Abstract
Nonlinear feedback shift registers (NFSRs) are the main building blocks in many convolutional decoders, and a stable NFSR can limit decoding error propagation. Due to lack of efficient algebraic tools, the stability of multi-valued NFSRs has been much less studied. This paper studies the stability of multi-valued NFSRs using a logic network approach. A multi-valued NFSR can be viewed as a logic network. Based on its logic network representation, some sufficient and necessary conditions are provided for globally (locally) stable multi-valued NFSRs, explicit forms are given for the set of basins, and the algorithm for obtaining the set of basins is provided as well. Finally, a new method is presented for constructing stable -stage NFSRs from stable -stage NFSRs by the properties of -morphism.
1. Introduction
Nonlinear feedback shift registers (NFSRs) are the main building blocks in many convolutional decoders. However, in the process of decoding, a decoding error tends to induce indefinitely long decoding errors. A stable NFSR is an alternative to limit this error propagation. Some studies have focused on the stability of NFSRs. In 1964, Massey and Liu [1] proposed that using a stable nonlinear feedback shift register (NFSR) as the main building block in a convolutional decoder is able to limit such an error propagation. In their NFSR-based decoder, the feedback function represents a decoding algorithm. They gave an example to highlight the application of the NFSR-based decoder. Mowle [2] proved that the number of -stage globally stable NFSRs is and also showed that all these NFSRs are binomially distributed. In [3, 4], the author gave the enumeration and classification of stable FSRs and an algorithm to generate all of them. A direct algorithm for the synthesis of stable NFSRs was proposed [5]. It is notable to point out that only binary NFSRs were concerned in the above work. In addition, Lempel [6] gave some results on -stable NFSRs. Since then, the stability of NFSRs has not been further studied due to lack of efficient mathematical tools, although numerous other efforts have been made on NFSRs over the past decades.
In [7–9], the authors studied the stability for binary NFSRs by viewing them as Boolean networks. A Boolean network is a finite state automaton evolving through Boolean functions. It was firstly introduced by Kauffman in 1969 to model a genetic network whose variables take only two possible values, “on" and “off" (or equivalently, and , resp.) [10]. Over the last decades, Boolean networks have attracted much attention in many communities, such as biology [11–13], physics [14–16], system sciences [17–21], and control theory [22, 23]. In the community of system sciences, Cheng and his collaborators [24] developed an algebraic framework for Boolean networks using a semitensor product approach. In the algebraic framework, a Boolean network can be equivalently converted into a conventional discrete-time linear system. A logic network is a generalization of a Boolean network. The variables of a logical network usually take multiple values. If they take only two values, say and , then the logical network is reduced to a Boolean network. The studies of multi-valued logical networks can refer to, for instance, [25–27].
Multi-valued NFSRs have been investigated in several studies. For example, some construction methods were given for de Bruijn sequences generated from multi-valued NFSRs [28–30]. A necessary and sufficient condition was given for the nonsingularity of multi-valued NFSRs [31]. Recently, the multi-valued NFSRs were studied in [32–34]. Some necessary and sufficient conditions were given for the stability of multi-valued NFSRs in [35].
In this paper, we study the stability of multi-valued NFSRs using a logic network approach. A multi-valued NFSR can be viewed as a logic network. Based on its logic network representation, we give the state transition matrix [34], which shows the simple relation with the truth table of the feedback function of the NFSR. From this viewpoint, it is more explicit than the state transition matrix introduced in [31], where the state transition matrices are expressed as the products of some structure matrices of the components of the vectorial function. In fact, from the cryptography perspective, it is very important and useful to show the explicit relation between the truth table of the feedback function and the state transition matrix in order to analyze and design an NFSR. This paper is an extension of our previous work [35], which is more complete and more substantial due to the following contributions:(1)Because the stability of an NFSR completely depends on the basin of the NFSR, we give the explicit forms for the set of basins, and the algorithm for obtaining the set of basins is provided as well.(2)A stable NFSR is an alternative to limit error propagation in the process of decoding; therefore we give a new method for constructing stable -stage NFSRs from stable -stage NFSRs over the binary field.
