RU2639661C1 - Method of multiplication and division of finite field elements - Google Patents

Abstract

FIELD: physics.SUBSTANCE: upon the multiplication of the finite field elements, first the finite field elements are transferred to the multiplicative form of representation from the additive forms of representation by using a table of preset functions through the elements of subfields, the indices of the subfield elements are found by the subfield index tables, that perform multiplication and division of the finite field elements using the indices of the subfield, for which first the indices of the factors are found by the subfield index tables, then the indices are folded modulo n-1, where n is the number of elements in the subfield, and the product is found by the table of the anti-indices. By dividing the elements of the subfields, first the indices of the divident and the divider are found by the subfield index tables, then the index of the divider is subtracted from the index of the divident, is resulted modulo n-1 and the quotient is found by the table of the anti-indices. Then the product and the quotient are converted from the multiplicative form of presentation of the finite field elements to additive form of presentation using the table of the specified functions.EFFECT: simplification of the method due to the use of the multiplicative form of representation of the finite field elements through the elements of the subfields and the reduction of the memory capacity.4 cl, 1 dwg, 6 tbl

Description

The invention relates to the field of computer technology and can be used to create specialized computers for encoding and decoding information protected by error-correcting code.

One of the main ways to improve the noise immunity of message transmission over error communication channels is to use noise immunity coding. For coding and decoding of algebraic noise-resistant codes of Bowse-Chowdhury-Hockvinham (BCH-codes), Reed-Solomon, Goppa and others, arithmetic operations with elements of finite fields are necessary. The structure of finite fields is different from the structure of ordinary infinite number fields, such as rational, real, and complex number fields. Performing multiplication and division of elements of finite fields differs from multiplication and division of ordinary numbers and often causes difficulties. To multiply and divide the elements of finite fields, especially with a large bit depth, it requires the execution of a large number of commands and the presence of a large amount of memory to store tables of logarithms and anti-logarithms (index and anti-index tables). The proposed method is based on the use of the additive and multiplicative form of representing the elements of finite fields in the form, respectively, of the sum and product of the elements of subfields. Multiplication and division of elements of finite fields is performed for the multiplicative form of their representation and is reduced to multiplication and division of elements of subfields, which allows to reduce the complexity of the method. This reduces the amount of memory required to multiply and divide the elements of the final fields, and thereby reduce the necessary computing resources. The method can be used in extensions of finite fields containing subfields, for example, in Galois fields GF (2 ^2m ), and others, the number of elements in which is decomposed into prime factors.

A known method of multiplying and dividing the elements of finite fields, in which to multiply the elements of finite fields, first both factors are represented as polynomials with binary coefficients, then the product of these two polynomials is calculated and the remainder of dividing the product by the generating polynomial of the field is determined, to divide the elements of the finite fields first from the table of inverse elements of a finite field, find the inverse element of the divisor, and then multiply the dividend by this inverse element [Kasami T., Tokura N., Ivadari E., Inagaki Y. Theo coding Ia. Per. from Japanese A.V. Kuznetsova. - M.: - The world. - 1978. - p. 72-78].

The disadvantage of this method is its great complexity due to the large number of instructions needed to calculate the product of polynomials and the remainder of dividing the product by the generating polynomial of the finite field, as well as the large amount of memory for storing the table of the inverse element of the finite field.

Closest to the proposed method is a method (prototype) of multiplying and dividing the elements of finite fields, which consists in the fact that when multiplying the elements of the final fields, first find the indices of the factors on the index tables of the final field, then add these indices modulo n-1, where n is the number of elements in the final field, and the product is found from the anti-index table. When dividing the elements of the final fields, first, the indices of the dividend and the divisor are found from the tables of indexes of the final field, then the index of the divisor is subtracted from the index of the divisor, given modulo n-1 and the quotient is found from the anti-index table (Mc Williams F.J., Sloane N.J .A. Theory of error correction codes. - Translated from English by I. Grushko and B. A. Zinoviev - M .: Mir. - 1979. - p. 95-98).

