I. INTRODUCTION
The recent superinformation and superconnected world and the fourth industry based on it will be implemented one after another, leading mankind to live a new paradigm that is completely different from what it has been before. This will be made possible by the development of ICT and by combing and converging it into a new theory and the realization underlying it [1][3]. In particular, the rapid development of hardware, superintensiveness of semiconductors and processors, maximization of IoT [4], activation of artificial intelligence, big data techniques and VRARMR are the most important items in the entry the fourth industry.
Based on the current digital system and its foundation, the hardware part of the computer structure, microprocessor etc. [5][8] has made rapid progress with the development of electronic engineering and especially with the development of VLSI/ULSI with very high concentration.[89] However, problems such as the connection between various devices and terminals, the limitations of the amount of the number of terminals, the time delay due to the large amount of information transmission, and the errors of information have begun to be derived. In order to solve this problem, research has been actively conducted to effectively analyze, interpret and synthesize digital systems based on finite fields [8][11]. If P=2, m=1, especially on GF(P^{m}), where P is a prime number and m is a positive integer, it forms the basis of binary, algebraically the Boolean algebra belongs to this category, and it is easy to apply to existing digital systems [11][15].
Unlike the combinational logic circuit implementation, the conventional sequential logic circuit must be considered current and next state. To this end, the next state function had to be derived. To implement this in a circuit, the final sequential logic circuit was realized using the basic component of sequential logic circuit, the flipflop, and a lot of time and computation is required to perform these procedures. In this paper, a method to make up for the aforementioned shortcomings and effectively construct a finite fields sequential digital system was proposed.
The course of this paper is described as follows: Chapter ? discussed the allocation of all elements of finite fields into digit codes, and Chapter III discussed the operating characteristics and properties of Tgate used in the construction of sequential digital systems in this paper.
Chapter IV discuss the process of the proposed method of composition of nonfeedback sequential digital systems of finite fields using example, after we compare the results. The conclusions of Chapter V summarized the features of the finitefields sequential digital system construction proposed in this paper.
II. ASSIGNMENT OF DIGIT CODE FOR ELEMENTS OVER FINITE FIELDS
The mathematical properties of finite bodies cited in this chapter are introduced and used without proof in the various published references.
Among them, we obtain mdimension irreducible polynomial expression after factorize expression (1), and we obtain the polynomial expression such as formula (2) can be obtained by α as long as the root of the equation is zero.
where ξ=P^{m} is the prime number and m is the positive integer.
where α is the root of the morder polynomial with the element of the integer field Z_{P} by law P and is a_{i} ∈ Z_{P} (i=0, 1, ……, P^{m}1).
The algorithm for code assignment of elements of finite fields is as follows.

STEP 1 : all elements are marked with e_{i} (i=0,1,2, …,P^{m}1).

STEP 2 : we represent all coefficient a_{m1} a_{m2} …a_{1} a_{0} according to order, next we mapping MSD to a_{m1} and LSD to a_{0}.

STEP 3 : we allocate the e_{0} to all coefficients are 0 case, also we called 0level.

STEP 4 : we allocate the e ^{ξ1}(ξ=P^{m}1) to all coefficients are (P1) case, and we called mlevel.

STEP 5 : The elements of _{m}C_{1} in 1level are filled by extending the P valued digit code from LSD to (P1) and assigning element e_{i} in order.

STEP 6 : The elements of _{m}C_{2} in 2level are filled by extending the P valued digit code from MSD from 1 to (P1), and then assigning e_{i} by combining P valued digit codes.