The remainder of this paper is organized as follows. Section 2 briefly reviews some related works on logic networks. Sections 3 and 4 are our main results. Some sufficient and necessary conditions are given for globally (locally) stable NFSRs in Section 3. In Section 4, we present the method to construct stable NFSRs, and examples are presented to show the effectiveness of the proposed method. The paper is concluded in Section 5.
2. Logic Network Representation of NFSR
In this section, we first briefly review the semitensor product of matrices and recall the multi-linear form of nonlinear logic function that is obtained by the semi-tensor product. Finally, we revisit the logic network representation of a multi-valued NFSR, which is very useful to investigate the stability of NFSRs.
For the statement ease, we first give some notations:(i); when , we denote , that is, binary field.(ii): -dimensional vectors over ; when , we denote , that is, -dimensional vector space over .(iii) the identity matrix of dimension .(iv): the -th column of the identity matrix .(v)(vi): the set of all -dimensional vectors over .(vii): the -th column of a matrix .(viii): the order of a square matrix of dimension , that is, the least power satisfying .(ix): the set of matrices, whose columns belong to . If , then it can be expressed as . For the sake of compactness, it is briefly denoted by .
2.1. Semi-Tensor Product and Multilinear Form of Logic Network
Semi-tensor product of matrices was introduced by Cheng [24]. It is a generalization of the conventional matrix product and works for any two matrices regardless of their sizes, while it retains all major properties of the conventional matrix product. Before reviewing the semitensor product, we first recall what the Kronecker product is.
Definition 1 (see [36]). Let and be matrices of dimensions and , respectively. The Kronecker product of and is defined as an matrix, given by
Definition 2 (see [24]). Let and be matrices of dimensions and , respectively, and let be the least common multiple of and The (left) semitensor product of and is defined as an matrix, given by
Clearly, if , the semi-tensor product is reduced to their conventional matrix product .
A logical function with variables is a mapping from to . Let be the decimal number corresponding to the tuple via the mapping . Then ranges from to . For the sake of simplicity, we denote . Then is the truth table of , arranged in the alphabet order, while is truth table of , arranged in the reverse alphabet order.
Identify a variable as a vector . Then a logic function with variable from to is changed to a function from to .
Lemma 3 (see [24]). For any logical function with , let be the truth table of , arranged in the reverse alphabet order. Then can be expressed as a multilinear form:where is called the structure matrix of .
Lemma 4 (see [24]). Supposewith , . Then . Moreover, For any , the state and the state , which satisfy , are one-to-one correspondent.
Definition 5 (see [37]). Let and be matrices of dimensions and , respectively, where and , , are the -th column of matrices and , respectively. The Khatri-Rao product of and is defined as an matrix, given by
and a logical network with -nodes can be described as the following system:where , and , with for all . Let be the structure matrix of the function for any . System (6) can be equivalently described as a linear system [21]:where is the state and is the state transition matrix.
2.2. Logic Network Representation of Multi-Valued NFSR
An -stage multi-valued Fibonacci NFSR can be described as Figure 1. It is a collection of storage devices, whose contents are denoted by the variables , taking values from the set Here the logical function is called the feedback function of the NFSR. For any , the content is shifted to at each periodic interval determined by a master clock. However, to obtain a new value for the variable , we compute the function of all the present contents in the shift register.
The state diagram of an -stage -valued NFSR is a directed graph consisting of nodes and directed edges. Each node represents a state of the NFSR, and an edge from state to state means that is shifted to the state . is called a predecessor of , and is called the successor of . Every state of an NFSR has a unique successor but may have no predecessor or a single predecessor or predecessors with a positive integer satisfying . The state with more than one predecessor is called a branch state, while the state without predecessors is called a starting state. A sequence of distinct states, , is called a cycle of length if is the successor of , and is a successor of for any Similarly, a sequence of distinct states, , is called a transient of length , if the following conditions are satisfied: none of them lies on a cycle; is a starting state; is a successor of for any ; the successor of lies on a cycle.
