14
supervisory symbols is used as the input of the NFSR, and the output of the NFSR is used
as the error estimation, so that the correct information symbols can be obtained.
Over the years, NFSRs have received extensive attention due to their wide range of
application scenarios. According to the structure, NFSRs can be divided into Fibonacci
NFSRs [
10
,
11
], Galois NFSRs [
11
,
12
], Grain-like cascade NFSRs [
13
,
14
], Trivium-like cascade
NFSRs [
15
,
16
] and so on. According to whether there is input, NFSRs can be divided into
autonomous NFSRs and non-autonomous NFSRs. According to the research problems,
in addition to stream cipher algorithm designs [
17
–
21
] and attacks [
22
,
23
], the theoretical
study can be summarized into three parts: (1) equivalence and decomposition [
24
–
26
],
irreducible NFSRs [
27
,
28
] and affine sub-families [
14
,
29
]; (2) nonsingularity [
10
,
30
,
31
]; (3)
period [
32
–
34
] and de Bruijn sequences [
35
–
38
] and so on. However, due to the lack of
suitable mathematical means, the research on NFSR still has a long way to go.
In order to break the limitation of the dimension of the traditional matrix product,
Cheng et al. [
39
,
40
] proposed a new matrix product—the semi-tensor product (STP), which
allows the multiplication of two matrices of any dimension. There are numerous significant
applications in this field, including the original Boolean network (BN) [
41
–
47
], game the-
ory [
48
–
50
], multi-agent synchronization and queue control [
51
,
52
], finite automata [
53
–
57
],
fault diagnosis and digital circuit design [
58
,
59
], and even network query and teleoper-
ation [
60
], internal combustion engine [
61
], intelligent Home [
62
] and other engineering
problems [
63
]. STP helps to solve many challenging issues in various fields. Professor
Guo [
64
] of the Chinese Academy of Sciences gave a high evaluation of STP, saying that
“the STP may become one of the new mathematical tools called for in the computer age
to realize the purpose of discovering new phenomena and solving new problems based
on calculation”.
Fortunately, in recent years, many scholars have discovered that STP can also be used
to model and study NFSRs fairly efficiently. Because NFSRs and BNs are both composed
of finite Boolean functions and finite states, this motivates many scholars to use STP
to model NFSR into a quasi-linear form similar to BN, and this quasi-linear form will
help to simplify many research problems. From this idea, many significant results were
born. These results are more novel than previous conclusions and are very helpful for the
design of practical stream cipher algorithms and decoding algorithms. According to the
specific research object, the main achievements can be roughly divided into the modeling
of NFSRs [
65
–
67
], the analysis of the structure of NFSRs [
68
–
81
], and the study of the
properties of NFSRs
[82–97]
. This paper will give an overview of the latest achievements in
NFSR research based on STP from these three aspects.
The content of this paper is organized as follows: First, the definition and properties
of STP are introduced, and several types of NFSR are briefly introduced. Then, the latest
results in NFSR research based on STP are introduced, mainly including modeling problems,
structural problems, properties and relevant criteria. Finally, a prospect and a summary of
this paper are given.
2. Preliminaries
This section provides some notations, the definition and some properties of the STP.
First of all, some notations used in this paper are listed as follows:
•Rm×n: set of real matrices of dimension m×n.
•F2: Galois field of two elements.
•Fn
2: set of all n-dimension vectors over F2.
•In: identity matrix of dimension n.
•δi
n: the ith column of the identity matrix In.
•∆2:={δi
2|i=1, 2}.
•∆n
2: set of all n-dimension vectors over ∆p.
•Ln×r
: set of
n×r
logical matrices satisfying
Col(L)⊆∆n
. For
L∈ Ln×r
, write
L= [δi1
nδi2
n. . . δir
n]or L=δn[i1i2. . . ir]for simplicity.
•Coli(M): the i-th column of matrix M.
Mathematics 2022,10, 3538 3 of 14
•Rowi(M): the i-th row of matrix M.
•M(i,j): the element on the i-th row, j-th column of matrix M.
