CN108268243A - A kind of compositum multiplier based on lookup - Google Patents

Abstract

The invention discloses a compound field multiplication device based on search, which includes an input port, an output port, a controller, a multiplication operation module and an addition operation module. The input ports include port a for input of the first operand a(x) of the composite field GF((2 ⁿ ) ² ), port a for input of the second operand b of the composite field GF((2 ⁿ ) ² ) Port b of (x), port clk for input clock signal t, port q for input irreducible polynomial q(x) of composite field GF((2 ⁿ ) ² ) and port q for input subfield GF(2 The port p of the irreducible polynomial p(x) of ⁿ ); the output port includes the port c for outputting the multiplication result c(x) of the composite field GF((2 ⁿ ) ² ); the controller includes the control input and output ports and A control circuit that schedules connected components to compute multiplications of the composite field GF((2 ⁿ ) ² ); the addition module includes lookup structures for computing GF(2 ⁿ ) additions; the multiplication module includes functions for computing GF(2 ⁿ ) multiplications lookup structure. The invention realizes the multiplication operation of the composite field based on the search, and is more efficient in calculating the multiplication operation on the GF((2 ⁿ ) ² ) than the existing multiplier of the composite field.

Description

A Composite Field Multiplication Device Based on Search

technical field

The invention relates to a device for multiplying two elements of a composite field, in particular to a search-based multiplication device for a composite field.

Background technique

Composite fields are a type of finite fields, also known as compound finite fields. Finite fields, first discovered by Galois, are number fields containing finite elements, and are widely used in communication, security, storage and other fields. Operations on finite fields are called finite field computations, including finite field addition, multiplication, inversion, division, etc.

The commonly used composite field is GF((2 ⁿ ) ² ), the size of the field is (2 ⁿ ) ² , and its subfield is GF(2 ⁿ ). Because the composite domain is GF((2 ⁿ ) ² ) operation includes subfield GF(2 ⁿ ) operation, so the operation efficiency of GF((2 ⁿ ) ² ) can be improved by optimizing the GF(2 ⁿ ) operation. Compound field multiplication is one of the most complex operations in compound fields, and it is the basic operation in cryptography, which plays an important role in cryptosystem and coding technology. The design method of composite field multiplication is generally based on the algebraic method, that is, the multiplication operation is carried out using algebraic theory. Algebra-based multiplication devices are relatively slow in terms of speed of operation.

Contents of the invention

In order to overcome the above-mentioned shortcomings and deficiencies of the prior art, the object of the present invention is to provide a compound field multiplication device based on search, which realizes the multiplication of the compound field by searching, and calculates the multiplication of GF ((2 ⁿ ) ² ) Compared with the existing finite field multiplier, it is more efficient.

The object of the present invention is achieved through the following technical solutions:

A look-up based compound field multiplication apparatus comprising:

Input ports, including port a for input to the first operand a(x) of the composite field GF((2 ⁿ ) ² ), the second operand for input to the composite field GF((2 ⁿ ) ² ) Port b of b(x), port clk for input clock signal t, port q for input irreducible polynomial q(x) of complex field GF((2 ⁿ ) ² ) and port q for input subfield GF( 2 ⁿ ) the port p of the irreducible polynomial p(x);

The output port is used to output the multiplication result c(x) of the composite field GF((2 ⁿ ) ² ), denoted as port c;

An addition operation module, including a search tree structure for calculating the addition of two known elements of GF(2 ⁿ );

A multiplication operation module comprising a search tree structure for calculating the multiplication of two known elements of GF(2 ⁿ );

The controller includes a control circuit for controlling the input and output ports and a control circuit for scheduling the addition operation module and the multiplication operation module to calculate the multiplication of the composite field GF((2 ⁿ ) ² ).

The addition search tree structure includes two search trees, each tree contains n layers, and the top layer is the layer where the root node is located, which is called the 0th layer; then the bottom layer, that is, the layer where the leaf nodes are located is the layer 0. n-1 layer;

The expansion layer is one layer below the leaf nodes of the search tree, and each node of the expansion layer is connected to three leaf nodes;

All tree nodes except leaf nodes have left child nodes and right child nodes;

The left root node and left child node represent the value 0, and the right root node and right child node represent the value 1;

Each path from the root node to a leaf node represents an element of GF(2 ⁿ ).