For the sake of statement simplicity, in the sequel an -stage NFSR means an -stage multi-valued Fibonacci NFSR over .
View the -stage NSFR in Figure 1 as a logic network. Then it can be expressed aswhere is the state and is the state transition function, satisfyingFor any positive integer , let , which indicates that the state is shifted times from .
Lemma 6 (see [34]). For an -stage NFSR with a feedback function , assume the truth table of to be , arranged in the reverse alphabet order. Then the NFSR can be equivalently expressed as a linear systemwhere is the state and is the state transition matrix, expressed aswhere
3. The Properties of Stable Multi-Valued NFSR
In this section, we first briefly review some existing basic concepts of the stability of NFSRs. Then we show that the error-propagation effect is closely related to the stability of an NFSR. Finally, we give some sufficient and necessary conditions for their stability.
3.1. Basic Concepts
Definition 7 (see [2]). A state is called an equilibrium state of the logic network (6), if . For a -valued NFSR, the equilibrium state of its logic network representation (8) is also called an equilibrium state of the NFSR.
Note that an equilibrium state of an NFSR forms a cycle of length , that is, unit cycle, in the state diagram of the NFSR.
Definition 8. The set is called the basin of an equilibrium state of an NFSR, if is a set of states eventually reaching the equilibrium state .
Definition 9 (see [1]). An -stage NFSR is globally stable to the equilibrium state , if, for any state , there exists a positive integer such that the state transition function of its logic network representation (8) satisfies ; that is, is the unique equilibrium state and there are no other cycles in the state diagram of the NFSR.
Definition 10 (see [1]). An -stage NFSR is locally stable to the equilibrium state , if there exists some state such that for some positive integer the state transition function of its logic network representation (8) satisfies .
Since an -stage multi-valued NFSR has an equivalent logic network representation in a linear system (10), accordingly, we give an equivalent definition of globally (locally) stable multi-valued NFSR as follows.
Definition 11. An -stage NFSR is globally stable to the equilibrium state , if, for any state , there exists a positive integer such that the state transition matrix of its logic network representation (8) satisfies .
Definition 12. An -stage NFSR is locally stable to the equilibrium state , if there exists some state such that for some positive integer the state transition matrix of its logic network representation (8) satisfies .
In the sequel, an NFSR is globally (resp., locally) stable meaning that an NFSR is globally (resp., locally) stable to the equilibrium state . From their definitions, it is easy to see that a globally stable NFSR must be locally stable but not the vice versa.
Definition 13 (see [2]). An NFSR is called a globally stable maximum transient NFSR if it is globally stable and has a single starting state.
3.2. Decoder NFSR
We show below that the error-propagation effect is closely related to the stability of an NFSR. The relevant portion of a decoder is shown in Figure 2 and is seen to constitute an NFSR. The first terms of the syndrome sequence are stored in the shift register, and the current input is , at the time when the decoder forms . Let the vector represent the shift register contents and let denote the all-zero vector. will be referred to as the state of the NFSR. The decoding algorithm is represented by the function ; that is, . For any reasonable decoding algorithm, , since this is the case where all parity checks are satisfied. From Figure 2, it should be clear that consecutive correct decoding decisions will clear the decoder of any spurious symbols introduced by a decoding error and hence will terminate the error propagation. The ability of the decoder to affect such a “reconvergence” is conveniently studied by considering the shift register to be loaded with some initial states and the syndrome input sequence to be all zeros; that is, all succeeding parity checks are satisfied. Finally, the shift register will enter state when reconvergence has been achieved. Thus the problem of studying error propagation will be reduced to the stability analysis of an NFSR in Figure 1.
3.3. Necessary and Sufficient Conditions for Stability
Theorem 14. An NFSR is locally stable if and only if the feedback function satisfies , and there is at least one such that
Proof. Necessity: Clearly, according to Definition 9, is a necessary condition for a locally stable NFSR. For any NFSR, the state has the possible predecessors: itself and with . If the NFSR is locally stable, then there exist some states such that, for some integers , . Thus, has a predecessor different to itself. Hence, there is at least one such that
Sufficiency: implies that is an equilibrium state of the NFSR. If there is at least an such that , and then is a predecessor of . In other words, there exists a state such that , which implies that the NFSR is locally stable.
