RU2058040C1 - Device for multiplication in finite fields - Google Patents

Abstract

FIELD: devices for coding and decoding cycle codes in systems for delivering and processing of digital information. SUBSTANCE: device has unit for shift of factor, high-bit multiplication unit, feedback setting unit, unit for generation of shift value, product shift unit. EFFECT: increased validity of information transmission. 10 dwg

Description

 The invention relates to the construction of encoding and decoding devices of cyclic codes intended for transmitting messages with high reliability in delivery systems and processing of discrete information.

 A device for multiplying, containing adder blocks modulo two, register cells, the first and second groups of multiplication blocks, and the input of the device is connected to the inputs of the multiplication units of the first group, the outputs of which are connected to the first inputs of the adder blocks, respectively, the second inputs of which are connected respectively to the outputs the corresponding multiplication blocks of the second group, the inputs of which, except the last, are interconnected, as well as with the output of the device and the output of the last multiplication block of the second group, the output of each the adder block, except the last one, is connected to the input of the corresponding register cell, the output of each register cell is connected to the third input of the corresponding adder block, the output of the last adder block is connected to the input of the last multiplication block of the second group.

 The disadvantage of this device is that the multiplication is done with great time and in a field with fixed parameters.

 A device for multiplication in finite fields, containing cells of the first, second and third registers, the first and second groups of adders modulo two and the elements OR, elements AND, multiplication blocks, moreover, the first group of outputs of the device is connected respectively to the first inputs of the elements OR of the first group, the outputs of which are connected respectively with the first inputs of the corresponding cells of the first register, the outputs of the cells of the first register, with the exception of the last, are connected respectively with the second inputs of the corresponding elements OR of the first group, the output of the last cell of the first register is connected to the second input of the corresponding OR element of the first group and to the first inputs of AND elements, the outputs of which are connected to the first inputs of the adders of the first group, the outputs of which are connected to the inputs of the cells of the third register, respectively, and the second inputs to the outputs of the cells the third register and with the outputs of the device, respectively, the second inputs of the elements And, except the last, are connected to the outputs of the cells of the second register and with the first inputs of the second adders, respectively, the second the input of the last AND element is connected to the output of the last cell of the second register, with the input of the first OR element of the second group, and with the inputs of the multiplication units, the outputs of which are connected respectively to the second inputs of the corresponding adders of the second group, the outputs of which are connected to the first inputs of the corresponding elements of the second group, the second inputs of which are connected to the inputs of the device of the second group, and the outputs with the inputs of the cells of the second register.

 The disadvantage of this device is that the multiplication in the final fields is a significant investment of time and with a fixed field dimension.

 Of the known multiplication devices in the final fields, the closest to the claimed combination of structural and functional features is the multiplication device, which contains the inverse element calculation unit, the summing unit, the polynomial calculation unit, the multiplexer, and the first and second inputs of the device modes are connected to the control inputs, respectively the inverse element calculation unit and the multiplexer, the first group of information inputs of the device are connected to the group of information inputs of the block the number of the inverse element and the first group of information inputs of the summing unit, and the second group of information inputs of the device is connected to the first group of information inputs of the block of polynomials and the second group of information inputs of the block of summation, the group of information outputs of the block of calculating the inverse element is connected to the second group of information inputs of the block of polynomial calculation , the group of information outputs of the summation unit and the polynomial calculation unit is connected respectively to the first and second th group of information inputs of the multiplexer, the group output of which is connected with a group of output devices.

 However, the known device has the disadvantage that multiplication can only be carried out in a field with a fixed dimension, that is, polynomials with a number of digits of at most some fixed value are multiplied. This determines the disadvantage of the known device, which consists in its limited functionality. In addition, in the known device, the multiplication is carried out sequentially, as the intermediate polynomials are calculated at each cycle, which reduces performance.

 The problem to which the invention is directed is the possibility of multiplying polynomials in a field, the dimension of which can be changed as necessary, as well as obtaining the result of multiplication in the shortest possible time for the purpose of further use of the device in specialized computers, as well as in solving problems of constructing coding and decoding devices.

) блоков вычисления полиномов соединены соответственно с группами информационных входов (i 1)-х блоков вычисления полиномов, дополнительно введены блок сдвига сомножителя, блок задания обратных связей, блок формирования величины сдвига и блок сдвига произведения, причем входы группы первых информационных входов устройства соединены соответственно с информационными входами блока сдвига сомножителя (a_e k)-e (k

) входы группы вторых информационных входов устройства соединены с информационными входами (k)-х блоков вычисления полиномов, а а_е-й вход группы вторых информационных входов устройства соединен с информационным входом блока сдвига сомножителя, группа управляющих входов устройства соединена с группами входов блока задания обратных связей и блока формирования величины сдвига, группы выходов блока формирования величины сдвига соединена с группами управляющих входов блока сдвига сомножителя и блока сдвига произведения, группа выходов блока задания обратных связей соединена с группами управляющих входов (а_е 1) блоков вычисления полиномов, у которых группы первых информационных входов соединены с группой выходов блока сдвига сомножителя и группой информационных входов блока умножения на старший разряд, группа выходов (а_е 1)-го блока вычисления полинома соединена с группой информационных входов блока сдвига произведения, группа выходов которого соединена с группой выходов устройства для умножения в конечных полях.To achieve a technical result, which consists in expanding the functionality of a multiplication device in finite fields due to the multiplication of polynomials in a field, the dimension of which can be changed, and improving performance, it is proposed to use a multiplication device in final fields containing a block of multiplication by senior digit and "a _e 1 "blocks for computing polynomials, and the group of outputs of the block of multiplication by the senior bit is connected to the group of second information inputs of the first block for computing polynomials, and the group second information inputs ix (i

) polynomial calculation blocks are connected respectively to groups of information inputs of (i 1) -th polynomial calculation blocks, a factor shifter block, a feedback setting block, a shift value generation block, and a product shift block are additionally introduced, and the inputs of the group of the device’s first information inputs are connected respectively to information inputs of the block shift factor (a _e k) -e (k

) Inputs of the second group of information inputs devices connected to data inputs (k) -x polynomial evaluation blocks, and a _G th group of second input information input apparatus connected to the data input of the shift factor unit, the group control device is connected to the input unit inputs the job groups inverse connections and a unit for generating a shear value, a group of outputs of a unit for generating a shear unit is connected to groups of control inputs of a unit for shifting a factor and a unit for shifting a product, a group of outputs b eye job feedbacks groups connected to the control inputs (a 1 _e) calculating blocks polynomials in which the first group of information inputs are connected to the group output unit shift factor and the group of information inputs to multiplier MSB, the group of outputs _(AU 1) -th the polynomial calculation unit is connected to the group of information inputs of the product shift unit, the output group of which is connected to the group of outputs of the device for multiplication in the final fields.

