Stability of Nonlinear Feedback Shift Registers with Periodic Input

The stability of Non-Linear Feedback Shift Registers (NFSRs) plays an important role in the cryptographic security. Due to the complexity of nonlinear systems and the lack of efficient algebraic tools, the theorems related to the stability of NFSRs are still not well-developed. In this paper, we view the NFSR with periodic inputs as a Boolean control network. Based on the mathematical tool of semi-tensor product (STP), the Boolean network can be mapped into an algebraic form. Through these basic theories, we analyze the state space of non-autonomous NFSRs, and discuss the stability of an NFSR with periodic inputs of limited length or unlimited length. The simulation results are provided to prove the efficiency of the model. Based on these works, we can provide a method to analyze the stability of the NFSR with periodic input, including limited length and unlimited length. By this, we can efficiently reduce the computational complexity, and its efficiency is demonstrated by applying the theorem in simulations dealing with the stability of a non-autonomous NFSR.

inputs of the controlled NFSRs [Zhong and Lin (2016)], as demonstrated by Grain [Hell, Johansson and Meier (2007)], Trivium and Mickey. Furthermore, NFSRs are of great interest because they show good statistical performance on the generation of pseudorandom sequences, which are hard to break by current cryptanalysis attacks [Zeng, Yang, Wei et al. (1991)]. Nonlinear systems have been applied in many fields [Li, Wu, Zhao et al. (2018); Han, Tian, Huang et al. (2018); Qiu, Chai, Liu et al. (2018); Wang, Liu, Qiu et al. (2018)] in which the stability analysis is a main component in the research of the controlled nonlinear system [Xu, Xiang and Sachnev (2018); Tang, Ling, Yao et al. (2018)]. Since stable NFSRs can limit error propagation during the process of decoding, the security of NFSRs can be well-developed through the use of the theories of stability of nonlinear systems. NFSRs consists of two basic kinds, autonomous NFSRs (without input) and non-autonomous NFSRs (with inputs). Some studies focus on the stability of autonomous NFSRs [Mowle (1966[Mowle ( , 1967; Ma, Qi and Tian (2013); Zhong and Lin (2015)], while less attention is given to the non-autonomous NFSRs. Massay and Liu introduced the concept of stability of non-autonomous NFSR, and designed a progress-driven stable version of an NFSR [Massey and Liu (1964)]. Due to the lack of efficient mathematical tools, research achievements in this field are rare. In 2012, Cheng and his co-workers introduced a new method to achieve the algebraic form of a Boolean network by using the semi-tensor product (STP) [Cheng and Qi (2010)]. Thanks to the mathematical tool, some problems, in fields such as physics [Gao, Li, Peng et al. (2013); Gao, Deng, Zhao et al. (2017); Li (2016)] and system science [Gao, Shi, Yang et al. (2014); Gao, Deng, Zhao et al. (2017); Li, Yan and Karimi (2018)], involving Boolean control networks, can be converted into algebraic problems. This is also very helpful for analyzing nonautonomous NFSRs [Gao, Liu, Lan et al. (2018)]. Based on these contributions, Zhong et al. [Zhong and Lin (2016)] proposed a novel way to study the stability of NFSRs, via a technique that has lower computational complexity than the exiting method. However, due to the complexity of controlled non-linear systems, the study of stability of the NFSR with different kinds of input has yet to reach a satisfactory conclusion. On the basis of these efficient mathematical tools, this paper mainly describes the state transition of the non-autonomous NFSR. In order to achieve a less computationally complex method, we focus on the transition matrix of the NFSR corresponding with different periodic input sequences during the process of expression. Based on these works, we can provide a method to analyze the stability of the NFSR with periodic input, including limited length and unlimited length. This paper is organized as follows. In Section 2, some basic concepts and related definitions are introduced. Section 3 provides the main results including a description of the stability of an NFSR with periodic input. We also discuss the stability of an NFSR with periodic input of limited length and unlimited length, respectively. Section 4 ends the paper with a brief conclusion.