Corollary 15. Let be the state transition matrix of the logic network representation (10) in a linear system of an -stage NFSR. Then the NFSR is locally stable if and only if there exists at least one such that .
Proof. Let the truth table of the feedback function of the NFSR be , arranged in the reverse alphabet order. According to Theorem 14, the NFSR is locally stable if and only if the feedback function satisfies , and there is at least one such that , that is, . Then the result follows from (12).
Proposition 16. Let be the state transition matrix of the logic network representation (10) in a linear system of an -stage NFSR. If , there exists an integer such that with some . Then the NFSR is locally stable to .
Proof. Let , and let the truth table of the feedback function of the NFSR be , arranged in the reverse alphabet order. Since , we have . According to (12), we have , that is, . Since there exists an integer , such that , we have . According to Definition 10, the NFSR is locally stable to .
Theorem 17. If the NFSR is globally stable maximum transient, then there exists a unique such that
Proof. If the NFSR is globally stable maximum transient, then except the starting state and the state , the other states have their own unique predecessor and unique successor. The state has the possible predecessors: itself and with Since the NFSR is globally stable maximum transient, has a unique predecessor different to itself. Then the result follows.
Remark 18. Theorem 14 shows that there is at least one such that is sufficient and necessary condition for a locally stable NFSR. However, Theorem 17 shows that there exists a unique such that for globally stable maximum transient NFSR, which is nothing but necessary condition.
Theorem 19. Let be the state transition matrix of the logic network representation (10) in a linear system of an -stage NFSR. The NFSR is globally stable, if and only if there exists an integer such that each column of is equal to Moreover, the NFSR is globally stable maximum transient, if and only if each column of is equal to
Proof. Necessity: As the equilibrium state is uniquely corresponding to the state , that an -stage NFSR is globally stale to the equilibrium state is equivalent to that the -stage NFSR is globally stable to the state Clearly, any state of an -stage globally stable NFSR with one more starting state must be shifted fewer times to reach the equilibrium state than the -stage globally stable maximum transient NFSR. For an -stage globally stable maximum transient NFSR, the starting state must shift times to go through all other states and finally reaches the state (or, equivalently, the state ) and keeps staying at this state. Therefore, is the largest power such that each column of is equal to .
Sufficiency: There exists an integer such that each column of is equal to Therefore, for the state with any , we have According to Definition 9, the NFSR is globally stable. In particular, means that the starting state for any eventually reaches the equilibrium state and keeps staying at this state. Thus, the result follows.
Theorem 20. Given a globally stable maximum transient -valued -stage NFSR, its starting state is with some
Proof. For a given globally stable maximum transient -valued -stage NFSR, the state has only two predecessors, itself and with some Assume that all states , for any , are not the starting state of the NFSR. Let the predecessor of any given state be , with some Then there exists with some such that , which implies that Note that the successor of is with , and the successor of is Then has two different successors, and with some , which is a contradiction that any state has a unique successor. Hence, for any given globally stable maximum transient -valued -stage NFSR, there exists some such that is the starting state of the NFSR.
Example 21. When and , we consider two nonlinear feedback shift registers, NFSR1 and NFSR2. Their feedback functions are, respectively, as follows:and
Computations show that the state transition matrices of the logic network representations of both NFSRs, respectively, areandWe use the same notations in previous sections. For the state transition matrix , According to Corollary 15, we get that NFSR1 is not locally stable. Meanwhile, for the state transition matrix , , and From Theorem 17 and Theorem 19, we obtain that NFSR2 is globally stable. Moreover , and according to in [34], the NFSR2 has two branch states: and . Actually, the NFSR2 has three starting states. All those features are consistent with their state diagrams, which are shown in Figures 3 and 4.
