KR101607812B1 - METHOD AND APPARATUS FOR PARALLEL MULTIPLICATION CALCULATION USING DICKSON BASIS ON GF(2^n) FINITE FIELD - Google Patents

METHOD AND APPARATUS FOR PARALLEL MULTIPLICATION CALCULATION USING DICKSON BASIS ON GF(2^n) FINITE FIELD Download PDF

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KR101607812B1
KR101607812B1 KR1020150103314A KR20150103314A KR101607812B1 KR 101607812 B1 KR101607812 B1 KR 101607812B1 KR 1020150103314 A KR1020150103314 A KR 1020150103314A KR 20150103314 A KR20150103314 A KR 20150103314A KR 101607812 B1 KR101607812 B1 KR 101607812B1
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wow
vector
rti
vectors
matrix
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홍도원
서창호
박선미
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공주대학교 산학협력단
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F7/38—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation
    • G06F7/48—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation using non-contact-making devices, e.g. tube, solid state device; using unspecified devices
    • G06F7/52—Multiplying; Dividing
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F7/00—Methods or arrangements for processing data by operating upon the order or content of the data handled
    • G06F7/38—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation
    • G06F7/48—Methods or arrangements for performing computations using exclusively denominational number representation, e.g. using binary, ternary, decimal representation using non-contact-making devices, e.g. tube, solid state device; using unspecified devices
    • G06F7/52—Multiplying; Dividing
    • G06F7/523—Multiplying only
    • G06F7/53—Multiplying only in parallel-parallel fashion, i.e. both operands being entered in parallel

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Abstract

The present invention relates to a parallel multiplication method and apparatus using a Dixon basis on a finite field GF (2 n )

Figure 112015070935442-pat00370
Element of
Figure 112015070935442-pat00371
Vector
Figure 112015070935442-pat00372
And a symmetric Toffler's matrix
Figure 112015070935442-pat00373
And the triangle topolitz procession
Figure 112015070935442-pat00374
; A vector output unit
Figure 112015070935442-pat00375
Element of
Figure 112015070935442-pat00376
Vector
Figure 112015070935442-pat00377
And the TOFLitz matrices < RTI ID = 0.0 >
Figure 112015070935442-pat00378
Wow
Figure 112015070935442-pat00379
Multiplication with (
Figure 112015070935442-pat00380
,
Figure 112015070935442-pat00381
,
Figure 112015070935442-pat00382
And outputting the vectors as vectors; The vector sum output section outputs the two calculated vectors (
Figure 112015070935442-pat00383
Wow
Figure 112015070935442-pat00384
)
Figure 112015070935442-pat00385
); And a vector conversion unit
Figure 112015070935442-pat00386
Wow
Figure 112015070935442-pat00387
), And the two elements
Figure 112015070935442-pat00388
Wow
Figure 112015070935442-pat00389
Of the
Figure 112015070935442-pat00390
Coordinate vector
Figure 112015070935442-pat00391
.

Figure R1020150103314

Description

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method and apparatus for parallel multiplication using a Dixon basis on a finite field GF (2 ^ n)

The present invention relates to a parallel multiplication method and apparatus using a Dixon basis on a finite field GN (2 n ), and more particularly, to a parallel multiplication method and apparatus using a Dixson ternary polynomial (irreducible Dickson trinomial) using Dickson basis the finite field relates to a parallel multiplication method and apparatus using the Dixon base on the finite field GF (2 n) which allows to reduce the complexity of the multiplication operation on GF (2 n).

The most important and basic multiplication operations on the finite field GF (2 n ) are public key cryptography such as elliptic curve cryptography, paring-based cryptography, (coding theory) and so on.

The efficiency of computation on finite fields is strongly influenced by the choice of basis used to represent the finite field elements. The normal basis of the base has a great advantage that squaring can be performed with bit cyclic shift so that it can be performed without any cost in hardware. However, the known multiplication Multiplication is inefficient compared to using other bases.

An optimal normal basis (hereinafter referred to as "ONB") has been proposed as a special form of regular basis, but an optimal regular basis does not always exist on an arbitrary finite element.

Recently, Mullin and Mahalanobis introduced the Dickson basis of Dickson polynomial, and then Ansari and Hasan used a simpler mathematical term to define Dickson polynomials and Dickson bases, Discussed

The definition of the Dixon polynomial is as follows.