The calculation process of the addition operation module is as follows:

For the addition h(x)=f(x)+g(x) of two known elements f(x) and g(x) of GF(2 ⁿ ), where,

f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ ,

g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ ,

h(x)=h _n-1 x ^n-1 +h _n-2 x ^n-2 +...+h ₀ ,

f _n-1 ,f _n-2 ,...,f ₀ ,g _n-1 ,g _n-2 ,...,g ₀ ,h _n-1 ,h _n-2 ,...,h ₀ is an element of the finite field GF(2);

If the addition h(x) of GF(2 ⁿ )=f(x)+g(x), and the path from layer 0 to node n _f of layer n-1 represents the element ^f ( x), the path from the 0th layer to the node n _g of the n-1th layer represents the element g(x) of GF(2 ⁿ ), then the nodes n _f and n _g of the n-1th layer and the nodes of the expansion layer n _s are connected; if the path from the 0th layer to the node n _h of the n-1th layer represents the element h(x) of GF(2 ⁿ ), then the node n h of the n-1th layer and the node _n of the expansion layer _s connected;

The steps to calculate h(x)=f(x)+g(x) are as follows:

First, for f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ , determine the path from layer 0 to node n _f of layer n-1 represents the element f(x) of GF(2 ⁿ );

Then, for g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ , the path of node n _g from layer 0 to layer n-1 represents element g(x) of GF(2 ⁿ );

If the nodes n _f and n _g of the n-1th layer are connected to the node n _s of the expansion layer, and the node n _h of the n-1th layer is connected to the node n _s of the expansion layer, then from the 0th layer to the n-th The element of GF(2 ⁿ ) represented by the path of node n _h in layer 1 is h(x)=f(x)+g(x), namely It is the calculation result of h(x)=f(x)+g(x).

The multiplication search tree structure includes two search trees, each tree contains n layers; the top layer is the layer where the root node is located, which is called the 0th layer; the bottom layer, that is, the layer where the leaf nodes are located is the layer 0 n-1 layer;

The expansion layer is one layer below the leaf nodes of the search tree, and each node of the expansion layer is connected to three leaf nodes;

All tree nodes except leaf nodes have left child nodes and right child nodes;

The left root node and left child node represent the value 0, and the right root node and right child node represent the value 1;

Each path from the root node to a leaf node represents an element of GF(2 ⁿ ).

The calculation process of the multiplication module is as follows:

The multiplication of two known elements f(x) and g(x) of GF(2 ⁿ ) h(x)=f(x)×g(x), where,

f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ ,

g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ ,

h(x)=h _n-1 x ^n-1 +h _n-2 x ^n-2 +...+h ₀ ,

f _n-1 ,f _n-2 ,...,f ₀ ,g _n-1 ,g _n-2 ,...,g ₀ ,h _n-1 ,h _n-2 ,...,h ₀ is an element of the finite field GF(2);

If the multiplication h(x) of GF(2 ⁿ )=f(x)×g(x), and the path from layer 0 to node n _f of layer n-1 represents the element ^f ( x), the path from the 0th layer to the node n _g of the n-1th layer represents the element g(x) of GF(2 ⁿ ), then the nodes n _f and n _g of the n-1th layer and the nodes of the expansion layer n _s are connected; if the path from the 0th layer to the node n _h of the n-1th layer represents the element h(x) of GF(2 ⁿ ), then the node n h of the n-1th layer and the node _n of the expansion layer _s connected;

The steps to calculate h(x)=f(x)×g(x) are as follows:

First, for f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ , determine the path from layer 0 to node n _f of layer n-1 represents the element f(x) of GF(2 ⁿ );

Then, for g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ , the path of node n _g from layer 0 to layer n-1 represents element g(x) of GF(2 ⁿ );

If the nodes n _f and n _g of the n-1th layer are connected to the node n _s of the expansion layer, and the node n _h of the n-1th layer is connected to the node n _s of the expansion layer, then from the 0th layer to the n-th The element of GF(2 ⁿ ) represented by the path of node n _h in layer 1 is h(x)=f(x)×g(x), namely It is the calculation result of h(x)=f(x)×g(x).

The polynomial expressions of the first operand a(x), the second operand b(x), and the multiplication result c(x) are respectively:

a(x)=a _h x+a _l ,

b(x)=b _h x+b _l ,

c(x)=c _h x+c _l ,

a _h , a _l , b _h , b _l are elements of finite field GF(2 ⁿ ); c _h , c _l are elements of finite field GF(2 ⁿ );

The coefficient forms of the first operand a(x), the second operand b(x), and the multiplication result c(x) are:

a(x)=a(a _h ,a _l ),

b(x)=b(b _h , b _l ),

c(x)=c(c _h ,c _l ),

a _h , a _l , b _h , b _l are elements of finite field GF(2 ⁿ ); c _h , c _l are elements of finite field GF(2 ⁿ ).