•dse: the largest integer less than s.
•|M|: the determinant of matrix M.
•ord(M): the order of matrix M.
•W[m,n]
: swap matrix, where
W[m,n]= [δ1
nδ1
m
,
δ2
nδ1
m
,
· · ·
,
δn
nδ1
m
,
· · ·
,
δ1
nδm
m
,
δ2
nδm
m
,
· · ·
,
δn
nδm
m]
.
Definition 1
([
98
])
.
Let
A∈Rm×n
,
B∈Rp×q
, the Kronecker product of matrices
A
and
B
is
defined as
A⊗B=
A(1,1)B A(1,2)B. . . A(1,n)B
A(2,1)B A(2,2)B. . . A(2,n)B
.
.
..
.
.....
.
.
A(m,1)B A(m,2)B. . . A(m,n)B
.
Definition 2
([
99
])
.
Let
A∈Rm×n
and
B∈Rp×n
. The Khatri–Rao product of matrices
A
and
B
is defined as an mp ×n matrix, given by
A∗B= [Col1(A)⊗Col1(B)Col2(A)⊗Col2(B). . . Coln(A)⊗Coln(B)]. (1)
Definition 3 ([39]).Let A ∈Rm×n, B ∈Rp×q. The STP of matrices A and B is defined as
AnB:= (A⊗Is/n)(B⊗Is/p), (2)
where s is the least common multiple of n and p.
Lemma 1
([
39
])
.
Any logical function
y=f(x1
,
x2
,
. . .
,
xn)
with Boolean variables
xi∈∆2
,
i=1, 2, . . . , n, can be expressed as a multi-linear form as
y=f(x1,x2, . . . , xn) = Mfx1x2. . . xn, (3)
where y ∈∆2, and M f∈ L2×2nis unique, called the structural matrix of f .
3. Research Status of NFSRs Based on STP
The invention of the STP has brought new vitality into the research of NFSRs. This
section will focus on the research status on applying STP to NFSRs .
3.1. Modeling Problems for NFSRs
This section will introduce several types of NFSRs and their algebraic forms established
by STP. Identify 1 and 0 as
δ1
2∈∆2
and
δ2
2∈∆2
, respectively. In the sequel, we say a variable
X∈F2
is in a scalar form, and call the corresponding variable
x∈∆2
is in a vector form.
Without loss of generality, we usually omit “n" in the following for simplicity.
3.1.1. Fibonacci NFSRs
Fibonacci NFSR is often adopted to design stream ciphers [
19
,
21
]. An
n
-stage Fibonacci
NFSR can be expressed as:
X1(t+1) = X2(t),
X2(t+1) = X3(t),
.
.
.
Xn−1(t+1) = Xn(t),
Xn(t+1) = f(X1(t),X2(t), . . . , Xn(t)).
(4)
Mathematics 2022,10, 3538 4 of 14
where the content of bit
i
is denoted as
Xi(t)
,
i=
1, 2,
. . .
,
n
. The state of the NFSR at time
t
is denoted by
X(t) = [X1(t)X2(t). . . Xn(t)]
, and
f
is a nonlinear feedback function.
Figure 1represents an n-stage Fibonacci NFSR.
Figure 1. An n-stage Fibonacci NFSR.
Using Lemma 1and the vector forms, for
i=
1, 2,
. . .
,
n
, there exists a structural matrix
Li
such that
xi=Lix(t)
, where
x(t) = nn
k=1xk(t)∈∆2n
. Then, an equivalently linear form
of Fibonacci NFSR [65] an be expressed as:
x(t+1) = Lx(t), (5)
where
x(t) = nn
k=1xk(t)∈∆2n
, and
L=L1∗L2∗ · · · ∗ Ln∈ L2n×2n
is the state transition
matrix of this NFSR.
More accurately, assume
[s1
,
s2
,
. . .
,
s2n]
to be the truth table of
f
arranged in reverse
alphabetical order and L=δ2n[η1. . . η2n−1η2n−1+1. . . η2n], then [66]
(ηi=2i−si,
η2n−1+i=2i−s2n−1+i
(6)
for all i=1, 2, . . . , 2n−1.