 The factor shift block contains groups of AND and OR elements, the first information input connected to the first inputs of the first, sixth and tenth elements AND, the second information input connected to the first inputs of the second, seventh and eleventh elements AND, the third information input connected to the first inputs of the third, of the eighth and twelfth elements AND, the fourth information input is connected to the first inputs of the fourth, ninth elements And, the fifth information input is connected to the first input of the fifth element And, the first the first input is connected to the second inputs of the tenth, eleventh and twelfth elements And, the second control input is connected to the second inputs of the sixth, seventh, eighth and ninth elements And, the third control input is connected to the second inputs of the first, second, third, fourth and fifth elements And, the outputs of the second, third, fourth and fifth AND elements are connected respectively to the first inputs of the first, second, third and fourth OR elements, the outputs of the sixth, seventh, eighth and ninth AND elements are connected respectively to torymi OR input group of elements, the outputs of the tenth, eleventh and twelfth AND gates respectively connected to the third input element or an output of first AND element and OR element outputs are connected respectively to the outputs of the group unit outputs.

 The product shift block contains groups of AND or OR elements, the first information input connected to the first input of the first element AND of the group, the second information input connected to the first inputs of the second and sixth AND elements, the third information input connected to the first inputs of the third, seventh and tenth AND , the fourth information input is connected to the first inputs of the fourth, eighth and eleventh elements And, the fifth information input is connected to the first inputs of the fifth, ninth and twelfth elements And, the first the control input is connected to the second inputs of the tenth, eleventh and twelfth elements And, the second control input is connected to the second inputs of the sixth, seventh, eighth and ninth elements And, the third control input is connected to the second inputs of the first, second, third, fourth and fifth elements And, the outputs of the first, second, third and fourth AND elements are connected respectively to the first inputs of the corresponding elements of the OR group, the outputs of the sixth, seventh, eighth and ninth AND elements are connected respectively to the second bubbled inputs of the corresponding element or group, the outputs of the tenth, eleventh and twelfth AND gates respectively connected to third inputs of the first, second and third elements or the group yields a group of elements OR and the output of the fifth AND gate connected respectively to the outputs of groups of multipliers outputs in finite fields.

) входы группы вторых информационных входов устройства соединены с информационными входами (k)-х блоков вычисления полиномов, а а_е-й вход группы вторых информационных входов устройства соединен с информационным входом блока сдвига сомножителя, группа управляющих входов устройства соединена с группами входов блока задания обратных связей и блока формирования величины сдвига, группы выходов блока формирования величины сдвига соединена с группами управляющих входов блока сдвига сомножителя и блока сдвига произведения, группа выходов блока задания обратных связей соединена с группами управляющих входов (а_е 1) блоков вычисления полиномов, у которых группы первых информационных входов соединены с группой выходов блока сдвига сомножителя и с группой информационных входов блока умножения на старший разряд, группа входов (а_е 1)-го блока вычисления полинома соединена с группой информационных входов блока сдвига произведения, группа выходов которого соединена с группой выходов устройства для умножения в конечных полях, обуславливает соответствие заявляемого технического решения критерию "новизна".The presence of features that distinguish the claimed technical solution from the prototype, namely: a block of shifting factors, a block for setting feedbacks, a block for generating a value for a shift, and a block for shifting products, the inputs of the group of the first information inputs of the device being connected respectively to the information inputs of the block for shifting the factor, (a _e k) -e (k

) Inputs of the second group of information inputs devices connected to data inputs (k) -x polynomial evaluation blocks, and a _G th group of second input information input apparatus connected to the data input of the shift factor unit, the group control device is connected to the input unit inputs the job groups inverse connections and a unit for generating a shear value, a group of outputs of a unit for generating a shear unit is connected to groups of control inputs of a unit for shifting a factor and a unit for shifting a product, a group of outputs b eye job feedback coupled to control inputs groups _(ae 1) blocks calculate polynomials whose group of first data inputs connected to a group of shift unit output factor and a group of information inputs multiplier for the MSB, the group of inputs _(ae 1) - of the polynomial computing unit is connected to the group of information inputs of the product shift unit, the group of outputs of which is connected to the group of outputs of the device for multiplication in the final fields, determines the conformity of the claimed technical Addressing the criterion of "novelty."

 To prove the presence of a causal relationship between the totality of the essential features of the claimed invention and the achieved technical result, we turn to the following theoretical premises.

возможно осуществить за один такт времени, если реализовать устройство, содержащее а ступеней, срабатывающих одновременно, причем на первой ступени будет получен результат S⁽¹⁾ g_a-1p(x). На второй ступени будет получен результат
S⁽²⁾(x) (g_a-1x + g_a-2)p(x) + b_a-2F(x), где b_a-2 первый коэффициент частного от деления g(x)p(x)/F(x). На i-й ступени будет получен результат
S⁽ⁱ⁾(x) (g_a-1x^a-1 + g_a-2x^a-2 + + g_a-i)P(x) +
+ (b_a-2x^a-2 + b_a-3x^a-3 + + b_a-i)F(x), где b_a-i (i 1)-й коэффициент частного. На а-й ступени будет получен результат умножения
r(x) S^(a)(x) (g_a-1x^a-1 + g_a-2x^a-2 + +
+ g_o)p(x) + (b_a-2x^a-2 + b_a-3x^a-3 + +
+ b_o)F(x) g(x)p(x) + b(x)F(x).In the Galois field GF (2), the multiplication of two polynomials
p (x) p _a-1 x ^a-1 + p _a-2 x ^a-2 +. + p ₁ x + p ₀ , and g (x)
g _a-1 x ^a-1 + g _a-2 x ^a-2 +. + g ₁ x + g ₀ , p _i g _i ∈ (0,1), i, j =

it is possible to carry out in one time step, if you implement a device containing a steps that are triggered simultaneously, and at the first stage, the result will be obtained S ⁽¹⁾ g _a-1 p (x). In the second stage, the result will be obtained.
S ⁽²⁾ (x) (g _a-1 x + g _a-2 ) p (x) + b _a-2 F (x), where b _{a-2 is the} first coefficient of the quotient of the division g (x) p (x ) / F (x). At the i-th stage, the result will be obtained
S ⁽ⁱ⁾ (x) (g _a-1 x ^a-1 + g _a-2 x ^a-2 + + g _ai ) P (x) +
+ (b _a-2 x ^a-2 + b _a-3 x ^a-3 + + b _ai ) F (x), where b _ai (i 1) is the quotient of the quotient. At the ith stage, the result of multiplication will be obtained
r (x) S ^(a) (x) (g _a-1 x ^a-1 + g _a-2 x ^a-2 + +
+ g _o ) p (x) + (b _a-2 x ^a-2 + b _a-3 x ^a-3 + +
+ b _o ) F (x) g (x) p (x) + b (x) F (x).

 The invention is implemented by computer technology. One of the possible options is proposed in this description.

 In FIG. 1 shows a functional diagram of the proposed device; in FIG. 2 is a functional diagram for an example implementation of a shift block of a factor; in FIG. 3 and 4 are functional diagrams of a block multiplying by a high order and a block of computing a polynomial, respectively; in FIG. 5 and 6 are functional diagrams for an example implementation of a feedback task unit and a shift value generating unit, respectively; in FIG. 7 is a functional diagram for an example implementation of a product shift block; in FIG. 8-10 is a functional diagram of a device for an example of implementing multiplication with a given field width.

