JPS6248812A - System of calculating inverse element - Google Patents

Abstract

PURPOSE:To attain comparatively high speed processing and to improve the processing efficiency such as an error correction code and economy by providing a means using an element as the first input, outputting the square of the input and repeating the square calculation and a multiplication means. CONSTITUTION:When an element A from which an inverse element is to be obtained is set to a register 10, the content of the register 10 is squared by a square circuit 11 and replaced with the preceding content in the register 10. Further, the output of the circuit 11 is inputted to a multiplication circuit 12 and the product with the content of a register 13 is calculated and the result is replaced with the preceding content (1 at first) of the register 13. The value set in the register 13 as the result by repeating the operation above by (m-1) times is the inverse element of the element A. In selecting the unity being the initial input to the circuit 12 as an optional element B, a quotient B/A is obtained as the output in place of the inverse element. Thus, the calculation of the inverse element is processed in comparatively high speed and the processing efficiency such as error correction node and the economy are improved.

Description

[Detailed description of the invention] 〔overview〕 This is a calculation method for the inverse element of the extended field GF (2''').

The inverse element calculation of element A can be obtained with a simple control configuration using m-1 square calculations and the product of all outputs of the square calculations.

[Industrial application field]

The present invention provides a method for calculating the inverse element of the extension field GF (2''').As is well known, the properties of the extension field GF (2''') are such that each element corresponds to an m-bit binary code. As a result, it has been effectively applied to research on coding theory.

As a result, in controlling error correction codes, for example, it is often necessary to calculate the inverse element of the extension field G F (2'').

[Prior art and problems to be solved by the invention] A conventional method is known as a method for calculating the inverse in the extension field G F (2').

In addition, in the following explanation, operations regarding the elements of the extended field GF(2'') are performed using the elements as polynomials having coefficients expressed as remainders modulo 2, and the results of operations between the polynomials being modulo the irreducible polynomial. The calculation is expressed as a remainder.

Let A be the element of the extension field G F (2'") for which the inverse is to be found, and in the first method, the extension field c F (2")
Any element A of is expressed as αJ, where α is the primordial light, and n=
2"-1, α'=1, so when element A is repeatedly multiplied by primitive light α and 1 is obtained by j multiplications, that is, when A×α'=1, Obtain the inverse element I=αj.

In the second method, an inverse element table for each element is prepared in advance, and the inverse element is obtained by indexing the inverse element table using element A as an address.

In the third method, the so-called Euclidean algorithm is used. As is well known, in this method, the greatest common divisor (which is always 1) of the irreducible polynomial F and the element A is calculated using Euclidean algorithm, and each aspect of the process is cumulated using a predetermined method. Obtain the inverse by calculating.

In the above method, the first method requires 1 to 2''-1 multiplications, 2@-1 multiplications on average, and therefore takes time to calculate. The second method requires only a table index, so the processing time is short, but it requires a storage capacity of at least (2" - 1) XI bits for the table, and as m increases, the required storage capacity increases. becomes a problem.

According to the third method, problems such as calculation time and storage capacity as in the case of the first and second methods are reduced, but the control is relatively complicated, so it can be realized by arithmetic control using a microprocessor, for example. There is a problem in that it is relatively expensive.

[Means for solving problems]

FIG. 1 is a block diagram showing the principle configuration of the present invention.

The figure shows the configuration of a circuit that calculates the inverse in the extended field G F (2''''), where 1-+, l-z~1-s-1 is 2
Multiplication circuits 2-1.2-t to 2-s-+ are multiplication circuits.

That is, an inverse element calculation system is constructed by connecting 11 pairs of squaring circuits and multiplier circuits in series.

0 action] Square circuit 1-. squares the element A input from the left side of the figure and inputs it to the next connected squaring circuit 1-2 and multiplication circuit 2-1.

The squaring circuit 1-2 outputs an output obtained by squaring the input, that is, A4.
It is input to the square circuit 1-3 and the multiplication circuit 2-2.

The multiplier circuit 2-8 applies the product A2 of the output A2 of the squaring circuit 1-3 and the l' input from the outside to the next multiplier circuit 2-8.
2, the multiplier circuit 2-t outputs A Z X A 4 as the product of the two human forces.

In this way, as is clear from the figure, the multiplier circuit 2-
1 to A2x84xA8, multiplier circuit 2-4 is A2
xA4 xA 8 ×AI & is output, and in this way, from the last multiplier circuit 2-m-+, the product of the outputs of the total square circuit is 2×A4×A8×−・−・A g @ −
1 is output. This output is equal to the inverse of element A.

As described above, the inverse element can be calculated using a circuit with a simple configuration and in the same calculation time as the conventional third method.

〔Example〕

The configuration shown in FIG. 1 can be used as is or can be configured as an embodiment. As shown in Fig. 1, the input source A is cumulatively squared by the 11 squaring circuits 1-1 to 1-s-1, so the final output is A2''-'', and the squared output between them is The product is calculated by the successive multiplication circuits 2-0 to 2-s-+, and finally the product A of the total square output from the multiplication circuit 2-pm-r
"XA'XA"X-...A2"-" is obtained.

As mentioned above, any element A of the extension field G F (2''')
is expressed as αj, where α is the primitive light, and n・2″I−1
, α”=1. Therefore, the nth power of any element A is 1
It is clear that

Here, n-2" -1-1+2+4+8+-・-+2
“-”, so AII=AI×8Z x Go to A4
A8
The inverse ■ is I = 1/A = A"XA'xA'x---xA"-
'It can be expressed as.

That is, the product of the above-mentioned total square outputs, which is the final output of the circuit of FIG.

FIG. 2 is a block diagram showing the configuration of another embodiment. In the figure, first, the element A for which the inverse element is to be found is in the register 10.
When set to , the contents of register 10 are squared by 2 times all, replacing the previous contents of register 10.

The output of the squaring circuit 11 is also input to the multiplication circuit 12 and multiplied by the contents of the register 13, which replaces the previous contents of the register 13. It is assumed that "1" is initially input to the register 13.

If the above operation is repeated once, the result will be the register 13
The value set is AZ X A 4 x 48 x・
-.x AZ-+, that is, the inverse element of element A.

In the above explanation, 1 is the first input to the multiplication circuit.
If ' is an arbitrary element B, it is clear that the output of FIGS. 1 and 2 is the quotient of B/A instead of the inverse element.

〔Effect of the invention〕

As is clear from the above description, according to the present invention, the calculation of the inverse element in the extension field G F (2''') can be processed at relatively high speed using a circuit with a relatively simple configuration. , has a significant industrial effect of improving the processing efficiency and economy of error correction codes and the like.

[Brief explanation of drawings]

FIG. 1 is a block diagram of the principle structure of the present invention, and FIG. 2 is a block diagram of the structure of an embodiment of the present invention. In the figure,

Claims (1)

[Claims] When calculating the inverse element of the m-th extension field GF(2^m) of the finite field GF(2), the element is used as the first input, the square of the input is output, and the square of the input is output. A means of repeating the square calculation m-1 times using the output as the next input (
1-_1 to 1-_m_-_1), and is configured to obtain an inverse element by the product of all outputs of the square calculation.