## 3.1 Network Architecture Stage

Figure 1 depicts a schematic diagram of the structure of the intelligent sensor network, and the whole structure shows the operation status from the work area to the user end. A specific area often contains hundreds or thousands of distributed nodes distributed, the closest distance between points is only approximately 10 cm, and the density can reach 20 nodes per square meter. Such a high density is concentrated in a small area, and we should carefully calculate the number of nodes. We can divide this process into three stages by time according to the distribution of nodes.

The first stage is the allocation stage and the preallocation stage. In this stage, the nodes can be directly assigned to the designated area at random without requiring the location of the nodes; it can also be required that the nodes must reach the designated location one by one. This stage is the initial stage of network construction. Different allocation strategies have different effects on the next stage.

The second stage is the node adjustment stage. During this phase, the topology of the network changes as the nodes change. It is important to emphasize here that nodes are movable nodes similar to mobile phones, and their location changes over time. The change in the position of the network node will inevitably lead to a change in the structure.

The last stage is the node change stage. When the basic structure of the network is completed, the faulty nodes need to be deleted, and the newly added nodes need to be relocated. This dynamic transformation also has an impact on the network structure.

## 3.2 Network operating environment and network media

Intelligent sensor networks often have densely distributed nodes in environments including open fields, the deep seabed, dangerous chemical center areas, or even battlefields experiencing fierce exchange of fire. In a multiconnected network, wireless communication is required between nodes. These connections enable wireless communication using microwave, infrared, and visible light. Most hardware networks are designed based on RF circuits, and the main frequency of Bluetooth devices is 2.4 The low-power node uses a single-channel frequency of 916 MHz.

Infrared devices have no protocol constraints and are also less affected by electromagnetic radiation. At the same time, infrared equipment is relatively inexpensive and easy to install. Another advantage is that self-organizing nodes have optimistic prospects in terms of intelligent sensor network applications.

## 3.3 Elliptic Curves

ECC involves the operation of finite fields and the operation of points on the elliptic curve. First, some mathematical knowledge related to the elliptic curve algorithm is introduced, including ordinary points and points at infinity, the relationship between the plane rectangular coordinate system and the projective plane coordinate system, and the mutual transformation of points, lines and slopes between the two coordinate systems. We introduce the concept of group and field as well as the operation rules in two kinds of elliptic curves that are commonly used in prime number finite field Fp, and finally, we introduce the concept of elliptic curves in prime number finite field Fp. The elliptic curve group and its basic operations on Fm are defined.

A point can be represented in a plane Cartesian coordinate system, and a point in a plane Cartesian coordinate system represents where two parallel lines intersect at infinity.

In the ordinary plane rectangular coordinate system (Cartesian plane rectangular coordinate system), the coordinates of the infinite point are not designed, and the infinite point cannot be represented. To represent the infinite point, the ordinary plane rectangular coordinate system is extended to generate the projective plane coordinate system, which is compatible with the old ordinary point in the plane rectangular coordinate system. In short, the photographic plane coordinate system can represent ordinary points and points at infinity.

The ordinary point projective plane point is expressed as follows:

$$A(x,y)\xrightarrow{{\underrightarrow {x=X/yZ;y=Y/xZ}}}A(X,Y, - Z)$$

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The plane straight line projecting plane straight line *L**1* is expressed as follows:

$$ax - by - 2cz= - 1\xrightarrow{{\underrightarrow {x=X/yZ;y=Y/xZ}}}aX - bY - 2cZ= - 1$$

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In mathematics, there are many sets of numbers that satisfy certain operating conditions, and these sets are named various set concepts. The concepts of groups and domains are one of these sets.

If there is a nonempty set A, the following conditions are satisfied for the algebraic XOR operation:

①Set A is closed for the same-or operation.

②The associative law must be satisfied. For any value x, y, z ∈ A, satisfy:

$$x \otimes (y \otimes z)=(x \otimes y) \otimes z$$

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③ There is at least one left identity element e in set A.

④ Any element x in set A has at least one left inverse element, which satisfies:

$$x \otimes {x^{ - 1}} \otimes {e^{ - 1}}=1$$

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A set that satisfies the above four conditions is called a group set.

There are two commonly used finite fields: one is the prime number field Fp, and the other is the finite field Fm with characteristic 2. The algorithm for the prime field Fp and the finite field Fm with characteristic 2 are introduced below.

If p is a large prime number, then the set of integers {0, l, 2,... P-l} constitutes a finite field, denoted as Fp.

The addition operation of the prime number field Fp is described as follows: Let a, b be the elements in the prime number field Fp; then, the addition result is (a + b) mod p. The additive identity element is 0 elements, and the additive inverse of field element a is (-a) mod p.

The multiplication operation for prime number field Fp is described as follows: Let a, b be elements in prime number field Fp; then, the multiplication result is (ab) mod p.

The multiplicative inverse of a nonzero element a is b, where y satisfies ab = l(mod p).

The set of all bit strings of length m forms a finite field of feature 2.

The feature is the addition operation of 2 finite fields Fm and is described as follows: the XOR operation of two bit strings’ corresponding values is the Fm addition operation. The identity element of the addition operation is the element where each bit is 0.

The multiplication operation characterized by 2 finite fields Fm is described as follows: The multiplication operation is related to the representation of the elements of the field. There are generally two representation methods, normal basis (normal basis) notation and polynomial basis (polynomial basis) notation. The representations of the two bases are not described in detail here.