- группу первых информационных входов; 2 блок сдвига сомножителя; 3₁-3

группу вторых информационных входов; 4 блок умножения на старший разряд; 5₁-5

блоки вычисления полиномов; 6₁-6_е группу управляющих входов; 7 блок задания обратных связей; 8 блок формирования величины сдвига; 9 блок сдвига произведения; 10 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -10

(i

) (a_e 1) групп информационных выходов блоков вычисления полиномов, 5₁-5

; 11₁-11

группу выходов устройства для умножения в конечных полях.Functional diagram of the inventive device (see Fig. 1) contains 1 ₁ -1

- a group of first information inputs; 2 block shift factor; 3 ₁ -3

a group of second information inputs; 4 block multiplication by senior level; 5 ₁ -5

polynomial calculation blocks; 6 ₁ -6 _e group of control inputs; 7 feedback job block; 8 unit for the formation of shear values; 9 block shift product; ten $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -10

(i

) (a _e 1) groups of information outputs of blocks for computing polynomials, 5 ₁ -5

; 11 ₁ -11

group of outputs of the device for multiplication in the final fields.

Functional diagram for an example implementation of the shift block of the factor 2 (see Fig. 2) contains: 1 ₁ -1 ₅ group of information inputs; 12 ₁ -12 ₁₂ group of elements And; 13 ₁ -13 ₃ group of control inputs; 14 ₁ -14 ₄ group of elements OR; 15 ₁ -15 ₅ group of outputs of block 2.

информационный вход; 15₁-15

группу информационных входов; 16₁-16

группу элементов И; 10 $\genfrac{}{}{0ex}{}{\text{0}}{\text{1}}$ -10

группу выходов блока 4.Functional diagram of the block of multiplication by the senior digit 4 (see Fig. 3) contains: 3

information input; 15 ₁ -15

group of information inputs; 16 ₁ -16

a group of elements And; ten $\genfrac{}{}{0ex}{}{\text{0}}{\text{1}}$ -10

group of outputs of block 4.

) (см. фиг. 4) содержит: 3_i информационный вход; 10 $\genfrac{}{}{0ex}{}{\text{i-1}}{\text{1}}$ -10

группу вторых информационных входов; 15₁-15

группу первых информационных входов; 18 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -18

первую группу элементов И; 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -19

первую группу сумматоров по модулю два; 20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -20

вторую группу сумматоров по модулю два; 21 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -21

вторую группу элементов И; 10 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -10

группу выходов блока 5.Functional block diagram for computing the polynomial 5 _i (i

) (see Fig. 4) contains: 3 _i information input; ten $\genfrac{}{}{0ex}{}{\text{i-1}}{\text{1}}$ -10

a group of second information inputs; 15 ₁ -15

a group of first information inputs; eighteen $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -eighteen

the first group of elements And; 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -19

the first group of adders modulo two; 20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -20

the second group of adders modulo two; 21 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -21

the second group of elements And; ten $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -10

group of outputs of block 5.

The functional diagram for an example implementation of the feedback task block 7 (see Fig. 5) contains: 6 ₁ -6 ₇ group of control inputs; 23 ₁ -23 ₄ group of elements OR; 24 element NOT; 22 ₁ -22 ₅ group of outputs of block 7.

Functional diagram for an example implementation of a unit for forming a shift value 8 (see Fig. 6) contains: 6 ₁ -6 ₇ group of control inputs; 25 ₁ -25 ₃ group of elements OR; 13 ₁ -13 ₃ group of outputs of block 8.

Functional diagram for an example implementation of the shift unit of product 9 (see Fig. 7) contains: 10 ₁ -10 _{5 a} group of information inputs; 13 ₁ -13 ₃ group of control inputs; 26 ₁ -26 ₁₂ group of elements And; 27 ₁ -27 ₄ group of elements OR; 11 ₁ -11 ₅ group of outputs.

) группу выходов блока вычисления полинома 5_i; 13₁-13₃ группу выходов блока формирования величины сдвига 8; 15₁-15₁₅ группу выходов блока сдвига сомножителя; 16₁-16₅ группу элементов И блока умножения на старший разряд; 18 $\genfrac{}{}{0ex}{}{\text{j}}{\text{1}}$ -18 $\genfrac{}{}{0ex}{}{\text{j}}{\text{5}}$ j-ю (j

) группу первых элементов И; 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{5}}$ i-ю (i

) группу первых сумматоров по модулю два; 20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{4}}$ i-ю (i

) группу вторых сумматоров по модулю два; 21 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{i}}{\text{5}}$ i-ю (i

) группу вторых элементов И; 11₁-11₅ группу информационных выходов устройства.Functional diagram (implementation example) of a device for multiplication in finite fields (see Fig. 8-10) contains: 1 ₁ -1 ₅ group of first information inputs; 2 block shift factor; 3 ₁ -3 ₅ group of second information inputs; 6 ₁ -6 ₇ group of control inputs of the device; 7 block for setting the dimension of the field; 8 unit for the formation of shear values; 9 block shift product; ten $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -10 $\genfrac{}{}{0ex}{}{\text{i}}{\text{5}}$ -i-th (i

) the group of outputs of the block for computing the polynomial 5 _i ; 13 ₁ -13 ₃ group of outputs of the unit for forming the magnitude of the shift 8; 15 ₁ -15 ₁₅ group of outputs of the block shift factor; 16 ₁ -16 ₅ group of elements AND block of multiplication by the senior digit; eighteen $\genfrac{}{}{0ex}{}{\text{j}}{\text{1}}$ -eighteen $\genfrac{}{}{0ex}{}{\text{j}}{\text{5}}$ j-th (j

) the group of the first elements And; 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{5}}$ i-th (i

) the group of the first adders modulo two; 20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{4}}$ i-th (i

) a group of second adders modulo two; 21 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{i}}{\text{5}}$ i-th (i

) a group of second elements And; 11 ₁ -11 ₅ group of information outputs of the device.

 The elements of the multiplication device in the final fields are interconnected as follows.

группы первых информационных входов соединены с соответствующими входами группы информационных входов блока сдвига сомножителя 2, а i-й вход группы вторых информационных входов 3₁-3

соединен с информационным входом блока умножения на старший разряд 4, (а_е j)-й вход группы вторых информационных входов 3₁-3

соединен с информационным входом блока вычисления полинома 5_j, входы 6₁-6_е группы управляющих входов соединены соответственно с группой входов блока задания обратных связей 7 и блока формирования величины сдвига 8, группа выходов блока формирования величины сдвига 8 соединена соответственно с группами информационных входов блока сдвига сомножителя 2 и блока сдвига произведения 9, группа выходов блока сдвига сомножителя 2 соединена с группой информационных входов блока умножения на старший разряд 4 и с группами первых информационных входов блоков вычисления полиномов 5₁-5

, входы группы вторых информационных входов блока вычисления полинома 5₁ соединены соответственно с выходами блока умножения на старший разряд 4, группа выходов блока задания обратных связей 7 соединена с группами управляющих входов блоков вычисления полиномов 5₁-5

, группы информационных выходов 10 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -10

i-х блоков вычисления полиномов 5_i (i

) соединены соответственно с группами вторых информационных входов (i + 1)-х блоков вычисления полиномов 5_i, а группа информационных выходов блока вычисления полинома 5