3.1.2. Galois NFSRs
Compared to Fibonacci NFSRs, Galois NFSRs may decrease the propagation time
and increase the throughput [
25
], so that they are employed in several stream cipher
designs [100]. An n-stage Galois NFSR can be expressed as:
X1(t+1) = f1(X1(t),X2(t), . . . , Xn(t)),
X2(t+1) = f2(X1(t),X2(t), . . . , Xn(t)),
.
.
.
Xn(t+1) = fn(X1(t),X2(t), . . . , Xn(t)).
(7)
where fiare nonlinear feedback functions, i=1, 2, . . . , n.
The Galois NFSR can also be equivalently expressed as a linear form using STP:
x(t+1) = Lx(t), (8)
where
x(t) = nn
i=1xi(t)∈∆2n
, and
L∈ L2n×2n
is called the state transition matrix of the
Galois NFSR. Figure 2represents an n-stage Galois NFSR.
Mathematics 2022,10, 3538 5 of 14
Figure 2. An n-stage Galois NFSR.
3.1.3. Grain-Like Cascade NFSRs
In a Grain-like cascade FSR, one LFSR is used to control another NFSR (see Figure 3).
Let
[X1(t). . . Xn(t)Y1(t). . . Ym(t)]
represents the state of the Grain-like cascade NFSR.
Then they have the following relation:
Y1(t+1) = Y2(t),
.
.
.
Ym−1(t) = Ym(t),
Ym(t+1) = g(Y1(t), . . . , Ym(t)),
X1(t+1) = X2(t),
.
.
.
Xn−1(t+1) = Xn(t),
Xn(t+1) = f(X1(t),X2(t), . . . , Xn(t)) ⊕Y1(t).
(9)
Using Lemma 1, the multi-linear form of (9) can be obtained [86]:
(y(t+1) = L1y(t),
x(t+1) = L2y1(t)x(t).(10)
Figure 3represents an
n
-stage Fibonacci NFSR. In general, a Grain-like cascade NFSR
can be considered as an NFSR with input, where the output of the LFSR is used as input of
the NFSR. From this idea, the algebraic form (9) can be reduced to
x(t+1) = Lux(t)u(t), (11)
where
x(t)∈∆2n
is the state at time instant
t
,
u(t)∈∆2
is the input at time instant
t
. More
precisely, assume
[s1
,
s2
,
. . .
,
s2n+1]
to be the truth table of
f
arranged in the reverse alphabet
order and L=δ2n[η1. . . η2nη2n+1. . . η2n+1], then [68]
(ηi=2die − si,
η2n+i=2die − s2n+i
(12)
for all i=1, 2, . . . , 2n.
Mathematics 2022,10, 3538 6 of 14
Figure 3. An m+n-stage Grain-like cascade NFSR.
3.1.4. Multi-Valued NFSRs
It can be seen that the NFSRs introduced above are all binary, but in practical applica-
tions, in order to consider software implementation, some studies are based on multi-valued
NFSRs. Consider a
k
-valued Fibonacci NFSR, whose form is the same as (4), except that
the value of
Xi
is from 0 to
k−
1 instead of {0,1}. Denote
L=δkn[η1η2. . . ηkn]
be the state
transition matrix of this NFSR, then [67]
ηm={[(m−1)mod kn−1] + 1}k−sm,m=1, 2, . . . , kn. (13)
Similarly, other types of NFSRs can also be correspondingly extended to multi-valued
NFSRs, which will not be repeated here.
3.2. Structural Problems of NFSRs
This section will introduce some researches about structural problems of NFSRs that
are studied by applying STP, including the equivalence, isomorphism, decomposition and
period of NFSRs. In-depth study of the structural problems of NFSR can aid in the creation
of stream cipher algorithms that are both more effective and safe.
3.2.1. The Equivalence Transition between Galois NFSRs and Fibonacci NFSRs
In Section 3.1, we introduced the structure of Galois NFSRs and Fibonacci NFSRs.