The disadvantage of this method is the great complexity due to the large amount of memory for storing index tables and anti-indices of the elements of the final field.

The aim of the invention is to reduce the complexity of the method by using the multiplicative form of representing the elements of the final field through the elements of the subfields, which reduces the amount of memory for storing table functions in comparison with the amount of memory for storing tables of indices and anti-indices of the elements of the final field.

To achieve the goal, a method for multiplying and dividing the elements of finite fields is proposed, which consists in the fact that when multiplying the elements of the final fields, first find the indices of the factors on the index tables of the final field, then add these indices modulo n-1, where n is the number of elements in the final field , and on the anti-index table find the product. When dividing the elements of finite fields, first, the indices of the dividend and the divisor are found from the tables of indexes of the final field, then the index of the divisor is subtracted from the index of the dividend, modulo n-1 is given, and the quotient is found from the anti-index table. What is new is that first the elements of the final fields from the additive representation form are converted into the multiplicative form of the representation through the elements of the subfields using table-defined functions, the indexes of the subfields are found from the tables of the indexes of the subfields, the multiplication and division of the elements of the final fields through the indices of the subfields is performed, and then they are translated using table-defined functions, the product and quotient from the multiplicative form of representing the elements of finite fields to the additive form of representation. At the same time, the table-defined functions for converting the elements of finite fields from the additive representation form to the multiplicative representation form and vice versa are calculated in advance and stored in memory. Moreover, the tables of indices and anti-indices of subfield elements are calculated in advance and stored in memory. In this case, the table-defined functions for converting the elements of finite fields from the additive representation form to the multiplicative form and vice versa are calculated by enumerating all the elements of the finite fields and calculations using tables of indices and anti-indices of the elements of the final fields.

The implementation of the method of multiplication and division of elements of finite fields will be considered using the example of the Galois field GF (2 ^2m ). The Galois field GF (2 ^2m ) contains the subfield GF (2 ^m ). Let α be a primitive element of GF (2 ^2m ), then the subfield GF (2 ^m ) consists of many elements

Field elements with e GF (2 ^2m ) are usually represented in additive form

where d, g∈EGF (2 ^m ),

α is a primitive element of GF (2 ^2m ).

With this representation, it is easy to add and subtract field elements. Suppose c ₁ = d ₁ + αg ₁ and c ₂ = d ₂ + αg ₂ , then the addition of field elements is written

The addition of field elements is reduced to a simpler addition of elements of the subfield. Similarly for subtracting field elements.

To multiply and divide, the elements of finite fields from the additive representation form using table-defined functions are first transferred to the multiplicative representation form through the elements of subfields. The multiplicative form for representing the elements of the finite field c∈GF (2 ^2m ) is the notation

where a is an element of the subfield GF (2 ^m ) of the field GF (2 ^2m ), and ∈GF (2 ^m ) ⊂GF (2 ^2m ),

α is a primitive element of GF (2 ^2m ),

b is the set of indices of the field elements, b = 0 ... 2 ^m , ∞, α ^∞ = 0, operations with the indices of the field GF (2 ^m ) are carried out modulo 2 ^m -1. The index of the zero element of the field does not exist; therefore, the symbol ∞ is used for it.

The transition from the additive form of representation of the elements of the finite field to the multiplicative form of representation is written as

Defining d and g uniquely defines a and b. Divide both sides by d and transform (5) to

, а

, d≠0.Where

, but

, d ≠ 0.

By virtue of uniqueness (6)

where r ₁ (h) is the function: GF (2 ^m ) → GF (2 ^m ),

ar ₂ (h) - function: GF (2 ^m ) → {0 ... 2 ^m , ∞}.