соединена с группой информационных входов блока сдвига произведения 9, выходы которого соединены соответственно с выходами 11₁-11

группы выходов устройства.Inputs 1 ₁ -1

the groups of first information inputs are connected to the corresponding inputs of the group of information inputs of the shift block of the factor 2, and the i-th input of the group of second information inputs 3 ₁ -3

connected to the information input of the block of multiplication by the senior digit 4, (a _e j) -th input of the group of second information inputs 3 ₁ -3

connected to the information input of the polynomial calculation unit 5 _j , inputs 6 ₁ -6 _{e of the} group of control inputs are connected respectively to the group of inputs of the feedback task unit 7 and the shear value generating unit 8, the group of outputs of the shear value generating unit 8 is connected respectively to the information input groups of the block the shift of factor 2 and the shift block of product 9, the group of outputs of the shift block of the factor 2 is connected to the group of information inputs of the multiplication block by the senior digit 4 and to the groups of the first information input s of blocks for computing polynomials 5 ₁ -5

, the inputs of the group of second information inputs of the block for computing the polynomial 5 _{1 are} connected respectively to the outputs of the block of multiplication by the senior bit 4, the group of outputs of the block for setting the feedback 7 is connected to the groups of control inputs of the blocks for computing the polynomials 5 ₁ -5

, groups of information outputs 10 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -10

i-blocks of computing polynomials 5 _i (i

) are connected respectively with groups of second information inputs of (i + 1) -th blocks of computing polynomials 5 _i , and a group of information outputs of a block of computing polynomial 5

connected to a group of information inputs of the shift unit of product 9, the outputs of which are connected respectively with outputs 11 ₁ -11

groups of device outputs.

In the shift block of the factor 2 (implementation example), input 1 _{1 is} connected to the first inputs of the elements And 12 ₁ , 12 ₆ , 12 ₁₀ groups, input 1 _{2 is} connected to the first inputs of the elements And 12 ₂ , 12 ₇ , 12 ₁₁ groups, input 1 ₃ connected to the first inputs of the elements 12 ₃ , 12 ₈ , 12 ₁₂ groups, input 1 ₄ connected to the first inputs of the elements And 12 ₄ , 12 ₉ groups, input 1 ₅ connected to the first input of the element And 12 ₅ , input 13 ₁ groups of control inputs of the block coupled to second inputs of AND gates 12 _{12 10} -12 group ₂ groups input 13 of control unit with inputs connected to second inputs of AND gates 12 ₆ to 12 ₉ group input 13 ₃ g uppy control unit inputs connected to second inputs of AND gates ₁₂ January ₅ -12 group elements and outputs February ₁₂ -12 ₅ are respectively connected to first inputs of OR element group January ₁₄ -14 _4, elements outputs and 12 _{6 to} 12 ₉ are connected respectively to the second inputs of the group of elements OR 14 ₁ -14 ₄ , the outputs of the elements AND 12 ₁₀ -12 _{12 are} connected respectively with the third inputs of the elements OR 14 ₂ -14 _{4 of the} group, the output of the element AND 12 ₁ and the outputs of the group of elements OR 14 ₁ -14 _{4 are} connected respectively with outputs 15 ₁ -15 ₅ groups of outputs of block 2.

группы информационных входов блока соединены соответственно с первыми входами группы элементов И 16₁-16

, вторые входы которых объединены и соединены с информационным входом 3

блока, выходы группы элементов И 16₁-16

соединены соответственно с выходами 10 $\genfrac{}{}{0ex}{}{\text{0}}{\text{1}}$ -10

группы выходов блока 4.In the block of multiplication by the senior digit 4 inputs 15 ₁ -15

the group of information inputs of the block are connected respectively to the first inputs of the group of elements And 16 ₁ -16

the second inputs of which are combined and connected to the information input 3

block, the outputs of the group of elements And 16 ₁ -16

connected respectively to outputs 10 $\genfrac{}{}{0ex}{}{\text{0}}{\text{1}}$ -10

output groups of block 4.

) вход 15_j (j

) группы первых информационных входов соединен с первым входом элемента И 18 $\genfrac{}{}{0ex}{}{\text{i}}{\text{j}}$ первой группы, вторые входы которых объединены и соединены с информационным входом 3_i блока, а выход элемента И 18 $\genfrac{}{}{0ex}{}{\text{i}}{\text{j}}$ соединен с первым входом соответствующего сумматора по модулю два 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{j}}$ первой группы, входы 10 $\genfrac{}{}{0ex}{}{\text{i-1}}{\text{1}}$ -10

группы вторых информационных входов соединены с вторыми входами соответствующих сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -20

второй группы, первые входы элементов И 21 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -21

второй группы объединены и соединены с входом 10

группы вторых информационных входов, входы 21₁-22

группы управляющих входов соединены соответственно с вторыми входами элементов И 21 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -21

второй группы, выходы которых соединены соответственно со вторыми входами сумматоров по модулю два 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -19

первой группы, а выходы сумматоров по модулю два 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{2}}$ -19

первой группы соответственно соединены с первыми входами сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -20

второй группы, выход сумматора по модулю два 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ первой группы и выходы сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -20

второй группы соединены соответственно с выходами 10 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -10

блока вычисления полинома 5_i.In each block of computing the polynomial 5 _i (i

) input 15 _j (j

) the group of the first information inputs is connected to the first input of the element And 18 $\genfrac{}{}{0ex}{}{\text{i}}{\text{j}}$ the first group, the second inputs of which are combined and connected to the information input 3 _{i of the} block, and the output of the element And 18 $\genfrac{}{}{0ex}{}{\text{i}}{\text{j}}$ connected to the first input of the corresponding adder modulo two 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{j}}$ first group, inputs 10 $\genfrac{}{}{0ex}{}{\text{i-1}}{\text{1}}$ -10

groups of second information inputs are connected to the second inputs of the respective adders modulo two 20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -20

second group, the first inputs of the elements And 21 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -21

second group combined and connected to input 10

groups of second information inputs, inputs 21 ₁ -22

groups of control inputs are connected respectively to the second inputs of the elements And 21 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -21

the second group, the outputs of which are connected respectively with the second inputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -19

the first group, and the outputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{2}}$ -19

the first group are respectively connected to the first inputs of the adders modulo two 20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -20

second group, the output of the adder modulo two 19 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ the first group and the outputs of the adders modulo two 20 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -20

the second group are connected respectively with outputs 10 $\genfrac{}{}{0ex}{}{\text{i}}{\text{1}}$ -10

block computing the polynomial 5 _i .

In the feedback setting block 7 (implementation example), input 6 _{1 is} connected to the first input of the third OR 23 ₃ element, input 6 _{2 is} connected to the first input of the fourth OR 23 ₄ element, input 6 _{3 is} connected to the first input of the second OR 23 ₂ element, input 6 _{4 is} connected to the second input of the second OR element 23 ₂ , with the input of the element NOT 24 and to the second input of the fourth OR element 23 ₄ , input 6 _{5 is} connected to the first input of the first OR element 23 ₁ , input 6 _{6 is} connected to the second input of the first OR element 23 ₁ , with the second input of the third element OR 23 ₃ and with the third input of the fourth element _4-coagulant or 23, input 6 ₇ is connected to a third input of the first OR gate January _23, with a third input of the second OR gate ₂₃ February and to the fourth input of the fourth element ₄ or 23, OR element outputs January ₂₃ -23 ₄ and an output of NOT 24, respectively connected to outputs 22 ₁ -22 _{5 of} block 7.