The steps of scheduling the addition module and the multiplication module to calculate the multiplication c(x)=a(x)×b(x) of GF((2 ⁿ ) ² ) are as follows:

Let a(x) = a _h x + a _l and b(x) = b _h x + b _l ,

Wait for the clock signal to turn from low level to high level; in the first clock cycle, call the multiplication module to calculate s ₀ =a _h b _h , s ₀ , a _h , b _h are the elements of the subfield GF(2 ⁿ ); In the second clock cycle, call the multiplication module to calculate s ₁ =a _h b _l , s ₁ , a _h , b _l are the elements of the subfield GF(2 ⁿ ); in the third clock cycle, call the multiplication module to calculate s ₂ = a _l b _h , s ₂ , a _l , b _h are the elements of the subfield GF(2 ⁿ ); in the fourth clock cycle, call the multiplication module to calculate s ₃ = a _l b _l , s ₃ , a _l , b _l is the element of the subfield GF(2 ⁿ ); in the fifth clock cycle, call the multiplication module to calculate s ₄ =s ₀ e, s ₄ , s ₀ , e are the elements of the subfield GF(2 ⁿ ); In the sixth clock cycle, call the addition module to calculate s ₅ =s ₄ +s ₃ , s ₅ , s ₄ , and s ₃ are elements of the subfield GF(2 ⁿ ); in the seventh clock cycle, call the addition module to calculate s ₆ ＝s ₀ +s ₁ , s ₆ , s ₀ , s ₁ are the elements of the subfield GF(2 ⁿ ); in the eighth clock cycle, call the addition operation module to calculate s ₇ =s ₆ +s ₂ , s ₇ , s ₆ , s ₂ are the elements of the subfield GF(2 ⁿ ); let c(x)= _ch x+c _l , _ch = s ₇ , _cl = s ₅ , _ch , c _l are the subfields An element of GF(2 ⁿ ), c(x) is an element of the composite field GF((2 ⁿ ) ² ), c(x) is a(x)=a _h x+a _l and b(x)=b _h The result of the multiplication operation of x+b _l .

The clock signal t is a single-bit signal, which takes a value of 0 or 1, representing a low level or a high level; turning from a low level to a high level represents the beginning of a clock cycle.

Compared with the prior art, the present invention has the following advantages and beneficial effects:

The addition operation module and the multiplication operation module of the present invention include two search trees, and then the controller dispatches the addition operation module and the multiplication operation module to calculate the multiplication of the composite field GF((2 ⁿ ) ² ). The present invention realizes the multiplication operation of the compound field by searching, and the multiplication operation of calculating GF((2 ⁿ ) ² ) can improve the performance compared with the existing finite field multiplier, and the speed of calculating the multiplication of the compound field is faster, which can It is widely used in cryptographic systems and data communications to increase the speed of encryption and decryption as well as the speed of data encoding and decoding.

Description of drawings

FIG. 1 is a schematic structural diagram of a search-based compound field multiplication device according to an embodiment of the present invention.

Figure 2 shows the search tree structure of the addition module as shown in Figure GF(2 ⁴ ).

Figure 3 shows the search tree structure of the multiplication module as shown in Figure GF(2 ⁴ ).

Detailed ways

The present invention will be described in further detail below in conjunction with the examples, but the embodiments of the present invention are not limited thereto.

Example

As shown in Figure 1, the search-based compound field multiplication device includes:

Input ports, including port a for input to the first operand a(x) of the composite field GF((2 ⁿ ) ² ), the second operand for input to the composite field GF((2 ⁿ ) ² ) Port b of b(x), port clk for input clock signal t, port q for input irreducible polynomial q(x) of complex field GF((2 ⁿ ) ² ) and port q for input subfield GF( 2 ⁿ ) the port p of the irreducible polynomial p(x);

The output port is used to output the multiplication result c(x) of the composite field GF((2 ⁿ ) ² ), denoted as port c;

A controller, including a control circuit for controlling the input and output ports and a control circuit for scheduling the addition module and the multiplication module to calculate the multiplication of the composite field GF((2 ⁿ ) ² );

An addition operation module, including a search tree structure for the addition of two known elements of GF(2 ⁿ ).

A multiplication module, including a search tree structure for computing the multiplication of two known elements of GF(2 ⁿ ).