(Definition 1)

Figure 112015070935442-pat00001
Is called a ring,
Figure 112015070935442-pat00002
To
Figure 112015070935442-pat00003
The first type of Dickson polynomial of the first kind,
Figure 112015070935442-pat00004
Is defined as follows.

Figure 112015070935442-pat00005

At this time,

Figure 112015070935442-pat00006
ego
Figure 112015070935442-pat00007
.

especially,

Figure 112015070935442-pat00008
about
Figure 112015070935442-pat00009
The Dickson basis is defined as follows.

(Definition 2)

Figure 112015070935442-pat00010
(Degree)
Figure 112015070935442-pat00011
sign
Figure 112015070935442-pat00012
Let's call it an irreducible polynomial. Then,
Figure 112015070935442-pat00013
silver
Figure 112015070935442-pat00014
Finite element
Figure 112015070935442-pat00015
. This base
Figure 112015070935442-pat00016
Is called the Dickson basis.

The Dickson basis is always present for arbitrary finite bodies and under appropriate conditions, the Dickson basis is the type II optimal normal basis permutation.

As a result, Dickson basis is emerging as an alternative for finite elements without optimal normal basis, and efficient multiplier design using Dickson basis is attracting attention.

Recently, Hasan and Negre used the Dixon basis as a finite element

Figure 112015070935442-pat00017
We show that the product of two finite elements can be expressed as the product of a Toeplitz matrix and a vector.

Using this

Figure 112015070935442-pat00018
The space complexity required to perform the multiplication operation on the
Figure 112015070935442-pat00019
The number of operations required on
Figure 112015070935442-pat00020
In this paper, we propose a parallel multiplier with sub-quadratic space complexity. However, there is no satisfactory result of parallel multiplier using Dickson basis. As a result,
Figure 112015070935442-pat00021
Parallel multipliers on the chip.

The background art of the present invention is disclosed in Korean Registered Patent No. 10-1094354 (Registered, December, 2011, Bit-Parallel Multiplication Method and Apparatus between Elements of a Finite Field).

According to an aspect of the present invention, there is provided a method of generating a finite field GF (refinement) defined by a Dickson tricon trinomial using a Dickson basis, 2 n ), which can reduce the complexity of a multiplication operation on a finite field GF (2 n ).

In the parallel multiplication method using the Dixon basis on the finite field GF (2 n ) according to one aspect of the present invention,

Figure 112015070935442-pat00022
Element of
Figure 112015070935442-pat00023
Vector
Figure 112015070935442-pat00024
And a symmetric Toffler's matrix
Figure 112015070935442-pat00025
And the triangle topolitz procession
Figure 112015070935442-pat00026
; A vector output unit
Figure 112015070935442-pat00027
Element of
Figure 112015070935442-pat00028
Vector
Figure 112015070935442-pat00029
And the TOFLitz matrices < RTI ID = 0.0 >
Figure 112015070935442-pat00030
Wow
Figure 112015070935442-pat00031
Multiplication with (
Figure 112015070935442-pat00032
,
Figure 112015070935442-pat00033
,
Figure 112015070935442-pat00034
And outputting the vectors as vectors; The vector sum output section outputs the two calculated vectors (
Figure 112015070935442-pat00035
Wow
Figure 112015070935442-pat00036
)
Figure 112015070935442-pat00037
); And a vector conversion unit
Figure 112015070935442-pat00038
Wow
Figure 112015070935442-pat00039
), And the two elements
Figure 112015070935442-pat00040
Wow
Figure 112015070935442-pat00041
Of the
Figure 112015070935442-pat00042
Coordinate vector
Figure 112015070935442-pat00043
Into a predetermined number of bits.

In the present invention, the two elements (

Figure 112015070935442-pat00044
Wow
Figure 112015070935442-pat00045
)
Figure 112015070935442-pat00046
In order to calculate the product of the polynomials by using the following equation (1)
Figure 112015070935442-pat00047
And then calculates a polynomial
Figure 112015070935442-pat00048
Is a polynomial
Figure 112015070935442-pat00049
To < RTI ID = 0.0 >
Figure 112015070935442-pat00050
Is calculated.