) соединен с выходом 13_i блока 8.In the unit for generating the shift value 8 (implementation example), inputs 6 ₁ and 6 _{2 are} connected respectively to the first and second inputs of the first OR element 25 ₁ , inputs 6 ₃ and 6 _{4 are} respectively connected to the first and second inputs of the second element OR 25 ₂ , inputs 6 ₅ -6 _{7 are} respectively connected to the first, second and third inputs of the third OR element 25 ₃ , the output of the OR element 25 _i (i

) is connected to the output 13 _{i of} block 8.

In the shift unit of product 9 (implementation example), input 10 _{1 is} connected to the first input _of group I element 26 ₁ , input 10 _{2 is} connected to the first inputs of group elements I 26 ₂ and 26 ₆ , input 10 _{3 is} connected to the first inputs of elements I 26 ₃ , 26 ₇ and 26 ₁₀ groups, input 10 ₄ connected to the first inputs of elements And 26 ₄ , 26 ₈ and 26 ₁₁ groups, input 10 ₅ connected to the first inputs of elements And 26 ₅ , 26 ₉ and 26 ₁₂ groups, input 13 ₁ groups of control input unit connected to the second inputs of aND gates _12, October ₂₆ -26 group ₂ groups while 13 control unit with inputs connected to second inputs of aND gates ₂₆ June ₉ -26 groups , Input 13 of control unit ₃ groups of inputs coupled to second inputs of AND gates ₂₆ January ₅ -26 group elements and outputs January _{26 4} -26 are respectively connected to first inputs of OR element group January ₂₇ -27 _4, the outputs of AND gates ₂₆ June -26 _{9 are} connected respectively to the second inputs of the group of elements OR 27 ₁ -27 ₄ , the outputs of the elements AND 26 ₁₀ -26 _{12 are} connected respectively to the third inputs of the elements of OR 27 ₁ -27 ₃ groups, the outputs of the group of elements OR 27 ₁ -27 ₄ and the output of the element AND 26 are connected respectively to the outputs 11 ₁ -11 _{5 of the} group of outputs of the device for multiplication in the final fields.

 The device for multiplication in the final fields works as follows.

) (i

), определенном полиномом F_i(x) (i

) степени a_i с коэффициентом из поля GF(2)- т.е.When describing the operation of the device in the GF field (2

) (i

) defined by the polynomial F _i (x) (i

) of degree a _i with a coefficient from the field GF (2) - i.e.

x

, F_ij∈GF(2), i=

,j=

,F_io=1, каждый элемент поля представляют в виде полинома над GF(2), степень которого меньше а_i, т.е. вместо элементов p,q,r∈GF(2

) рассматривают полиномы
p_i(x)

p_ijx^j, p_ij∈ GF(2), i=

, j=

,
q_i(x)

q_ijx^j, q_ij∈ GF(2), i=

, j=

,
r_i(x)

r_ijx^j, r_ij∈ GF(2), i=

, j=

.F _i (x) F _io + F _i1 x +. + F

x

, F _ij ∈GF (2), i =

, j =

, F _io = 1, each element of the field is represented as a polynomial over GF (2), the degree of which is less than a _i , i.e. instead of elements p, q, r∈GF (2

) consider polynomials
p _i (x)

p _ij x ^j , p _ij ∈ GF (2), i =

, j =

,
q _i (x)

q _ij x ^j , q _ij ∈ GF (2), i =

, j =

,
r _i (x)

r _ij x ^j , r _ij ∈ GF (2), i =

, j =

.

Then the multiplication of the elements of GF (2 ^ai ), i.e. p _i q _i r _i , is performed according to the rules of multiplication of the polynomials representing these elements modulo F _i (x), i.e.

r _i (x) p _i (x) q _i (x) + b _i (x) F _i (x), (1) where b _i (x) is a polynomial of degree less than a _i 1.

упорядочены в порядке неубывания индексов, т.е.In the future, for definiteness, we assume that the quantities a _i , i

ordered in non-decreasing order of indices, i.e.

a ₁ ≅ a ₂ ≅ ≅ a _e .

To calculate the polynomial r _e (x), we can use the recurrence equation S $\genfrac{}{}{0ex}{}{\text{(k)}}{\text{e}}$ (x) q _l1ae-k-1 p _e (x) + S $\genfrac{}{}{0ex}{}{\text{(k-1}}{\text{e}}$ ) (x) x + b _l1aek-1F (x), k 1, 2, (2) where b _l1aek-1 q _l1ae-k p _l1ae-1 S $\genfrac{}{}{0ex}{}{\text{(k-1)}}{\text{l1ae-1}}$ ,
S $\genfrac{}{}{0ex}{}{\text{(o)}}{\text{e}}$ (x) q $\genfrac{}{}{0ex}{}{\text{p}}{\text{lae-ke}}$ (x)
Then
r _e (x) S $\genfrac{}{}{0ex}{}{\text{(ae-1) (}}{\text{e}}$ x).

(x) q_i(x)

(x)+b_i(x)

(x),
где

(x) r_i(x)x

,

(x) p_i(x)x

,

(x) F_i(x)x

Поскольку степень полинома

(x) равна ае, то для вычисления

(x) можно использовать рекурентное уравнение, аналогичное уравнению (2), т.е.The polynomial r _i (x) in the case a _i <a _e can be calculated by transforming relation (1) as follows

(x) q _i (x)

(x) + b _i (x)

(x)
Where

(x) r _i (x) x

,

(x) p _i (x) x

,

(x) F _i (x) x

Since the degree of the polynomial

(x) is equal to ae, then to calculate

(x) a recurrence equation similar to equation (2) can be used, i.e.

(x) q

(x)+

(x)x+b

(x), k=1,2, (3)
где

(x) q

(x).

(x) q

(x) +

(x) x + b

(x), k = 1,2, (3)
Where

(x) q

(x).

(x)

(x).Then

(x)

(x).

(x) на х ^ае^-ai
На a_i, i

входов группы первых информационных входов 1₁-1

устройства в параллельном коде подаются коэффициенты полинома p_i(x). Аналогичным образом на a_i, i

входов группы вторых информационных входов 3₁-3

устройства подаются коэффициенты полинома q_i(x).To obtain r _i (x) it is enough to divide

(x) for x ^and f ^-a i
On a _i , i

inputs of the group of first information inputs 1 ₁ -1

devices in parallel code are given coefficients of the polynomial p _i (x). Similarly, on a _i , i

inputs of the group of second information inputs 3 ₁ -3

devices are given coefficients of the polynomial q _i (x).

Using the information supplied from the outputs of the unit for generating the shift value 8 to the control inputs of the shift unit of the factor 2, the coefficient vector of the polynomial p _i (x) is shifted by (a _e a _i ) bits towards the higher bits.