Semi-tensor product (STP)
The STP is a more general form of a matrix product that allows us to perform a matrix product when the size of the column of one matrix is not equal to the size of the row of the other. The definition is defined as follows: Definition 2.1.1. Assume that M is a matrix of dimensions m n × , N is a matrix of dimensions p q × , and let a be the least common multiple of n and p. The STP of M and N is defined as where ⊗ is the Kronecker product and k I is an identity matrix of dimension k .
Obviously, if n p = , then the STP of M and N in Definition 2.1 will result ina conventional matrix product. For the sake of convenience, the symbol "  " can be omitted from the definition unless specifically required.

Boolean network (BN) and Boolean control network (BCN)
These autonomous NFSRs can be regarded as a Boolean network, which consists of Boolean functions with finite logical variables. The variables in a Boolean network can be classified as "1" and "0". So an autonomous NFSR can be described as a Boolean network as follows: Non-autonomous NFSR is the autonomous NFSR with an input sequence, which can be expressed as a Boolean control network with n nodes and m inputs as 1 2 1 ( 1) ( ( ), ( ),..., ( ), ( ),..., ( ))( 1, 2,... ) where (  In order to effectively describe the state of the network in the time t, we transform the network into an algebraic form, and introduce the definitions related to the transfer of the logic-based problems into algebraic problems below. Assume that k I can be represented as } { 1, 2,... , and M is the structure matrix of g , which is expressed as Through the use of the STP, a Boolean function can be represented as a muti-linear form. We then denote the structure matrix corresponding to the node i x by a 2 2 n × matrix i M . Through Definition 2.2.1, any Boolean function in NFSR (2) can be represented as where 1, 2,..., i n =

.
Based on the properties of the STP, a further transformation of the algebraic form of the Boolean (control) network is shown as follows, such that a Boolean network can be converted into a conventional discrete-time linear system. ( 1) ( ) ( )... (2), a 2 2 n n × matrix, satisfying When the input sequence is added into the autonomous NFSR, the study is turned into a non-autonomous NFSR.
Similarly, if we assume that , the algebraic representation of the nonautonomous NFSR is performed as shown in the following cases. We begin with discuss the algebraic representation of a non-autonomous NFSR with multi-steps.
Case 1: If there exists a series of input sequence ( ) i u t in a period 1,..., i l = , after l steps, the state of non-autonomous NFSR in the l-th step can be solved as where ( ) x t is state of the system in the time t, the ( 1) In most cases, there only exists one input per step in the NFSR. Thus, NFSR (10) can be rewritten as where L  is a 1 2 2 n n + × matrix, which consists of transition matrix L and matrix ( ) , and ( ) u t is the only input in one step.
Case 2 is a special situation of Case 1. In this paper, we will focus on the state transition under Case 1.

State of the NFSR with input
In this subsection, we mainly focus on the description of the transition matrix so that we can describe the state transition of a stable NFSR with different input sequences. Since the logical variables in a system can be classified as "0" and "1", the corresponding inputs and transition matrices are also different. Next, we analyze the state transition in the circumstances of the two different kinds of input. δ correspond to the logical variables "0" and "1", respectively, and denote the corresponding transition matrix of logical variables "0" and "1" by 0 L  and 1 L  . When ( 1) Here, we note that Conversely, if the input is 1 2 δ , then NFSR (11) can be rewritten as . Similarly, we note that 1 1 Proof. Let us consider NFSR (11) δ can be defined as The input "0" is a invalid input.
Case 2: If the input is "1", then the product between L  and 1 2 δ is defined as The input "1" is a valid input. Next, we provide an example to illustrate the different effects for the inputs of "0" and "1". It should be noted that, in most cases, the logical relationship between the state nodes and inputs is exclusive-or A B + . Thus, we assume that the logical relationship between the state nodes and inputs is the exclusive-or in the following examples. ( 1) ( ) where ( )( 1, 2,3, 4) i x t i = , are states of the nodes.