(1) Input port

The input ports include port a for inputting the first operand a(x) of the composite field GF((2 ⁿ ) ² ), the second operand a(x) for inputting the composite field GF((2 ⁿ ) ² ) Port b for number b(x), port clk for input clock signal t, port q for input irreducible polynomial q(x) of composite field GF((2 ⁿ ) ² ) and port q for input subfield GF port p of the irreducible polynomial p(x) of (2 ⁿ );

The first operand a(x) and the second operand b(x) of the input port can be expressed as a polynomial form:

a(x)=a _h x+a _l ,

b(x)=b _h x+b _l ,

a _h , a _l , b _h , b _l are elements of the finite field GF(2 ⁿ ).

The first operand a(x) and the second operand b(x) of the input port can be expressed in the form of coefficients:

a(x)=a(a _h ,a _l ),

b(x)=b(b _h , b _l ),

a _h , a _l , b _h , b _l are elements of the finite field GF(2 ⁿ ).

The clock signal t of the input port is a single-bit signal, and its value is 0 or 1, representing low level or high level; turning from low level to high level represents the beginning of a clock cycle.

The irreducible polynomial q(x) of the composite field GF((2 ⁿ ) ² ) of the input port can be expressed as a polynomial form:

q(x)= ^x2 +x+e,

e is a constant of the finite field GF(2 ⁿ ).

The irreducible polynomial p(x) of the subfield GF(2 ⁿ ) of the input port can be expressed as a polynomial form:

p(x)=x ⁿ +p _n-1 x ^n-1 +p _n-2 x ^n-2 +...+p ₁ x+1,

p _n-1 ,p _n-2 ,...,p ₁ are elements of the finite field GF(2), that is, one of the binary numbers (0) ₂ and (1) ₂ .

(2) Output port

The multiplication result c(x) of the complex field GF((2 ⁿ ) ² ) at the output port can be expressed in the form of coefficients:

c(x)=c(c _h ,c _l ),

c _h , c _l are elements of the finite field GF(2 ⁿ ).

(3) Addition operation module

As shown in Figure 2, the addition operation module includes an addition search tree structure, which is used to calculate the addition of two known elements f(x) and g(x) of GF(2 ⁿ ) h(x)=f( x)+g(x), where,

f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ ,

g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ ,

h(x)=h _n-1 x ^n-1 +h _n-2 x ^n-2 +...+h ₀ ,

f _n-1 ,f _n-2 ,...,f ₀ ,g _n-1 ,g _n-2 ,...,g ₀ ,h _n-1 ,h _n-2 ,...,h ₀ is an element of the finite field GF(2);

Calculating h(x)=f(x)+g(x) uses addition to find the tree structure, described as follows:

The search tree structure contains two search trees, and each tree contains n layers. The top layer, that is, the layer where the root node is located is called the 0th layer, and the bottom layer, that is, the layer where the leaf nodes are located is the nth- 1 story;

The expansion layer is one layer below the leaf nodes of the search tree, and each node of the expansion layer is connected to three leaf nodes;

All tree nodes except leaf nodes have left child nodes and right child nodes;

The left node (left root node and left child node) represents the value 0, and the right node (right root node and right child node) represents the value 1;

Each path from the root node to a leaf node represents a GF(2 ⁿ ) element; for example, starting from the left root node, including the left child node of the left root node, the left child node of the left child node of the left root node Waiting for nodes until the path ending at the leftmost leaf node (the leftmost node of the n-1th layer) represents the element (00...00) ₂ of GF(2 ⁿ );

If the addition h(x) of GF(2 ⁿ )=f(x)+g(x), and the path from layer 0 to node n _f of layer n-1 represents the element ^f ( x), the path from the 0th layer to the node n _g of the n-1th layer represents the element g(x) of GF(2 ⁿ ), then the nodes n _f and n _g of the n-1th layer and the nodes of the expansion layer n _s are connected; if the path from the 0th layer to the node n _h of the n-1th layer represents the element h(x) of GF(2 ⁿ ), then the node n h of the n-1th layer and the node _n of the expansion layer _s connected;

The steps to calculate h(x)=f(x)+g(x) are as follows:

First, for f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ , determine the path from layer 0 to node n _f of layer n-1 represents the element f(x) of GF(2 ⁿ );

Then, for g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ , the path of node n _g from layer 0 to layer n-1 represents element g(x) of GF(2 ⁿ );

If the nodes n _f and n _g of the n-1th layer are connected to the node n _s of the expansion layer, and the node n _h of the n-1th layer is connected to the node n _s of the expansion layer, then from the 0th layer to the n-th The element of GF(2 ⁿ ) represented by the path of node n _h in layer 1 is h(x)=f(x)+g(x), namely It is the calculation result of h(x)=f(x)+g(x).