(x) p_i(x)х ^ае^-ai и с а_е-го входа группы вторых информационных входов устройства коэффициент a_i,ae-1. На выходах блока умножения на старший разряд 4 будет сформирован результат вектор коэффициентов полинома
S $\genfrac{}{}{0ex}{}{\text{(o)}}{\text{i}}$ (x) q

(x).The inputs of the block of multiplication by the senior digit 4 is fed from the outputs of the block shift of the factor 2 vector of coefficients of polynomial

(x) p _i (x) x ^a f ^-a i and a _G th group of second information input unit inputs the coefficient a _{i, ae-1.} At the outputs of the block of multiplication by senior digit 4, the result will be the vector of coefficients of the polynomial
S $\genfrac{}{}{0ex}{}{\text{(o)}}{\text{i}}$ (x) q

(x).

) подается вектор коэффициентов полинома

(x), на вторые информационные входы подается вектор коэффициентов полинома S $\genfrac{}{}{0ex}{}{\text{(k-1)}}{\text{i}}$ (x), на информационный вход подается с (a_e k)-го входа группы вторых информационных входов 3₁-3

устройства коэффициент q_i, на управляющие входы подается код с выходов блока задания обратных связей 7. На выходах 10 $\genfrac{}{}{0ex}{}{\text{k}}{\text{1}}$ -10

блока вычисления полинома 5_k будет сформирован результат вектор коэффициентов полинома

(x) q

(x)+

(x)x+b

(x).At the first information inputs of the block for computing the polynomial 5 _k (k

) the vector of coefficients of the polynomial

(x), the vector of coefficients of the polynomial S $\genfrac{}{}{0ex}{}{\text{(k-1)}}{\text{i}}$ (x), the information input is fed from the (a _e k) -th input of the group of second information inputs 3 ₁ -3

the device coefficient q _i , the control inputs are supplied with the code from the outputs of the feedback task unit 7. At the outputs 10 $\genfrac{}{}{0ex}{}{\text{k}}{\text{1}}$ -10

of the polynomial calculation unit 5 _k , the result will be the polynomial coefficient vector

(x) q

(x) +

(x) x + b

(x).

(x) (первого слагаемого) формируются с помощью первой группы элементов И 18 $\genfrac{}{}{0ex}{}{\text{k}}{\text{1}}$ -18

блока 5_k. Формирование коэффициентов полинома

(x) обеспечивает блок задания обратных связей 7. Коэффициенты полинома b

(x) (третьего слагаемого) формируются с помощью второй группы элементов И 21 $\genfrac{}{}{0ex}{}{\text{k}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{k}}{\text{ae}}$ блока 5_k. Суммирование всех трех слагаемых осуществляется с помощью первой группы 19 $\genfrac{}{}{0ex}{}{\text{k}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{k}}{\text{ae}}$ и второй группы 20 $\genfrac{}{}{0ex}{}{\text{k}}{\text{1}}$ -20

сумматоров по модулю два.Coefficients of the polynomial q

(x) (first term) are formed using the first group of elements AND 18 $\genfrac{}{}{0ex}{}{\text{k}}{\text{1}}$ -eighteen

block 5 _k . Formation of polynomial coefficients

(x) provides a feedback assignment block 7. Coefficients of the polynomial b

(x) (third term) are formed using the second group of elements And 21 $\genfrac{}{}{0ex}{}{\text{k}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{<figure-callout id="5" label="k ae block" filenames="00000150.png,00000155.png" state="{{state}}">k </figure-callout>}}{\text{ae}}$ block 5 _k . The summation of all three terms is carried out using the first group 19 $\genfrac{}{}{0ex}{}{\text{k}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{k}}{\text{ae}}$ and second group 20 $\genfrac{}{}{0ex}{}{\text{k}}{\text{1}}$ -20

modulo two adders.

(x) S_i(^ae-1)(х) с выходов блока вычисления полинома 5

будут подаваться на информационные входы блока сдвига произведения 9. С помощью кода, подаваемого на управляющие входы блока сдвига произведения с выходов блока формирования величины сдвига 8, в блоке сдвига произведения 9 будет осуществлен сдвиг вектора коэффициентов полинома

(x) на (a_e-a_i) разрядов в сторону младших разрядов. На выходах 11_e-11

блока сдвига произведения 9 будут получены коэффициенты искомого полинома r_i(x).Polynomial Coefficients

(x) S _i ( ^ae-1) (x) from the outputs of the block for computing polynomial 5

will be fed to the information inputs of the product shift unit 9. Using the code supplied to the control inputs of the product shift unit from the outputs of the unit for forming the amount of shift 8, the shift of the product of unit 9 will shift the polynomial coefficient vector

(x) on (a _e -a _i ) bits towards the lower bits. At outputs 11 _e -11

block shift product 9 will be obtained coefficients of the desired polynomial r _i (x).

Consider the operation of the device for the case when l 7,
F ₁ (x) x ³ + x + 1, F ₂ (x) x ³ + x ² + 1,
F ₃ (x) 'x ⁴ + x + 1,
F ₄ (x) x ⁴ + x ³ + 1, F ₅ (x) x ⁵ + x ² + 1,
F ₆ (x) x ⁵ + x ⁴ + x ³ + x ² + 1,
F ₇ (x) x ⁵ + x ⁴ + x ² + x + 1.

Moreover, a ₁ a ₂ 3, a ₃ a ₄ 4, and ₅ a ₆ a ₇ 5.

(x) F_i(x)х ^ае^-ai

(x) F₁(x)x² x⁵ + x³ + x²,

(x) F₂(x)x² x⁵+ x⁴ + x²,

(x) F₃(x)x x⁵ + x² + x,

(x) F₄(x)x x⁵ + x⁴ + x,

(x) F₅(x),

(x) F₆(x),

(x) F₇(x) составим матрицу синтеза блока задания обратных связей 7, в которой столбцы соответствуют входам, а строки выходам блока задания обратных связей 7.Based on polynomial coefficient values

(x) F _i (x) x ^a e ^-a i

(x) F ₁ (x) x ² x ⁵ + x ³ + x ² ,

(x) F ₂ (x) x ² x ⁵ + x ⁴ + x ² ,

(x) F ₃ (x) xx ⁵ + x ² + x,

(x) F ₄ (x) xx ⁵ + x ⁴ + x,

(x) F ₅ (x),

(x) F ₆ (x),

(x) F ₇ (x) we compose the synthesis matrix of the feedback unit 7, in which the columns correspond to the inputs and the rows to the outputs of the feedback unit 7.

b₄= Y₁∨Y₆, b₅= Y₂∨Y₄∨Y₆∨Y₇
Построенный в соответствии с полученными выражениями блока задания обратных связей 7 приведен на фиг. 5.Y ₁ Y ₂ Y ₃ Y ₄ Y ₅ Y ₆ Y ₇
0 0 0 0 1 1 1 b ₁
0 0 1 1 0 0 1 b ₂
1 1 1 0 1 1 1 b ₃
1 0 0 0 0 1 0 b ₄
0 1 0 1 0 1 1 b ₅
Then
b ₁ = Y ₅ ∨ Y ₆ ∨ Y ₇ , b ₁ = Y ₃ ∨ Y ₄ ∨ Y ₇ , b ₃ =

b ₄ = Y ₁ ∨ Y ₆ , b ₅ = Y ₂ ∨ Y ₄ ∨ Y ₆ ∨ Y ₇
Constructed in accordance with the obtained expressions of the feedback task unit 7, is shown in FIG. 5.