The functions r ₁ (h) and r ₂ (h) can be determined in advance and stored in a table form. Then

After translating the elements of the final fields into the multiplicative form of the presentation, the indexes of the elements of the subfields are found from the tables of indexes of the subfields, the multiplication and division of the elements of the final fields through the indexes of the subfields is performed. Let two operands involved in multiplication and division, in a multiplicative form of representation, be written as

The multiplication and division of the field elements is performed through the indices of the elements of the subfield i ₁ and i ₂ in accordance with the formulas

- примитивный элемент поля GF(2^m),Where

- a primitive element of the field GF (2 ^m ),

Addition and subtraction of the indices of the field and subfield elements is carried out by modulo reduction of the number of elements in the field or subfield, respectively, minus 1.

Then the product and the quotient are transferred from the multiplicative form of representation of the elements of finite fields to the additive form of representation. Such a translation is expressed as

Or you can write

и

, а≠0.Where

and

, and ≠ 0.

From uniqueness (12) it follows

и

: {0…2^m, ∞}→GF(2^m).where are the functions

and

: {0 ... 2 ^m , ∞} → GF (2 ^m ).

Then

и

можно определить заранее и хранить в табличном виде.Functions

and

can be determined in advance and stored in tabular form.

The figure shows a diagram of the operation of the multiplication over the field GF (2 ^2m ). For division, the scheme will be similar, only the indices of the subfield elements will not add up, but subtracted.

Thus, the operations of multiplication and division over the elements of the field GF (2 ^2m ) are reduced to the calculation of functions with m bit input and arithmetic operations on the elements of the subfield GF (2 ^m ). The tabular definition of functions with m bit input requires significantly less memory than the performance of multiplication and division operations on the field GF (2 ^2m ) using, for example, index tables and anti-indices of this field. In the first case, the amount of memory is estimated at O (2 ^m ), and in the second, O (2 ^2m ).

,

, r₁ (x), r₂ (x) может выполняться заранее, и, поэтому не критично к числу операций и объему памяти. Например, составление таблиц этих функций можно выполнять заранее перебором элементов поля GF(2^2m) и используя таблицы индексов и антииндексов для элементов поля GF(2^2m).Table assignment of functions

,

, r ₁ (x), r ₂ (x) can be performed in advance, and therefore it is not critical to the number of operations and the amount of memory. For example, the compilation of tables of these functions can be performed in advance by enumerating the elements of the field GF (2 ^2m ) and using tables of indices and anti-indices for elements of the field GF (2 ^2m ).

и

необходимо выполнение следующих шагов.To define function tables

and

The following steps are required.

,

,

Step 1. Put

,

,

,

,

,

,

Step 2. Put b = 2, a = 1,

Step 3. Put g ₁ = 1

Step 4. Put d ₁ = 1

Step 5. Check d ₁ + αg ₁ = a α ^b , if satisfied, go to 7

, идти к 8, иначе d₁:=βd₁, идти к 5Step 6. If

, go to 8, otherwise d ₁ : = βd ₁ , go to 5

,

, идти к 9Step 7

,

go to 9

Step 8. g _1: = βg ₁ go to 4

Step 9. If b = 2 ^m -2, go to 10, otherwise b: = b + 1, go to 3

Step 10. The End

The tables of functions r ₁ (x) and r ₂ (x) are compiled in a similar way.

Example.

The field GF (2 ⁴ ) with the generating polynomial g (x) = x ⁴ + x + 1

The subfield GF (2 ² ) with the generating polynomial g (x) = x ² + x + 1

Elements of the field GF (2 ⁴ ), which are elements of the subfield GF (2 ² )

0, α ° = β ° = l, α ⁵ = β = α + α ² , α ¹⁰ = β ² = 1 + α + α ²

Multiplication of elements of the field GF (2 ⁴ ).

Let field elements be given in the form (2)

c ₁ = α ¹⁰ + αα ⁵ , and c ₂ = α ⁵ + αα ¹⁰ .