(4) Multiplication operation module

As shown in Figure 3, the multiplication operation module includes a multiplication search tree structure, which is used to calculate the multiplication of two known elements f(x) and g(x) of GF(2 ⁿ ) h(x)=f( x)×g(x), where,

f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ ,

g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ ,

h(x)=h _n-1 x ^n-1 +h _n-2 x ^n-2 +...+h ₀ ,

f _n-1 ,f _n-2 ,...,f ₀ ,g _n-1 ,g _n-2 ,...,g ₀ ,h _n-1 ,h _n-2 ,...,h ₀ is an element of the finite field GF(2);

Calculate h(x)=f(x)×g(x) using multiplication to find the tree structure, described as follows:

The search tree structure contains two search trees, and each tree contains n layers. The top layer, that is, the layer where the root node is located is called the 0th layer, and the bottom layer, that is, the layer where the leaf nodes are located is the nth- 1 story;

The expansion layer is one layer below the leaf nodes of the search tree, and each node of the expansion layer is connected to three leaf nodes;

All tree nodes except leaf nodes have left child nodes and right child nodes;

The left node (left root node and left child node) represents the value 0, and the right node (right root node and right child node) represents the value 1;

Each path from the root node to a leaf node represents a GF(2 ⁿ ) element; for example, starting from the left root node, including the left child node of the left root node, the left child node of the left child node of the left root node Waiting for nodes until the path ending at the leftmost leaf node (the leftmost node of the n-1th layer) represents the element (00...00) ₂ of GF(2 ⁿ );

If the multiplication h(x) of GF(2 ⁿ )=f(x)×g(x), and the path from layer 0 to node n _f of layer n-1 represents the element ^f ( x), the path from the 0th layer to the node n _g of the n-1th layer represents the element g(x) of GF(2 ⁿ ), then the nodes n _f and n _g of the n-1th layer and the nodes of the expansion layer n _s are connected; if the path from the 0th layer to the node n _h of the n-1th layer represents the element h(x) of GF(2 ⁿ ), then the node n h of the n-1th layer and the node _n of the expansion layer _s connected;

The steps to calculate h(x)=f(x)×g(x) are as follows:

First, for f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ , determine the path from layer 0 to node n _f of layer n-1 represents the element f(x) of GF(2 ⁿ );

Then, for g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ , the path of node n _g from layer 0 to layer n-1 represents element g(x) of GF(2 ⁿ );

If the nodes n _f and n _g of the n-1th layer are connected to the node n _s of the expansion layer, and the node n _h of the n-1th layer is connected to the node n _s of the expansion layer, then from the 0th layer to the n-th The element of GF(2 ⁿ ) represented by the path of node n _h in layer 1 is h(x)=f(x)×g(x), namely It is the calculation result of h(x)=f(x)×g(x).

(5) Controller

The controller is connected to the input port, monitors the input port a, and receives the port a of the first operand a(x) of the composite finite field GF((2 ⁿ ) ² ); monitors the input port b, and receives the composite finite field GF( Port b of the second operand b(x) of (2 ⁿ ) ² ); monitor input port clk, receive clock signal t; monitor input port q, receive irreducible polynomial of composite finite field GF((2 ⁿ ) ² ) q(x); monitor the input port p, and receive the irreducible polynomial p(x) of the subfield GF(2 ⁿ ). The controller is connected to the addition module and the multiplication module, and the controller is connected to the output port, after the multiplication c(x)=a(x)×b(x) of GF((2 ⁿ ) ² ) is calculated , outputs c(x) to output port c.

Let a(x)=a _h x+a _l and b(x)=b _h x+b _l , calculate the multiplication of GF((2 ⁿ ) ² ) c(x)=a(x)×b(x) The steps are as follows:

Wait for the clock signal to change from low level to high level;

In the first clock cycle, call the multiplication module to calculate s ₀ =a _h b _h , s ₀ , a _h , and b _h are the elements of the subfield GF(2 ⁿ );

In the second clock cycle, call the multiplication module to calculate s ₁ =a _h b _l , s ₁ , a _h , b _l are the elements of the subfield GF(2 ⁿ );

In the third clock cycle, call the multiplication module to calculate s ₂ =a _l b _h , where s ₂ , a _l , and b _h are elements of the subfield GF(2 ⁿ );

In the fourth clock cycle, call the multiplication module to calculate s ₃ =al b _l , s ₃ , a _l , b _l are the elements of the subfield GF(2 ⁿ ₎ ;