Since both types have their own advantages and disadvantages, such as the period of
output sequences in Fibonacci NFSR being equal to that of state sequences and Galois
NFSR has higher speed of output sequences generation [
25
], it is necessary to study the
equivalence of the two. If a Galois NFSR is euivalent to a Fibonacci NFSR, then this special
Galois NFSR can have the advantages of Fibonacci NFSR in addition to its own advantages.
(1) Equivalence
Definition 4
([
101
])
.
Two NFSRs are equivalent if the sets of their output sequences are the same.
There are several studies on equivalence condition of Galois NFSR and Fibonacci NFSR
using STP [
74
] was disclosed therein that if a Galois NFSR is equivalent to a Fibonacci
NFSR, then its stage number is no less than that of the Fibonacci NFSR. The number of
n
-stage Galois NFSRs that are equivalent to a given
n
-stage Fibonacci NFSR is
(
2
n−1!)2
.
The literature [
75
,
76
] gave a series of necessary and sufficient conditions for equivalence
from the perspective of observability matrix and output tuple, respectively.
In addition, there are also some special cases of equivalence that have been studied.
An
n
-stage
τ
-terminal-bit Galois NFSR is equivalent to an
n
-stage Fibonacci NFSR when
the feedback functions of the Galois NFSR satisfy certain conditions. If the output sequence
set of an
n
-stage Fibonacci NFSR equal to its complementary set, then there are 2
×(
2
n−1!)2
Galois NFSRs are equivalent to this Fibonacci NFSR. For details, see [73].
A property closely related to equivalence is isomorphism. Two NFSRs are said to be
isomorphic if their state diagrams are isomorphic. Two state diagrams
G= (V
,
A)
and
G= (V
,
A)
are isomorphic means that there exists a bijection mapping
φ:V→V
such
that for any edge
E∈A
from state
X
to
Y
, there exists an edge
E∈A
from
φ(X)
to
φ(Y)
.
About the relationship between isomorphism and equivalence, if an
n
-stage Fibonacci
NFSR and an
n
-stage Galois NFSR are equivalent, then their state transition diagrams are
isomorphic [
70
]. In more depth, the literature [
73
] explored the conditions that the feedback
Mathematics 2022,10, 3538 7 of 14
functions need to satisfy to make two NFSRs achieve anti-isomorphism, dual isomorphism
and dual anti-isomorphism, respectively.
(2) Weak equivalence
Weak equivalence is a relation weaker than equivalence.
Definition 5
([
80
])
.
Fibonacci NFSR (5) is said to be weakly equivalent to Galois NFSR (8) if
for any output sequence, denoted by
Y
of Fibonacci NFSR (5), there always exists an initial state
denoted by
z
, such that the output sequence of Galois NFSR (8) with initial state
z
is the same as
Y
.
According to the definition of weak equivalence, (5) is weakly equivalent to (8) if the
set of output sequences of (5) is a subset of that of (8). If Fibonacci NFSR (5) is weakly
equivalent to Galois NFSR (8), then (5) is also equivalent to (8). Conversely, if Galois NFSR
(8) is weakly equivalent to Fibonacci NFSR (5), then (8) is not necessarily equivalent to
(5). Moreover [
80
], pointed out that given any
n
-stage Fibonacci NFSR, their method can
construct
(
2
n−1)!2−
1 weakly equivalent
n
-stage Galois NFSRs, and conversely, given any
n
-stage Galois NFSR, it is possible to construct a weakly equivalent
m
-stage Fibonacci NFSR
where m<n.
3.2.2. The Equivalence and Decomposition between Cascade NFSRs
Stream cipher design can benefit from examining the features of two equivalent
cascade NFSRs, for example, to choose a better NFSR based on its quality metrics [
69
]
showed that for any given cascade connection of an
m
-stage NFSR1 into an
n
-stage NFSR2,
there exists only another one equivalent cascade connection of an
m
-stage NFSR3 into an
n-stage NFSR4.