(1)

Figure 112015070935442-pat00051

Here, the expression (1)

Figure 112015070935442-pat00052
About
Figure 112015070935442-pat00053
Called Dixon polynomial
Figure 112015070935442-pat00054
Lt; / RTI >< RTI ID = 0.0 >

procession

Figure 112015070935442-pat00055
Wow
Figure 112015070935442-pat00056
The
Figure 112015070935442-pat00057
The size of the toplex matrices

Figure 112015070935442-pat00058
ego,

procession

Figure 112015070935442-pat00059
silver
Figure 112015070935442-pat00060
The Hankel matrix of size

Figure 112015070935442-pat00061

Figure 112015070935442-pat00062
Vector
Figure 112015070935442-pat00063
to be.

In the present invention,

Figure 112015070935442-pat00064
Dickson's ternary polynomial
Figure 112015070935442-pat00065
, The finite element
Figure 112015070935442-pat00066
Any two elements of (
Figure 112015070935442-pat00067
Wow
Figure 112015070935442-pat00068
) Is a Dickson base
Figure 112015070935442-pat00069
Is expressed as < RTI ID = 0.0 >
Figure 112015070935442-pat00070
At this time, the two elements (
Figure 112015070935442-pat00071
Wow
Figure 112015070935442-pat00072
) Is a coordinate vector
Figure 112015070935442-pat00073
Wow
Figure 112015070935442-pat00074
Respectively.

The parallel multiplication device using the Dixon basis on the finite field GF (2 n ) according to another aspect of the present invention,

Figure 112015070935442-pat00075
Element of
Figure 112015070935442-pat00076
Vector
Figure 112015070935442-pat00077
And a symmetric Toffler's matrix
Figure 112015070935442-pat00078
And the triangle topolitz procession
Figure 112015070935442-pat00079
A matrix generator for generating a matrix; The finite element
Figure 112015070935442-pat00080
Element of
Figure 112015070935442-pat00081
Vector
Figure 112015070935442-pat00082
And the TOFLitz matrices < RTI ID = 0.0 >
Figure 112015070935442-pat00083
Wow
Figure 112015070935442-pat00084
Multiplication with (
Figure 112015070935442-pat00085
,
Figure 112015070935442-pat00086
,
Figure 112015070935442-pat00087
And outputting the vectors as vectors; The calculated two vectors (
Figure 112015070935442-pat00088
Wow
Figure 112015070935442-pat00089
)
Figure 112015070935442-pat00090
And outputs the vector sum output; And the vectors (
Figure 112015070935442-pat00091
Wow
Figure 112015070935442-pat00092
), And the two elements
Figure 112015070935442-pat00093
Wow
Figure 112015070935442-pat00094
Of the
Figure 112015070935442-pat00095
Coordinate vector
Figure 112015070935442-pat00096
And a vector conversion unit for converting the vector data into the vector data.

In the present invention, the vector conversion unit may convert the two elements (

Figure 112015070935442-pat00097
Wow
Figure 112015070935442-pat00098
)
Figure 112015070935442-pat00099
In order to calculate the product of the polynomials by using the following equation (1)
Figure 112015070935442-pat00100
And then calculates a polynomial
Figure 112015070935442-pat00101
Is a polynomial
Figure 112015070935442-pat00102
Lt; RTI ID = 0.0 >
Figure 112015070935442-pat00103
Is calculated.

(1)

Figure 112015070935442-pat00104

Here, the expression (1)

Figure 112015070935442-pat00105
About
Figure 112015070935442-pat00106
Called Dixon polynomial
Figure 112015070935442-pat00107
Lt; / RTI >< RTI ID = 0.0 >

procession

Figure 112015070935442-pat00108
Wow
Figure 112015070935442-pat00109
The
Figure 112015070935442-pat00110
The size of the toplex matrices

Figure 112015070935442-pat00111
ego,

procession

Figure 112015070935442-pat00112
silver
Figure 112015070935442-pat00113
The Hankel matrix of size

Figure 112015070935442-pat00114

Figure 112015070935442-pat00115
Vector
Figure 112015070935442-pat00116
to be.