In the fifth clock cycle, call the multiplication module to calculate s ₄ =s ₀ e, s ₄ , s ₀ , and e are elements of the subfield GF(2 ⁿ );

In the sixth clock cycle, call the addition module to calculate s ₅ =s ₄ +s ₃ , s ₅ , s ₄ , and s ₃ are elements of the subfield GF(2 ⁿ );

In the seventh clock cycle, call the addition module to calculate s ₆ =s ₀ +s ₁ , s ₆ , s ₀ , and s ₁ are elements of the subfield GF(2 ⁿ );

In the eighth clock cycle, call the addition module to calculate s ₇ =s ₆ +s ₂ , s ₇ , s ₆ , and s ₂ are elements of the subfield GF(2 ⁿ );

Let c(x)=ch _x +c _l , c _h =s ₇ , c _l =s ₅ , c _h , c _l are elements of the subfield GF(2 ⁿ ), c(x) is the compound field GF( (2 ⁿ ) ² ), c(x) is the multiplication result of a(x)=a _h x+a _l and b(x)=b _h x+b _l .

In this embodiment, n=4 is taken as an example to illustrate the working process of calculating multiplication in the present invention.

The first operand a(x) of the input port is an element of the compound field GF((2 ⁴ ) ² ), which can be expressed as a polynomial:

a(x)=a _h x+a _l ,

a _h , a _l are the elements of the finite field GF(2 ⁴ );

The second operand b(x) of the input port is an element of the composite field GF((2 ⁴ ) ² ), which can be expressed as a polynomial:

b(x)=b _h x+b _l ,

b _h , b _l are the elements of the finite field GF(2 ⁴ );

The operand c(x) of the output port is an element of the compound field GF((2 ⁴ ) ² ), which can be expressed as a polynomial:

c(x)=c _h x+c _l ,

c _h , c _l are the elements of the finite field GF(2 ⁴ );

The clock signal t of the input port is a single-bit signal, and the clock period is 20 nanoseconds;

The steps for the controller to calculate the multiplication c(x)=a(x)×b(x) of GF((2 ⁴ ) ² ) are as follows:

The operation controller receives and inputs the first operand a(x), the second operand b(x), the clock signal t, and the irreducible polynomial q(x)=x ² +x+9 of GF((2 ⁴ ) ² ) , the irreducible polynomial p(x) of GF(2 ⁴ )=x ⁴ +x+1, waiting for the clock signal t to change from low level to high level (from 0 to 1);

In the first clock cycle, call the multiplication module to calculate s ₀ =a _h b _h , s ₀ , a _h , and b _h are the elements of the subfield GF(2 ⁿ );

In the second clock cycle, call the multiplication module to calculate s ₁ =a _h b _l , s ₁ , a _h , b _l are the elements of the subfield GF(2 ⁿ );

In the third clock cycle, call the multiplication module to calculate s ₂ =a _l b _h , where s ₂ , a _l , and b _h are elements of the subfield GF(2 ⁿ );

In the fourth clock cycle, call the multiplication module to calculate s ₃ =al b _l , s ₃ , a _l , b _l are the elements of the subfield GF(2 ⁿ ₎ ;

In the fifth clock cycle, call the multiplication module to calculate s ₄ =s ₀ e, s ₄ , s ₀ , and e are elements of the subfield GF(2 ⁿ );

In the sixth clock cycle, call the addition module to calculate s ₅ =s ₄ +s ₃ , s ₅ , s ₄ , and s ₃ are elements of the subfield GF(2 ⁿ );

In the seventh clock cycle, call the addition module to calculate s ₆ =s ₀ +s ₁ , s ₆ , s ₀ , and s ₁ are elements of the subfield GF(2 ⁿ );

In the eighth clock cycle, call the addition module to calculate s ₇ =s ₆ +s ₂ , s ₇ , s ₆ , and s ₂ are elements of the subfield GF(2 ⁿ );

Let c(x)=ch _x +c _l , c _h =s ₇ , c _l =s ₅ , c _h , c _l are elements of the subfield GF(2 ⁴ ), c(x) is the compound field GF( (2 ⁴ ) ² ), c(x) is the multiplication result of a(x)=a _h x+a _l and b(x)=b _h x+b _l ; the controller outputs c(x) to output port c.

The above-mentioned embodiment is a preferred embodiment of the present invention, but the embodiment of the present invention is not limited by the embodiment, and any other changes, modifications, substitutions and combinations made without departing from the spirit and principle of the present invention , simplification, all should be equivalent replacement methods, and are all included in the protection scope of the present invention.