Decomposability and equivalence go hand in hand for cascaded NFSR. An NFSR is
said to be decomposable if it is equivalent to a cascade connection of to NFSRs. By multi-
plying the state transition matrix by the permutation matrix, Zhong and Lin [
70
] obtained
a sufficient and necessary condition that a Fibonacci NFSR can be decomposed into a
cascade NFSR.
3.2.3. Minimum Period and Maximum Period
Minimum period is a concept in Grain-like cascade NFSRs. In 2011, Hu and
Gong [102]
confirmed that the period of the sequence generated by a Grain-like cascade NFSR is a
multiple of the period of the sequence generated by its LFSR if the initial state of the LFSR
is nonzero, and meanwhile proposed an open question: for fixed feedback functions of
an NFSR and an LFSR, determine whether the sequences generated by the NFSR in a
Grain-like structure can achieve the minimum period, i.e., the period of the LFSR. This
question has attracted some scholars to discuss. Zhong and Lin [
68
] converted this open
question into a problem of solving an integer equation based on the framework of STP. They
verified that for any given initial state of an
n
-stage NFSR and any given nonzero initial
state of an
m
-stage LFSR, the probability that the sequence produced by the Grain-like
cascade NFSR achieves the minimum period 2m−1 is very small, which is at most 2−n.
As for maximum period, the sequences with maximum period can keep more cryp-
tographical security than other sequences. The NFSR which can generate sequences with
maximum period is usually called full-length NFSR. Literature [
66
,
71
] revealed that an
n
-stage NFSR is a full-length NFSR if and only if the state transition matrix
L
in (5) sat-
isfies
ord(L) =
2
n
, and for an
n
-stage full-length NFSR ,
|L|=−
1 holds. The full-length
NFSRs can be constructed through the cycles joining algorithm [
65
], and this algorithm can
construct 22n−2−1different n-stage full-length NFSRs.
Other studies on cycles via STP include cycle reconstruction [
78
], cycle decomposi-
tion [82], etc.
For the convenience of readers, the relevant results are presented in Table 1in the
order of the year of publication of the mentioned literature.
Mathematics 2022,10, 3538 8 of 14
Table 1. The existing results on period of NFSRs based on STP.
Year Literature Innovation Points Object
2014 [65] cycles joining algorithm maximum period
2015 [66,71]ord(L) = 2niff the NFSR is full-length maximum period
2015 [82] multi-valued NFSRs cycle decomposition
2018 [78] NFSRs with single input cycle reconstruction
2018 [68] the probability to achieve minimum period ≤2−nminimum period
3.3. Properties and Correlative Criteria of NFSRs
The property of NFSRs plays a crucial role in reflecting its performance. Studying the
properties of NFSRs can provide reference for stream cipher designers. This subsection will
introduce the research status of different properties of NFSRs in recent years.
3.3.1. Nonsingularity
Nonsingularity is a fundamental demand to guarantee that the NFSRs avoid gener-
ating equivalent keys in stream cipher designing. The definition of nonsingularity is as
follows:
Definition 6
([
103
])
.
An NFSR is said to be nonsingular if its state transition diagram contains
only cycles.
Lemma 2
([
103
])
.
An NFSR is nonsingular if and only if each state has only one successor and
one predecessor.
Regarding nonsingularity, there are some classical theories, such as the proof in [
4
] that
a binary NFSR is nonsingular if and only if its feedback function
f(x1
,
x2
,
. . .
,
xn) = x1⊕
f0(x2
,
. . .
,
xn)
, where
f0
is independent of the variable
x1
. However, the paper [
82
] pointed
that this method cannot be applied directly with multi-valued NFSRs. Therefore [
82
],
proposed another method to judge nonsingularity, that is, an
n
-stage
k
-valued NFSR
is nonsingular if and only if
|Mi| 6=
0,
i=
1, 2,
. . .
,
kn−1
, where
Mi∈ Lk×k
satisfying
LW[k,k]. . . W[k,kn−1]= [M1M2. . . Mkn−1]
and
L
is the state transition matrix of the NFSR
obtained by STP.