Example 22. When , we consider two nonlinear feedback shift registers, NFSR3 and NFSR4. Their feedback functions are, respectively, as follows:and
Computations show that the state transition matrices of the logic network representations of both NFSRs, respectively, areandWe use the same notations in previous sections. For the state transition matrix and , we obtain . According to Theorem 19, both NFSR3 and NFSR4 are globally stable maximum transient. Their state diagrams are shown in Figure 5.
Clearly, (resp., ) is the starting state of NFSR3 (resp., NFSR4), which is consistent with the result in Theorem 20. It also shows that different globally stable maximum transient -valued NFSRs with may have different starting states, which is unlike the globally stable maximum transient binary NFSRs whose starting states are the same, that is, .
3.4. Basin of the Equilibrium State of NFSRs
Definition 23. The set is called the basin of an equilibrium state of an NFSR, if is a set of states eventually reaching the equilibrium state .
We let be the basin of the equilibrium state . The stability of an NFSR in Figure 1 completely depends on the basin . In the following, we will focus on how to get the basin of the equilibrium state. Reference [34] gives a way to find all starting states of an NFSR, shown in the following lemma.
Lemma 24 (see [34]). Let be a state transition matrix of an -stage NFSR. is a starting state if and only if is not a column of the state transition matrix , where .
Theorem 25. Let be a state transition matrix of an -stage NFSR. Then the basin of the equilibrium state is , is the smallest satisfying , and with some positive integer .
Proof. The result follows from Lemmas 6 and 24.
In fact, it is easy to get the whole state transition graph of an NFSR when its state transition matrix is known. For any , we have that is the predecessor state of and is the successor state of ; that is, For example, we consider the NFSR2 in Example 21. Its state transition matrix . Obviously, only , and according to Lemma 24, they are all starting states of NFSR2. For the state , it is easy to see that , and and . Thus, according to Theorem 25, we have Similarly, for the state , we can also use the same method. Finally, we have All those features are consistent with the logic network expression of its state diagrams, which are shown in Figure 6. In the following, we strive to give an algorithm to obtain the basin . We define to be the set of elements that represent the positions of the entry s of all elements in . Precisely speaking, if , then . For the sake of convenience, we also called the basin. For an -stage NFSR, we first find its starting states according to Lemma 24. Let be a set of starting states, and denote its cardinality as Suppose that the starting state set has been obtained in terms of Lemma 24, and its elements are denoted by Let , which is a set of the positions of the entry s of all the elements in the starting states Second, we assume that the state transition matrix of the NFSR is known. Following by , we define a mappingActually, Note that any starting state of an NFSR eventually reaches a cycle and keeps staying on it. is constituted by the starting states that eventually reach the state and the states that those starting states go through. Finally, we need to take away repeat states.
Finally, we give Algorithm 1 to obtain the basin for an -stage NFSR based on the mapping and the set if we knew the starting states of the NFSR.
| ||||||||||||||
4. The Construction of Stable Feedback Shift Registers over the Binary Field
An -stage -valued NFSR can be described as Figure 1. Let the present state of the NFSR be , and then the successor of can be ; that is, the state can have different successors. Then we construct directly the stable -stage -valued NFSRs from the stable -stage -valued NFSRs by the properties of -morphism, which is not a trivial work. We will consider it in another new work, in which we will define a new mapping. Therefore, in this section, we first give a new method for constructing stable -stage NFSRs from stable -stage NFSRs by the properties of -morphism over the binary field.
4.1. -Morphism
In this subsection, we will give an overview of the -morphism. Let be a positive integer, and let . We define , the conjugate of , and , the dual of , byandwhere denotes the Boolean complement of . We define a mapping , , as follows. For and ,and, in the sequel, we denote (24) as .
For and , we haveEquations (25) and (26) imply that the mapping is a -to- mapping, and it maps and to the same element. Conversely, each has two preimages in under , which are given byIn the sequel, we denote (27) as , .