составим матрицу синтеза блока формирования величины сдвига 8, в которой столбцы соответствуют входам, а строки выходам блока формирования величины сдвига 8.Based on the values of a _i , i

we compose the synthesis matrix of the unit for forming the amount of shear 8, in which the columns correspond to the inputs and the rows for the outputs of the unit for forming the amount of shear 8.

Y ₁ Y ₂ Y ₃ Y ₄ Y ₅ Y ₆ Y ₇
1 1 0 0 0 0 0 C ₁
0 0 1 1 0 0 0 C ₂
0 0 0 0 1 1 1 C ₃
Then
C ₁ Y ₁ VY ₂ , C ₂ Y ₃ VY ₄ , C ₃ Y ₅ VY ₆ VY ₇ .

 Constructed in accordance with the obtained expressions, the shear value generating unit 8 is shown in FIG. 6.

Consider the operation of the device when multiplying the elements of the field GF (2 ³ ), defined by the polynomial F ₁ (x) x ³ + x + 1. In this case, the signal "1" is fed to the control input of the device. The signal "1" appears at the outputs 22 ₃ and 22 ₄ of the feedback setting block 7 (at the remaining outputs of the feedback setting block, the signal is "0"), as well as at the output 13 ₁ of the shift value generating unit 8 (at the other outputs of the shift value forming unit 8 signal "0").

Let p _i (x) x ² + 1, q _i (x) x ² + x. Moreover, the binary code 00101 corresponding to the coefficient of the polynomial p _i (x) is supplied to the information inputs 1 ₁ -1 ₅ of the shift block of the factor 2. As a result, at the outputs 15 ₁ -15 ₅ of the shift block of the factor 2, the code 10100 appears, which corresponds to the coefficients of the polynomial p _i (x). The binary code 00110 corresponding to the coefficients of the polynomial q _i (x) is supplied to inputs 3 ₁ –3 ₅ .

(x) q

(x)=0(x⁴+x²)=0, т.е. двоичный код 00000.At outputs 10 $\genfrac{}{}{0ex}{}{\text{0}}{\text{1}}$ -10 $\genfrac{}{}{0ex}{}{\text{0}}{\text{5}}$ groups of elements And 16 ₁ -16 ₅ block multiplication by senior digit 4 appear polynomial coefficients

(x) q

(x) = 0 (x ⁴ + x ² ) = 0, i.e. binary code 00000.

(x) 0 (x³ + x²) 0, т.е. двоичный код 00000. На выходах группы элементов И 21 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{1}}{\text{5}}$ (вторых входах сумматоров по модулю два 19 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{1}}{\text{5}}$ ) появляются коэффициенты полинома b

(x) 0(x³ + x²) 0, т.е. двоичный код 00000. На выходах сумматоров по модулю два 19 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{1}}{\text{5}}$ появляется двоичный код 00000, который поступает (кроме младшего разряда коэффициента при х^о 0) на первые входы сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{1}}{\text{4}}$ . На вторые входы сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{1}}{\text{4}}$ поступают коэффициенты полинома

(x)x 0 * x 0, т. е. двоичный код 0000. В результате на выходах 10 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -10 $\genfrac{}{}{0ex}{}{\text{1}}{\text{5}}$ блока вычисления полинома 5₁ появляются коэффициенты полинома

(x) 0 + 0 0, т.е. двоичный код 00000.At the outputs of the group of elements And 18 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -eighteen $\genfrac{}{}{0ex}{}{\text{1}}{\text{5}}$ (the first inputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{1}}{\text{5}}$ ) of the polynomial 5 calculation unit, the coefficients of the polynomial q

(x) 0 (x ³ + x ² ) 0, i.e. binary code 00000. At the outputs of the group of elements And 21 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{1}}{\text{5}}$ (the second inputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{1}}{\text{5}}$ ) the coefficients of the polynomial b

(x) 0 (x ³ + x ² ) 0, i.e. binary code 00000. At the outputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{1}}{\text{5}}$ It appears binary code 00000, which enters (except LSB ^of the coefficient of x 0) at the first inputs of two adders modulo 20 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{1}}{\text{4}}$ . The second inputs of the adders modulo two 20 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{1}}{\text{4}}$ polynomial coefficients arrive

(x) x 0 * x 0, that is, the binary code is 0000. As a result, outputs 10 $\genfrac{}{}{0ex}{}{\text{1}}{\text{1}}$ -10 $\genfrac{}{}{0ex}{}{\text{1}}{\text{5}}$ block computing polynomial 5 ₁ polynomial coefficients appear

(x) 0 + 0 0, i.e. binary code 00000.

(x) 1(x⁴ + x²) x⁴ + x², т.е. двоичный код 10100. На выходах группы элементов И 21 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{2}}{\text{5}}$ (вторых входах сумматоров по модулю два 19 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{2}}{\text{5}}$ ) появляются коэффициенты полинома b

(x) 0(x³ + x²) 0, т.е. двоичный код 00000. На выходах сумматоров по модулю два 19 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{2}}{\text{5}}$ появляется двоичный код 10100, который поступает (кроме младшего разряда) на первые входы сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{2}}{\text{4}}$ . На вторые входы сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{2}}{\text{4}}$ поступают коэффициенты полинома

(x)x 0 * x 0, т.е. двоичный код 0000. В результате на выходах 10 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -10 $\genfrac{}{}{0ex}{}{\text{2}}{\text{5}}$ блока вычисления полинома 5₂появляется вектор коэффициентов полинома S $\genfrac{}{}{0ex}{}{\text{(2)}}{\text{i}}$ (x) x⁴ + x² + 0 x⁴ + x², т.е. двоичный код 10100.At the outputs of the group of elements And 18 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -eighteen $\genfrac{}{}{0ex}{}{\text{2}}{\text{5}}$ (the first inputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{2}}{\text{5}}$ ) of the block for computing the polynomial 5 ₂ , the coefficients of the polynomial q

(x) 1 (x ⁴ + x ² ) x ⁴ + x ² , i.e. binary code 10100. At the outputs of the group of elements And 21 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{2}}{\text{5}}$ (the second inputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{2}}{\text{5}}$ ) the coefficients of the polynomial b

(x) 0 (x ³ + x ² ) 0, i.e. binary code 00000. At the outputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{2}}{\text{5}}$ a binary code 10100 appears, which arrives (except for the least significant bit) at the first inputs of the adders modulo two 20 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{2}}{\text{4}}$ . The second inputs of the adders modulo two 20 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{2}}{\text{4}}$ polynomial coefficients arrive

(x) x 0 * x 0, i.e. binary code 0000. As a result, outputs 10 $\genfrac{}{}{0ex}{}{\text{2}}{\text{1}}$ -10 $\genfrac{}{}{0ex}{}{\text{2}}{\text{5}}$ block computing the polynomial 5 ₂ appears the coefficient vector of the polynomial S $\genfrac{}{}{0ex}{}{\text{(2)}}{\text{i}}$ (x) x ⁴ + x ² + 0 x ⁴ + x ² , i.e. binary code 10100.