For Grain-like cascade NFSR (10), Lu et al. [
86
] regarded it as an NFSR with input (11)
and put forward some sufficient conditions for nonsingularity, for example, NFSR (11) is
nonsingular if
Lu= [Lu1Lu2]
with
Lu1
and
Lu2
are both nonsingular. Lu also pointed that if
in NFSR (11)
Col(Lu)6=∆2n
, then this NFSR is singular. Meanwhile, some other properties
of Luare also given in this paper.
3.3.2. Stability and Driven Stability
During the decoding process of convolutional codes, decoding errors may occur due
to channel attacks and other reasons, and decoding errors may propagate to cause more
decoding errors. To limit error propagation, Massey [
104
] found that stability as well as
driven stability helps a lot in this regard. Stability is a concept in autonomous NFSRs,
indicating that NFSR has a kind of “reconvergence" ability, which can make NFSRs with
decoding error return to correct decoding. Driven stability is a property weaker than
stability, defined in non-autonomous NFSRs with input. Driven stability only requires
that the states which can be reached from the equilibrium point driven by input can
achieve reconvergence.
Definition 7
([
84
])
.
An
n
-stage autonomous NFSR is globally stable to the equilibrium state 0
(
δ2n
2n
), if for any state
X∈Fn
2
(
x∈∆2n
), there exists an integer
N>
0, such that
FN(X) =
0
(LNx=δ2n
2n).
Mathematics 2022,10, 3538 9 of 14
Definition 8
([
85
])
.
An
n
-stage NFSR with input is driven stable to the equilibrium state 0(state
δ2n
2n
), if for every state
X
(state
x
) that can be reached from 0(
δ2n
2n
) by driving the NFSR with an
input sequence, there exists a positive integer
N
such that
FN(X) =
0(
LNx=δ2n
2n
), where
F
and
L
are the state transition function and state transition matrix of the corresponding autonomous
NFSR.
The stability of NFSRs can be evaluated by iteration of the state transition matrix
L
via
STP [84].
Theorem 1
([
84
])
.
An
n
-stage Fibonacci NFSR (5) is globally stable if and only if there exists a
positive integer N 62n−1such that each column of the matrix LNis equal to δ2n
2n.
In addition to this, there are several works that studied the stability in special cases
based on STP. Gao et al. [
78
] researched the stability of NFSRs with periodic input, including
limited length and unlimited length. Lu and his team [
87
] focused on the stability in the case
where NFSR is monotonous, which is called reliable NFSR. Their method for constructing
reliable FSRs indicated that the number of reliable FSRs is
22n−4
Φ(n)(n>
5
)
times of that
constructed by the previous method. The methods given in [
91
,
92
] can be applied to
multi-valued NFSRs, and can construct stable NFSRs. Furthermore, the literature [
93
,
95
]
addressed the stability issues of (n,k)NFSR and Grain-like cascade NFSRs, respectively.
As for the driven stability, the researchers found that it is only necessary to examine
whether the state that the equilibrium point can reach can achieve reconvergence, which
is also the origin of the driven stability [
104
]. We can turn the driven stability question
into whether the reachable set of the equilibrium point is a subset of the basin [
85
] pro-
vided algorithms for finding the reachable set and the basin using STP, which reduces the
computational complexity of the previous algorithms.
For the convenience of readers, the relevant results are presented in Table 2in the
order of the year of publication of the literature. These results are helpful to the theoretical
analysis of NFSRs, and have reference significance for the design of decoding algorithm.
Table 2. The existing results on stability of NFSRs based on STP.
Year Literature Innovation Points Object
2016 [84] NFSRs, iteration method shown in Theorem 1stability
2016 [85] NFSRs, reduce the computational complexity driven stability
2017 [93](n,k)NFSRs stability
2018 [78] NFSRs with periodic input stability
2019 [91,92] multi-valued NFSRs, construct stable NFSRs stability
2020 [95] Grain-like cascade NFSRs stability
2021 [87] monotonous FSRs, construct reliable FSRs stability
3.3.3. Observability
Observability is a fundamental property in control theory which can ensure that any
two distinct initial states can be uniquely determined by their outputs. That is, starting
from two distinct initial states, the NFSR does not produce two identical outputs.