Claims (8)

1. A compound field multiplication device based on search, characterized in that, comprising: Input ports, including port a for input to the first operand a(x) of the composite field GF((2 ⁿ ) ² ), the second operand for input to the composite field GF((2 ⁿ ) ² ) Port b of b(x), port clk for input clock signal t, port q for input irreducible polynomial q(x) of complex field GF((2 ⁿ ) ² ) and port q for input subfield GF( 2 ⁿ ) the port p of the irreducible polynomial p(x); The output port is used to output the multiplication result c(x) of the composite field GF((2 ⁿ ) ² ), denoted as port c; An addition operation module, including a search tree structure for calculating the addition of two known elements of GF(2 ⁿ ); A multiplication operation module comprising a search tree structure for calculating the multiplication of two known elements of GF(2 ⁿ ); The controller includes a control circuit for controlling the input and output ports and a control circuit for scheduling the addition operation module and the multiplication operation module to calculate the multiplication of the composite field GF((2 ⁿ ) ² ). 2. The compound field multiplication device based on search according to claim 1, wherein the addition search tree structure comprises two search trees, each tree comprises n layers, and the top layer is the layer where the root node is located , called the 0th layer; then the bottom layer, that is, the layer where the leaf nodes are located is the n-1th layer; The expansion layer is one layer below the leaf nodes of the search tree, and each node of the expansion layer is connected to three leaf nodes; All tree nodes except leaf nodes have left child nodes and right child nodes; The left root node and left child node represent the value 0, and the right root node and right child node represent the value 1; Each path from the root node to a leaf node represents an element of GF(2 ⁿ ). 3. the compound field multiplication device based on searching according to claim 2, is characterized in that, the computing process of described addition operation module is as follows: For the addition h(x)=f(x)+g(x) of two known elements f(x) and g(x) of GF(2 ⁿ ), where, f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ , g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ , h(x)=h _n-1 x ^n-1 +h _n-2 x ^n-2 +...+h ₀ , f _n-1 ,f _n-2 ,...,f ₀ ,g _n-1 ,g _n-2 ,...,g ₀ ,h _n-1 ,h _n-2 ,...,h ₀ is an element of the finite field GF(2); If the addition h(x) of GF(2 ⁿ )=f(x)+g(x), and the path from layer 0 to node n _f of layer n-1 represents the element ^f ( x), the path from the 0th layer to the node n _g of the n-1th layer represents the element g(x) of GF(2 ⁿ ), then the nodes n _f and n _g of the n-1th layer and the nodes of the expansion layer n _s are connected; if the path from the 0th layer to the node n _h of the n-1th layer represents the element h(x) of GF(2 ⁿ ), then the node n h of the n-1th layer and the node _n of the expansion layer _s connected; The steps to calculate h(x)=f(x)+g(x) are as follows: First, for f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ , determine the path from layer 0 to node n _f of layer n-1 represents the element f(x) of GF(2 ⁿ ); Then, for g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ , the path of node n _g from layer 0 to layer n-1 represents element g(x) of GF(2 ⁿ ); If the nodes n _f and n _g of the n-1th layer are connected to the node n _s of the expansion layer, and the node n _h of the n-1th layer is connected to the node n _s of the expansion layer, then from the 0th layer to the n-th The element of GF(2 ⁿ ) represented by the path of node n _h in layer 1 is h(x)=f(x)+g(x), namely It is the calculation result of h(x)=f(x)+g(x). 4. The compound field multiplication device based on search according to claim 1, wherein the multiplication search tree structure comprises two search trees, and each tree comprises n layers; the topmost layer is the layer where the root node is located , called the 0th layer; then the bottom layer, that is, the layer where the leaf nodes are located is the n-1th layer; The expansion layer is one layer below the leaf nodes of the search tree, and each node of the expansion layer is connected to three leaf nodes; All tree nodes except leaf nodes have left child nodes and right child nodes; The left root node and left child node represent the value 0, and the right root node and right child node represent the value 1; Each path from the root node to a leaf node represents an element of GF(2 ⁿ ). 