Some properties of -morphism are recalled below. The -th-order de Bruijn graph is a directed graph with vertices, labeled by the elements of . The vertices and of , , are jointed by an arc, directed from to . A factor of is a partial graph of , and it includes all the vertices of . For example, the state graph of every nonsingular -stage NFSR in is a factor of .
Lemma 26 (see [38]). Let and be a pair of conjugate states in . Then are two conjugate pairs in .
The mapping induces a graph homomorphism (called -morphism) from the -th-order de Bruijn graph to the -th-order de Bruijn graph [38]. If is a subgraph of , then its -morphism image is a subgraph of . Obviously, the state diagram of an -stage NFSR is a subgraph of .
4.2. Synthesis Theory of Stable FSRs
Lemma 27 (see [39]). Let be a cycle in , and let be a state on Then the state is on one of the cycles and , and is on the other one.
By Lemma 27, it is easy to obtain the following corollary.
Corollary 28. Let be a factor of , and let be a state on . Then the state is on one of and , and is on the other one.
Theorem 29. Let be the state diagram of an -stage NFSR, and let be pairwise different states on If and are two predecessors of , then and are two predecessors of one of and , and and are two predecessors of the other one; or and are two predecessors of one of and , and and are two predecessors of the other one.
Proof. Since and are two predecessors of , we have Then, we have Thus, if , we have that and are two predecessors of , and and are two predecessors of ; if , we have that and are two predecessors of , and and are two predecessors of
By Theorem 29, it is easy to obtain the following corollary.
Corollary 30. is the state diagram of a stable -stage NFSR; then there exists an -stage NFSR such that is the state diagram of NFSR. Moreover, is two self-dual in
Theorem 31. If is the state diagram of an -stage NFSR with the feedback function , then there exists an -stage NFSR such that is its state diagram. Moreover, if the feedback function of the -stage NFSR is , then
Proof. We define that the mappings and are induced by the functions and , respectively; that is, for any given , , According to Corollary 30, there exists a one-to-one correspondence between and the self-dual More explicitly, for any given ,whereThen, the result follows from (31), (32), and (33); that is,
Let be the feedback function of an -stage NFSR1, and letbe the feedback function of an -stage NFSR2, where and and denote and , respectively. Note that if and only if for Then, the values of two functions and have different values only at the state
Example 32. Consider a -stage stable NFSR with a feedback functionAccording to Theorem 31, we obtain a -variant function, which is the feedback function of the -NFSR as follows:According to (35), we obtain a -stage stable NFSR with a feedback functionThe state diagrams of the NFSRs with the feedback functions , and are shown in Figures 7, 8, and 9, respectively.
In summary, the theorems and corollaries in Section 4.1 presented a procedure for constructing stable -stage NFSRs from stable -stage NFSRs. Step 1 determines the feedback function of the -stage NFSR from stable -stage NFSR according to Theorem 31. Step 2 is used in finding the feedback function of -stage stable NFSR from -stage NFSR obtained by step .
5. Conclusion
A stable NFSR is an alternative to limit this error propagation. This paper studied the stability of multi-valued NFSRs using a logic network approach. A multi-valued NFSR can be viewed as a logic network. Based on its logic network representation, we first gave some sufficient and necessary conditions for globally (locally) stable multi-valued NFSRs. Then, explicit forms have been given for the set of basins, and the algorithm for obtaining the set of basins is provided as well. The approach used in this paper is helpful to theoretically analyze multi-valued NFSRs. Finally, the method of constructing stable NFSRs is presented, so that we can get a stable -stage NFSR from stable -stage NFSR by the properties of -morphism. Nonlinear feedback shift registers are subject to impulsive effects and time-delay effects, which might be interesting to be considered in the future work.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Joint Funds of the National Natural Science Foundation of China under Key Program Grant U1713212, the National Natural Science Foundation of China under Grants 61702341, 61876110, 61836005, and 61672358, the Natural Science Foundation of Guangdong Province under Grant 2017A030313338, and the Fundamental Research Project in the Science and Technology Plan of Shenzhen under Grant JCYJ20170817102218122.