Limited length of periodic input
The periodic input sequence can be classified as limited or unlimited in length according to their length properties. In this subsection, we analyze the problem of the limited length of periodic inputs, where the short-period repeats finite times. Then, based on the stability of the NFSR, we focus on the state transition after periodic input in a stable NFSR.
The definition for representing the method of limited length of periodic input is shown as follows: Definition 3.2.1. Assume that there exists a series of periodic input sequences, written as where 1 2 , ,..., c i i i are all the logical numbers in a period, and k is the number of times the period repeats. For convenience, we will denote the periodic input sequence by then the sequence of periodic input is denoted by (0, 0,1,1, 0) _ 8 .
Next, we analyze the state transition generated by limited length of periodic input sequence.
where 0 1 i or α = , 1, 2,..., c α = , are logical numbers in a complete short-period, k is the number of times the short-period repeats, and Θ represents the transition matrix of NFSR (9). NFSR (9) with input sequence 1 2 ( , ,..., ) _ c i i i k can be further expressed as where k is a positive integer.

Proof:
Beginning with the state (0) x , the state transition attached to a periodic input of one fold is performed as If the input sequence increases to k folds, namely, there exists a series of periodic input sequence 1 2 ( , ,..., ) _ c i i i k , based on the Eq. (30) and starting from the state 0 x , the corresponding state transition can be constructed as ...
where the Θ satisfies where, The matrix Θ is the transition matrix of the network, which is helpful in transferring the network into a linear representation. The value of the matrix Θ is influenced by the periodic input and the length of the attractor is also affected by the length of these inputs. After periodic inputs of length-c in a fold, the NFSR can generate length-kc attractors (k=1,2,...c). According to Theorem 3.2.2, we can determine the state after kc steps in an NFSR with periodic inputs. In the calculation of the k Θ , we compute 2 Θ =Θ Θ  at first, then compute 4 2 2 Θ =Θ Θ  ,..., and by this analogy, we can get the value of the matrix k Θ .
In the process of the calculation, the computational complexity is 2 (log ) O l .
Next, we provide an example to illustrate Definition 3. ( 1) ( ) x t x t x t x t x t x t x t x t x t x t u t Based on Theorem 3.2.2, we begin by calculating the value of the matrix Θ as follows: .
In Example 3.1.2, we found that, 16  (36) When the period repeats k times, which means there exists a periodic input sequence (111101000110010) _ k , the state transition can be expressed as ( 1) x t δ δ Based on Theorem 3.2.2, when k=5, we note that 1 5 1 16 16 δ δ = Θ . We then find the state 1 16 δ can return to the initial state after 75 state transitions, and the length of the corresponding attractor is 75. At this moment, the NFSR described in the system (34) reaches stability.

Unlimited length of periodic input
In the previous section, we analyzed the periodic input with limited length, but in the actual application, unlimited-time input sequences are more likely. Therefore, we now consider the stability of NFSR with unlimited length of periodic input. Then we introduce a method for representing the unlimited length of periodic input sequence.
where L  is a 1 2 2 n n + × matrix, and 1 2 , ,..., c i i i are logical variables in the one period.
Accordingly, we can construct a corresponding representation of the NFSR with a periodic input of unlimited length as follows, x t x t x t x t where ( ) x t and ( 1) x t + are states of the network in the time t and 1 t + .
From the point of cryptographical security in NFSRs, the stability is a fundamental condition, and the maximum period is preferable. Next, employing Definition 3.3.1, we provide a method to check the stability of the NFSR with periodic input of unlimited length so that we can enhance the cryptographical security.
The above equations demonstrate that the different starting nodes attach to different transition matrices. If we then choose the 1 16 δ as the initial state, and apply Algorithm 3.3.4, the state is stable after the state transitions as follows: Thus, the modified non-autonomous NFSR mentioned in this example is stable.

Conclusions
This paper applies the Boolean network to reduce the computational complexity about the method of description, in cases of the state transition in an NFSR with different kinds of periodic input. Notably, we developed a novel method to analyze the stability of the system with unlimited length of periodic input. For a non-autonomous NFSR, the more stable it is, and the larger the period generated, the more security it will offer when used. Developing effective algorithms or approximate techniques for the present approach will be a challenging problem in future work.