STEP 7 : Assign element e_{i} to each element up to (m1)level in turn in the same way as above STEP6.
III. SEQUENTIAL CIRCUIT IMPLEMENTATION
Unlike combinatorial digital systems, the output of sequential digital systems is determined by past inputs as well as current inputs. Thus, the current input must be sent to the input terminal by a delay device or a memory device, and act as input along with the input of the next time. Generally, sequential digital systems are described in 5tuple, as shown in Equation (3).
where S is the state, I is the input, Z is the output, δ is the sequential state function, λ is the output function, and S, I, Z=e_{i}∈GF(P^{m}) (i=0, 1, 2, ……, P^{m}1).
On the other hand, expression (3) has the same mapping relationship as Equation (4).
where S_{t} is in its present state and S_{t+1} is in its next state. Meanwhile, λ is mapping function according to the Mealy and Moore models.
As seen in Equations (5) and (6), the output from the Moore model is only a function of the present state. This paper deals with the Moore model.
In this paper, the basic logic element of building blocks used in the construction of sequential digital systems is Tgate, and the block diagram for this is shown in Figure 1.
In Figure 1, I_{i} is the input, Z is the output, I_{i}, Z=e_{i}∈GF(P^{m}) (i=0, 1, 2, ……, P^{m}1) and a_{j}(j=0, 1, 2, ……, m1) is the control digit code, a_{j}∈{0, 1, 2 …… (P1)}, also V_{k} is the status digit code, where k=0, 1, 2, …… , m1.
When implementing a sequential digital system, the next state function is important.
Also, since sequential digital systems must handle the current state and the next state at the same time, expressing the next state function using time t can be expressed as Equation (7).
where S_{t+1}, S_{t}, I_{t}=e_{i}∈GF(P^{m})(i=0, 1, 2, …… , P^{m}1).
Meanwhile, state S can be assigned as state digit code V_{k}(k=0, 1, 2, ……, m1) according to the algorithm for assigning finite elements of Chapter II to digit codes, similar to the concept of coding each state into bit codes in the existing sequential digital system in binary. Therefore, after each state is assigned the status digit code, the state expression corresponding to the value from 1 to (P1) is obtained for each state digit code. The sum of the state and state is the modP sum. In particular, the addition of a GF (2^{m}) of P=2 is mod2 sum, which is possible by the EXOR calculation. Therefore, the state digit code V_{k} in finite fields is generated by V_{m1}, V_{m2}, ……, V_{1}, V_{0}, and the state function according to the V_{k} value is obtained as Equation (8).
where ζ =P^{m}1, k=0, 1, 2, ……, m1, and + is the modP sum. In addition, the sequential status function can be obtained based on the state function (8) obtained earlier from the statement of the previous status indicating the relationship between the present and the previous state. The above content is expressed in an Equation (9).
Where, B=H=P^{m} and ζ is P^{m}1, and S_{i} is the state according to the value of V_{k}. Also, V_{k} is the state digit code(k=0, 1, 2, ……, m1), V_{k}∈{0, 1, 2, ……, P1}, and S_{i}=e_{i}∈GF(P^{m}) (i=0, 1, 2, ……, P^{m}1). In addition, ∑, + is the modP sum.
The circuits in sequential digital systems are illustrated in blocks as shown in Figure 2.
Sequential circuit realization algorithm is as follow.

STEP 1: The state in the state transition table is tabulated by assigning it as digital code V_{k} by digit code allocation algorithms.

STEP 2: In the table obtained from STEP1, obtain the state function according to the state digit code V_{k}.

STEP 3: Obtain the previous table from state transition table.

STEP 4: Based on STEP 2's state function from the previous table obtained from STEP 3, the sequential state function is obtained and the output function is obtained from the goal state of the state transition

STEP 5: Realize contents obtained from STEP 4 as circuit of final sequential digital system
IV. APPLICATION AND COMPARISON AND REVIEW
The advanced sequential circuit realization of finite fields, proposed in this paper, was applied for example, and the results were compared and reviewed. From the state transition of finite fields GF (23) as shown in Figure 3, the nonfeedback sequential circuit, which is part II of Chapter III, is as follows.
STEP 1 : The state shown in Figure 4 is S0, S1, S2, S3 and S6, and allocating them as state digit code Vk is shown in Table 1.
present state\state digit code  V_{2}  V_{1}  V_{0} 

S_{0}  0  0  0 
S_{1}  0  0  1 
S_{2}  0  1  0 
S_{3}  1  0  0 
S_{6}  0  1  1 
STEP 2 : we obtain the state function are as following Equations (10), (11) and (12).
STEP 3 : Table 2 shows the previous state table from Figure 3.
STEP 4 : Based on Table 2 which is the previous state table and state function, we are able to obtain the next transition function (13), (14) and (15), also the output function is the Equation (16) from goal state in state transition diagram.
STEP 5 : Using the contents up to STEP4 and Figure 3, the sequential circuits are realized as shown in Figure 4.
The method of the reference[10] is to implement the concept of universal structure of binary logic by applying it to sequential logic, which has the disadvantages of being applied to all I/O relationships. In generally sequential logic circuits have feedback and are designed using flipflops, the basic circuitbuilding elements that can be implemented. The construction of sequential logic circuit over finite fields proposed in this paper has no feedback and circuit components have designed the circuit using Tgate. This reduces the time required to perform the feedback and has the advantage of directly obtaining the required output. The hardware comparison of the digital system of the method proposed in this paper and the method cited in this paper is difficult to directly compare because the use element is not the same, but considering the blocks in each sequential digital system as one single element, the comparison is as follows. The reference use the 5 ULM(Universal Logic Module), but this paper use 3 Tgates and 1 decoder. Therefore the proposed method is more compact and expansible.
V. CONCLUSION
In this paper, all elements of finite body were allocated as digit codes using the mathematical properties of finite fields to construct sequential digital system. These digit codes are used as Tgate's control digit code arc(k=0, 1, 2, ………., m2, m1), and state digit code Vk(k=0, 1, 2, …….) when configuring sequential digital systems without trajectory. The nontransferable sequential digital system construction method proposed in this paper is summarized as follows. This method allocates the status of the finite body elements presented in Chapter II by the digit code allocation algorithm and obtains the state equation from the conditions according to their values by these state digit codes. Based on the status equation obtained earlier after obtaining the whole table from the status transition, the sequential digital system is configured using Tgate and the device by obtaining the output function from the Goal state. The proposed method does not require complex calculations compared to the computational methods required for the composition of the cited paper. In addition, the number of Tgates is more compact when we construct sequential logic digital systems. For the future, a method (or algorithm) research to implement a more effective sequential logic digital circuit system is needed and is currently under study.