5. The compound domain multiplication device based on search according to claim 4, characterized in that, The calculation process of the multiplication module is as follows: The multiplication of two known elements f(x) and g(x) of GF(2 ⁿ ) h(x)=f(x)×g(x), where, f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ , g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ , h(x)=h _n-1 x ^n-1 +h _n-2 x ^n-2 +...+h ₀ , f _n-1 ,f _n-2 ,...,f ₀ ,g _n-1 ,g _n-2 ,...,g ₀ ,h _n-1 ,h _n-2 ,...,h ₀ is an element of the finite field GF(2); If the multiplication h(x) of GF(2 ⁿ )=f(x)×g(x), and the path from layer 0 to node n _f of layer n-1 represents the element ^f ( x), the path from the 0th layer to the node n _g of the n-1th layer represents the element g(x) of GF(2 ⁿ ), then the nodes n _f and n _g of the n-1th layer and the nodes of the expansion layer n _s are connected; if the path from the 0th layer to the node n _h of the n-1th layer represents the element h(x) of GF(2 ⁿ ), then the node n h of the n-1th layer and the node _n of the expansion layer _s connected; The steps to calculate h(x)=f(x)×g(x) are as follows: First, for f(x)=f _n-1 x ^n-1 +f _n-2 x ^n-2 +...+f ₀ , determine the path from layer 0 to node n _f of layer n-1 represents the element f(x) of GF(2 ⁿ ); Then, for g(x)=g _n-1 x ^n-1 +g _n-2 x ^n-2 +...+g ₀ , the path of node n _g from layer 0 to layer n-1 represents element g(x) of GF(2 ⁿ ); If the nodes n _f and n _g of the n-1th layer are connected to the node n _s of the expansion layer, and the node n _h of the n-1th layer is connected to the node n _s of the expansion layer, then from the 0th layer to the n-th The element of GF(2 ⁿ ) represented by the path of node n _h in layer 1 is h(x)=f(x)×g(x), namely It is the calculation result of h(x)=f(x)×g(x). 6. The compound domain multiplication device based on search according to claim 1, wherein the polynomial of the first operand a(x), the second operand b(x), and the multiplication result c(x) The expressions are: a(x)=a _h x+a _l , b(x)=b _h x+b _l , c(x)=c _h x+c _l , a _h , a _l , b _h , b _l are elements of finite field GF(2 ⁿ ); c _h , c _l are elements of finite field GF(2 ⁿ ); The coefficient forms of the first operand a(x), the second operand b(x), and the multiplication result c(x) are: a(x)=a(a _h ,a _l ), b(x)=b(b _h ,b _l ), c(x)=c(c _h ,c _l ), a _h , a _l , b _h , b _l are elements of finite field GF(2 ⁿ ); c _h , c _l are elements of finite field GF(2 ⁿ ). 7. The ^compound field multiplication device based on search according to claim 6, characterized in that, said scheduling addition module and multiplication module calculate the multiplication c( ^x )=a(x )×b(x) steps are as follows: Let a(x) = a _h x + a _l and b(x) = b _h x + b _l , Wait for the clock signal to turn from low level to high level; in the first clock cycle, call the multiplication module to calculate s ₀ =a _h b _h , s ₀ , a _h , b _h are the elements of the subfield GF(2 ⁿ ); In the second clock cycle, call the multiplication module to calculate s ₁ =a _h b _l , s ₁ , a _h , b _l are the elements of the subfield GF(2 ⁿ ); in the third clock cycle, call the multiplication module to calculate s ₂ = a _l b _h , s ₂ , a _l , b _h are the elements of the subfield GF(2 ⁿ ); in the fourth clock cycle, call the multiplication module to calculate s ₃ = a _l b _l , s ₃ , a _l , b _l is the element of the subfield GF(2 ⁿ ); in the fifth clock cycle, call the multiplication module to calculate s ₄ =s ₀ e, s ₄ , s ₀ , e are the elements of the subfield GF(2 ⁿ ); In the sixth clock cycle, call the addition module to calculate s ₅ =s ₄ +s ₃ , s ₅ , s ₄ , and s ₃ are elements of the subfield GF(2 ⁿ ); in the seventh clock cycle, call the addition module to calculate s ₆ ＝s ₀ +s ₁ , s ₆ , s ₀ , s ₁ are the elements of the subfield GF(2 ⁿ ); in the eighth clock cycle, call the addition operation module to calculate s ₇ =s ₆ +s ₂ , s ₇ , s ₆ , s ₂ are the elements of the subfield GF(2 ⁿ ); let c(x)= _ch x+c _l , _ch = s ₇ , _cl = s ₅ , _ch , c _l are the subfields An element of GF(2 ⁿ ), c(x) is an element of the composite field GF((2 ⁿ ) ² ), c(x) is a(x)=a _h x+a _l and b(x)=b _h The result of the multiplication operation of x+b _l . 8. The compound domain multiplication device based on searching according to claim 1, wherein the clock signal t is a single-bit signal, and the value is 0 or 1, representing low level or high level; low level Going high indicates the beginning of a clock cycle.