(x) 1(x⁴ + x²) x⁴ + x², т.е. двоичный код 10100. На выходах группы элементов И 21 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{3}}{\text{5}}$ (вторых входах сумматоров по модулю два 19 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{3}}{\text{5}}$ ) появляются коэффициенты полинома b

(x) 1(x³ + x²) x³ + x², т. е. двоичный код 01100. На выходах сумматоров по модулю два 19 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{3}}{\text{5}}$ появляется двоичный код 11000, который поступает (кроме младшего разряда) на первые входы сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{3}}{\text{4}}$ . На вторые входы сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{3}}{\text{4}}$ поступают коэффициенты полинома

(x)x (x⁴ + x²)x x⁵ + x³, т. е. двоичный код 0100. В результате на выходах 10 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -10 $\genfrac{}{}{0ex}{}{\text{3}}{\text{5}}$ блока вычисления полинома 5 появляются коэффициенты полинома

(x) (x⁴ + x³) + +x³= x⁴, т.е. двоичный код 10000.At the outputs of the group of elements And 18 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -eighteen $\genfrac{}{}{0ex}{}{\text{3}}{\text{5}}$ (the first inputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{3}}{\text{5}}$ ) of the block for calculating the polynomial 5 ₃ the coefficients of the polynomial q

(x) 1 (x ⁴ + x ² ) x ⁴ + x ² , i.e. binary code 10100. At the outputs of the group of elements And 21 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{3}}{\text{5}}$ (the second inputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{3}}{\text{5}}$ ) the coefficients of the polynomial b

(x) 1 (x ³ + x ² ) x ³ + x ² , that is, binary code 01100. At the outputs of the adders, there are two modulo 19 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{3}}{\text{5}}$ a binary code 11000 appears, which arrives (except for the least significant bit) at the first inputs of the adders modulo two 20 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{3}}{\text{4}}$ . The second inputs of the adders modulo two 20 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{3}}{\text{4}}$ polynomial coefficients arrive

(x) x (x ⁴ + x ² ) xx ⁵ + x ³ , that is, binary code 0100. As a result, outputs 10 $\genfrac{}{}{0ex}{}{\text{3}}{\text{1}}$ -10 $\genfrac{}{}{0ex}{}{\text{3}}{\text{5}}$ block computing the polynomial 5 appear polynomial coefficients

(x) (x ⁴ + x ³ ) + + x ³ = x ⁴ , i.e. binary code 10000.

(x) 0(x⁴ + x²) 0, т.е. двоичный код 00000. На выходах группы элементов И 21 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{4}}{\text{5}}$ (вторых входах сумматоров по модулю два 19 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{4}}{\text{5}}$ ) появляются коэффициенты полинома b

(x) 1(x³ + x²) x³ + x², т.е. двоичный код 01100. На выходах сумматоров по модулю два 19 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{4}}{\text{5}}$ появляется двоичный код 01100, который поступает (кроме младшего разряда) на первые входы сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{4}}{\text{4}}$ . На вторые входы сумматоров по модулю два 20 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{4}}{\text{4}}$ поступают коэффициенты полинома

(x) x⁴ * x x⁵, т.е. двоичный код 0000. В результате на выходах 10 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -10 $\genfrac{}{}{0ex}{}{\text{4}}{\text{5}}$ блока вычисления полинома 5₄появляются коэффициенты полинома

(x)

(x) x³+x²+0 x³+x², т.е. двоичный код 01100.At the outputs of the group of elements And 18 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -eighteen $\genfrac{}{}{0ex}{}{\text{4}}{\text{5}}$ (the first inputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{4}}{\text{5}}$ ) of the block for computing the polynomial 5 ₄ the coefficients of the polynomial q

(x) 0 (x ⁴ + x ² ) 0, i.e. binary code 00000. At the outputs of the group of elements And 21 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -21 $\genfrac{}{}{0ex}{}{\text{4}}{\text{5}}$ (the second inputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{4}}{\text{5}}$ ) the coefficients of the polynomial b

(x) 1 (x ³ + x ² ) x ³ + x ² , i.e. binary code 01100. At the outputs of the adders modulo two 19 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -19 $\genfrac{}{}{0ex}{}{\text{4}}{\text{5}}$ a binary code 01100 appears, which arrives (except for the least significant bit) at the first inputs of the adders modulo two 20 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{4}}{\text{4}}$ . The second inputs of the adders modulo two 20 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -20 $\genfrac{}{}{0ex}{}{\text{4}}{\text{4}}$ polynomial coefficients arrive

(x) x ⁴ * xx ⁵ , i.e. binary code 0000. As a result, outputs 10 $\genfrac{}{}{0ex}{}{\text{4}}{\text{1}}$ -10 $\genfrac{}{}{0ex}{}{\text{4}}{\text{5}}$ the polynomial computing unit 5 ₄ the polynomial coefficients appear

(x)

(x) x ³ + x ² +0 x ³ + x ² , i.e. binary code 01100.

(x)/х ^ае^-ai (x³ + x²)/x² x + 1, т.е. двоичный код 00011.At the outputs 11 ₁ -11 ₅ of the shift unit of product 9, the coefficients of the polynomial r _i (x) appear

(x) / х ^а е ^-a i (x ³ + x ² ) / x ² x + 1, i.e. binary code 00011.

 The technical and economic efficiency of the proposed device in relation to the known can be determined from the increase in speed.

If in a known device time is spent on multiplication
T ₁ t _n at, where t _{o is the} clock frequency, t _{n is} the preparation time for the multiplication procedure, then in the proposed device, the multiplication will take time
T ₂ t _n + t _o .

The effectiveness of the device is determined by the formula
E T ₁ / T ₂ .

Claims (1)

DEVICE FOR MULTIPLICATION IN FINITE FIELDS, containing a block for multiplying by the senior digit and a _l 1 polynomial calculation blocks (where a _l is the maximum bit depth of the factors, l is the number of different polynomials defining the final field), and the outputs of the multiplication block to the senior bit are connected to information inputs the first group of the first block for computing polynomials, information inputs of the first group

the polynomial calculation unit is connected to the information outputs of the (i 1) -th polynomial calculation unit, characterized in that it includes a factor shift unit, a feedback setting unit, a shift value generation unit and a product shift unit, the information inputs of the first group of devices being connected to corresponding information inputs of the shift block of the factor,

the information input of the second group of the device is connected to the information input of the (a _l i) -th polynomial block, and the _{l l} information input of the second group of the device is connected to the information input of the block multiplying by the senior digit, the control inputs of the device are connected to the corresponding inputs of the inverse job unit links and the unit for generating the amount of shift, the outputs of which are connected to the control inputs of the shift unit of the factor and the unit of shift of the product, the outputs of the unit for setting feedbacks are connected to the control inputs 1 s a _l blocks polynomial evaluation, data inputs of the second group are connected to the outputs of the block shift factor and the group of information inputs to multiplier MSB, the outputs (a _l 1) -th polynomial calculation unit connected to data inputs of the shift unit product, which outputs connected to the outputs of the device.

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