Definition 9.
(1) ([
105
]) Two initial states
x06=x0
0∈∆2n
are said to be indistinguishable, if their
corresponding output sequences are equal. Otherwise, the two distinct initial states are said to
be distinguishable.
(2) An NFSR is said to be observable if every two distinct initial states are distinguishable.
In [
76
] the authors investigated the equivalence transformation between Galois NFSRs
and Fibonacci NFSRs based on observability matrix based on STP. According to the defini-
tion of observability of sequence generators, the NFSR-based stream ciphers should avoid
unobservable Galois NFSRs from the security viewpoint and select observable ones. In [
90
],
the authors studied the observability of binary Galois NFSRs using a new observability
Mathematics 2022,10, 3538 10 of 14
matrix which is constructed by putting the output sequences generated by the initial states
on the same branch and its concatenated cycles into a block via STP. Further, [
97
] researched
the observability of multi-value Galois NFSRs and gave the relevant criterion using the
state pair table method and the matrix method, respectively.
Theorem 2
([
90
])
.
Let
Nk
be the number of distinct columns of the observability matrix
Ok
of
an
n
-stage Galois NFSR (8). Then (8) is observable if and only if
Nk+1− Nk>
1for all positive
integer k satisfying Nk<2n.
4. Summary and Prospect of NFSRs
With the help of STP, NFSR research is getting ever more complete and comprehensive,
and the results on theoretical problems have significant reference value for the design
of stream cipher algorithms and decoding algorithms. However, there are still many
unresolved theoretical issues.
(1) Reduce the computational complexity. Using STP to process these actual network
models, the more network nodes, the higher the dimension of the system and the higher
the computational performance requirements of the computer. How to reduce the computa-
tional complexity of the system is a big challenge in theoretical research of NFSR. At present,
there are some methods to reduce computational complexity: approximation method [
106
],
network aggregation method [
107
], logic matrix decomposition [
108
], pinning control [
109
],
model order reduction [
110
] and block decoupling [
111
], etc. Based on these methods,
it is a future research direction to continue to explore more effective methods to reduce
computational complexity.
(2) The existing related research is not perfect, and many issues have no exact result yet,
such as the study of observability and driven stability in multi-valued NFSRs, the modeling
of Trivium-like cascade NFSR and study of its related properties, the maximum and
minimum period problems in multi-valued NFSRs. In addition, there are very few studies
on non-autonomous NFSR, which can also provide a very valuable research idea to improve
related research.
(3) Explore more comprehensive theoretical issues of NFSRs, and provide theoretical
support for the design of stream cipher algorithms and decoding algorithms. Thanks
to the STP as a tool, many indicators can be easily studied. In addition to nonsingular-
ity, observability, and stability, whether there are more indicators that can characterize
the performance and security strength of NFSRs can also be used as follow-up research
directions.
(4) Based on the above theoretical analysis, it is the ultimate goal to give more effective
stream cipher algorithms with higher security.
This paper reviews and summarizes the most recent developments in NFSR based on
STP at this stage. The combination of STP and NFSR is still slowly maturing, and there are
still a lot of problems to be solved. We believe this paper can provide researchers interested
in STP and NFSR with a thinking direction.
Author Contributions:
Conceptualization, Z.G. and J.-e.F.; methodology, Z.G.; validation, Z.G.;
formal analysis, Z.G.; investigation, Z.G.; resources, Z.G.; writing—original draft preparation, Z.G.;
writing—review and editing, Z.G..; visualization, Z.G.; supervision, J.-e.F.; project administration,
J.-e.F.; funding acquisition, J.-e.F. All authors have read and agreed to the published version of
the manuscript.
Funding:
This document is the results of the research project funded by the National Natural Science
Foundation (NNSF) of China under Grant 61877036.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
Mathematics 2022,10, 3538 11 of 14
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