Step 1. Transfer of field elements from form (2) to form (1)

c ₁ = α ¹⁰ + αα ⁵ , a = α ¹⁰ r ₁ (α ¹⁰ ) = α ⁵ , b = r ₂ (α ¹⁰ ) = 2, c ₁ = α ⁵ α ²

c ₂ = α ⁵ + αα ¹⁰ , a = α ⁵ r ₁ (α ⁵ ) = 1, b = r ₂ (α ⁵ ) = 3, c ₂ = 1α ³

Step 2. Multiplication of field elements

c = c ₁ c ₂ = -α ⁵ α ² 1α ³ = α ¹⁰ α ⁰

Step 3. Translation of the work into form (2)

Division of elements of the field GF (2 ⁴ ).

Let field elements be given in the form (2)

c ₁ = α ¹⁰ + αα ⁰ , and c ₂ = α ⁵ + αα ⁵ .

Step 1. Transfer of field elements from form (2) to form (1)

c ₁ = α ¹⁰ + αα ⁰ , a = α ¹⁰ r ₁ (α ⁵ ) = α ⁵ , b = r ₂ (α ⁵ ) = 3, c ₁ = α ⁵ α ³

c ₂ = α ⁵ + αα ⁵ , a = α ⁵ r ₁ (1) = α ⁵ , b = r ₂ (1) = 4, c ₂ = α ⁵ α ⁴

Step 2. Division of field elements

Step 3. Transfer private to form (2)

The proposed method can be used not only in extensions of Galois fields GF (2 ^2m ) of characteristic 2, but also in other extensions of finite fields with characteristics other than 2, which can be decomposed into subfields. Many important technical applications, for example, coding and decoding of error-correcting codes, can be implemented only with a reduction in memory for storing tabular functions by which multiplication and division of finite field elements is performed. For small values of bit depth of elements of a finite field m, multiplication and division can be performed using index tables and anti-indices. However, for large m (≥8), difficulties arise due to an increase in the volume of index tables and anti-indices. The proposed method requires less memory for its implementation and simplifies the multiplication and division of the Galois field elements GF (2 ^2m ). The amount of memory for storing index tables and anti-indices is estimated by O (2 ^2m ), and for the proposed method, the amount of memory for storing table functions is estimated by O (2 ^m ). For example, with m = 8, the amount of memory for storing index tables and anti-indices is 262144 bytes, and for the proposed method, the required memory size is only 1540 bytes. Moreover, with increasing m, the advantage of the proposed method increases exponentially, that is, very quickly.

Achievable technical result of the method of multiplication and division of elements of finite fields is to reduce the amount of memory for implementing the method and its simplification.

Claims (4)

1. The method of multiplying and dividing the elements of finite fields, which is that when multiplying the elements of finite fields, first find the indices of the factors from the tables of indexes of the final field, then add these indices modulo n-1, where n is the number of elements in the final field, and find the product from the anti-index table, when dividing the elements of the final fields, first find the indices of the dividend and the divisor from the index tables of the final field, then subtract the divisor index from the index of the dividend, give modulo n-1 and the anti-index table by there is a quotient, characterized in that first the elements of the final fields from the additive representation form are converted into multiplicative representations via the elements of the subfields using table-defined functions, the indexes of the subfields are found from the tables of the indexes of the subfields, the multiplication and division of the elements of the final fields through the indices of the subfields are performed, and then the product and the quotient from the multiplicative representation form of the elements of the finite fields are translated into the additive representation form using table-defined functions. 2. The method according to p. 1, characterized in that the tabularly defined functions for converting the elements of finite fields from the additive presentation form to the multiplicative presentation form and vice versa are calculated in advance and stored in memory. 3. The method according to p. 1, characterized in that the tables of indices and anti-indices of the elements of the subfields are calculated in advance and stored in memory. 4. The method according to claim 1, characterized in that the tabularly defined functions for converting the elements of finite fields from the additive representation form to the multiplicative form and vice versa are calculated by enumerating all the elements of the finite fields and calculations using tables of indices and anti-indices of the elements of the final fields.

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