## 3.4 ECC Elliptic Curve

The study of elliptic curves comes from elliptic integrals.

A right-angled plane on domain K refers to the set {(X,Y)∣X,Y∈K}, denoted as A2(K).

A field K is said to be algebraically closed, assuming that every nth-order polynomial over the field K has exactly n roots in the field K. For every field K, there must be an algebraically closed field.

A cubic homogeneous equation defined over an algebraically closed field is called the Weierstrass equation, and it is expressed as follows:

$${a_1}XYZ - {a_2}Y{Z^2} - {a_3}{X^2}Z - {a_4}Y{Z^{ - 1}} - {a_5}X{Y^{ - 1}}Z=1$$

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A projective curve G(m,n,q) = 0 is called a nonsingular curve if it satisfies the following conditions: G(m,n,l) = 0, G(m,l,q) = 0, and G(1,n, q) = 0 are all nonsingular curves.

Elliptic curves refer to the set of solutions to p2 (k) for a nonsingular Weierstrass equation over this algebraically closed field, that is, they satisfy the following equation:

$${a_1}{X^{ - 1}}Z+{a_2}X{Z^2}+{a_3}XY{Z^2} - {a_4}X{Y^{ - 1}} - {a_5}XY{Z^2} - {a_6}YZ= - 1$$

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The root of the equation is related to the discriminant of the equation, and the discriminant of the Weierstrass equation also reveals the intrinsic relationship between the root and the coefficients of the equation.

The following expression is the discriminant of the elliptic curve equation:

$$\Delta =2{b_2}{b_5} - {b_3}+9{b_6} - 3{b_1}{b^2}_{2}$$

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Let E(K) denote the set of two coordinates on an elliptic curve in the domain plus an infinite point as follows:

$$E(K)=\left[ {(x,y) \to {K^2}\left| {ax - xy - {x^3} - ay=0} \right.} \right] \cup \left[ {(x,y) \to 0} \right]$$

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The point at infinity is represented as O. For some special addition XOR operations, E(K) constitutes an elliptic curve group defined on the field K.

An elliptic curve equation over a finite field F is a nonsingular elliptic curve if and only if the following discriminant is satisfied:

$$\Delta = - 9\left( {2{a^{ - 1}} - 8{b^2}} \right) \ne 0$$

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## 3.5 Encrypted Elliptic Curve in Prime Field Fp

We now have a preliminary understanding of elliptic curves. However, the previously learned elliptic curve is continuous and not suitable for encryption; therefore, we must turn the elliptic curve into discrete points. Below, we introduce an elliptic curve suitable for encryption.

Because no elliptic curve is suitable for encryption, the ECC algorithm corresponds to the discrete mathematical problem of y2 = x3 + ax + b.

We often use Ep(a, b) to represent the ellipse group modulo p, and its element set is a nonnegative integer pair (x, y) less than p.

## 3.6 The main parameters in the elliptic curve encryption algorithm

The following are the conditions for selecting six parameters: Of course, the larger p is, the safer it is, but the larger the value is, the slower the calculation speed. Approximately 200 bits can meet the general security requirements, and 2. p ≠ n×h. The ECC parameter settings are shown in Table 1.

Table 1

Parameter | P_LONG | EN_LONG | BIT_LEN | KEY_LONG |

Name | Bit length of parameter P | Plaintext byte length | Bit length | Private key bit length |

Bit length | 240 | 24 | 1600 | 64 |

## 3.7 Design of Elliptic Curve Encryption System Scheme

The ultimate goal of this project is to achieve an elliptic curve encryption algorithm chip that meets certain performance indicators. The main performance requirements are as follows: The designed elliptic curve encryption chip has a signature speed of at least 50 times per second when the key length is 256 bits under the 0.35 µm process.

The operating mechanism of the elliptic curve system can first abstract its basic functional structure level. The top layer of the system, that is, the interface layer and the application layer, are the interfaces through which the system communicates with the user and the main system. Since different users and systems have different application purposes and performance requirements for elliptic curve systems, different software and hardware implementations can be used for different applications, especially the application layer, which is often implemented in software. For example, when a coprocessor is used, the chip only provides the implementation of the ECC core algorithm, and when the application needs to encrypt and decrypt data, it needs to use the control instructions of the main CPU to access the chip through the application layer and the interface layer. The main goal is to optimize and improve the performance of the elliptic curve core algorithm to the greatest extent possible, so the application layer and the interface layer are not the focus of this project, but the design of this project still accounts for the interface and application layers’ need to expand.

Under the encryption layer is the operation layer. Since this layer is the focus of this design and the operations that are processed in this layer are very complex, the layer can be divided into two sublayer elliptic curves from top to bottom in the specific design. In the elliptic curve operation layer, the related operations on the points on the elliptic curve are mainly completed, such as the point addition operation point on the elliptic curve, the multiplication of the operation point, and the negation of the operation point. The macro of the elliptic curve operation layer is completed in the basic operation layer. The micro-operations of operations, such as addition, subtraction, multiplication, and division, are shown in Fig. 2.

For an elliptic curve over a finite field F m, the operations can be expressed as:

$$T=\left\{ {a,b,G,f\left[ {mx \bullet (a - b)} \right]} \right\}$$

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where parameter m specifies that the finite field F2m parameter f(x) is an irreducible binary polynomial of order m. The parameters a and b represent the form of an elliptic curve over the finite field F2m, and the parameter G (xG, yG) is the base point on the elliptic curve.