In the present invention,

Figure 112015070935442-pat00117
Dickson's ternary polynomial
Figure 112015070935442-pat00118
, The finite element
Figure 112015070935442-pat00119
Any two elements of (
Figure 112015070935442-pat00120
Wow
Figure 112015070935442-pat00121
) Is a Dickson base
Figure 112015070935442-pat00122
Is expressed as < RTI ID = 0.0 >
Figure 112015070935442-pat00123
At this time, the two elements (
Figure 112015070935442-pat00124
Wow
Figure 112015070935442-pat00125
) Is a coordinate vector
Figure 112015070935442-pat00126
Wow
Figure 112015070935442-pat00127
Respectively.

According to one aspect of the present invention, the present invention can reduce the complexity of a multiplication operation on a finite field GF (2 n ) defined by the Dickson tricon trinomial, which is a dictation using Dickson basis , And can be applied to all hardware designs based on Dixon's ternary polynomial, since it can be applied to all finite fields using polynomials.

Figure 1 is a block diagram of an embodiment of the present invention,

Figure 112015070935442-pat00128
≪ / RTI > FIG.
FIG. 2 is an exemplary diagram illustrating the process 200 of FIG. 1 in more detail.
FIG. 3 is a block diagram of a TOEFLITS matrix using four blocks,
Figure 112015070935442-pat00129
And vector
Figure 112015070935442-pat00130
Fig. 3 is a diagram illustrating an example of a calculation process for a product of a multiplication factor;
Fig. 4 is a cross-sectional view of the light-
Figure 112015070935442-pat00131
In the form of
Figure 112015070935442-pat00132
), A finite element
Figure 112015070935442-pat00133
FIG. 2 is a block diagram of a parallel multiplier according to an embodiment of the present invention; FIG.

Hereinafter, an embodiment of a parallel multiplication method and apparatus using a Dixon basis on a finite field GF (2 n ) according to the present invention will be described with reference to the accompanying drawings.

In this process, the thicknesses of the lines and the sizes of the components shown in the drawings may be exaggerated for clarity and convenience of explanation. In addition, the terms described below are defined in consideration of the functions of the present invention, which may vary depending on the intention or custom of the user, the operator. Therefore, definitions of these terms should be made based on the contents throughout this specification.

Figure 1 is a block diagram of an embodiment of the present invention,

Figure 112015070935442-pat00134
FIG. 4 is a control flowchart showing a method of performing a multiplication operation on an input signal; FIG.

Fig.

Figure 112015070935442-pat00135
Element of
Figure 112015070935442-pat00136
Vector
Figure 112015070935442-pat00137
And a symmetric Toffler's matrix < RTI ID = 0.0 >
Figure 112015070935442-pat00138
And the upper triangular Toffler's matrix
Figure 112015070935442-pat00139
(100) for forming the finite element
Figure 112015070935442-pat00140
Element of
Figure 112015070935442-pat00141
Vector
Figure 112015070935442-pat00142
And the TOFLitz matrices < RTI ID = 0.0 >
Figure 112015070935442-pat00143
Wow
Figure 112015070935442-pat00144
Multiplication with (
Figure 112015070935442-pat00145
,
Figure 112015070935442-pat00146
,
Figure 112015070935442-pat00147
(200), calculating the two vectors (
Figure 112015070935442-pat00148
Wow
Figure 112015070935442-pat00149
)
Figure 112015070935442-pat00150
(300), calculating the vectors (
Figure 112015070935442-pat00151
Wow
Figure 112015070935442-pat00152
), And the two elements (
Figure 112015070935442-pat00153
Wow
Figure 112015070935442-pat00154
)
Figure 112015070935442-pat00155
Coordinate vector
Figure 112015070935442-pat00156
(Step 400).

Here,

Figure 112015070935442-pat00157
The transpose matrix of
Figure 112015070935442-pat00158
And
Figure 112015070935442-pat00159
ego
Figure 112015070935442-pat00160
to be.

Each step of FIG. 1 will be described in detail as follows.

Finite element

Figure 112015070935442-pat00161
This appointment, the Dickson ternary polynomial
Figure 112015070935442-pat00162
. ≪ / RTI >

Then,

Figure 112015070935442-pat00163
Any two elements of
Figure 112015070935442-pat00164
Wow
Figure 112015070935442-pat00165
Dickson base
Figure 112015070935442-pat00166
Is expressed as follows.

Figure 112015070935442-pat00167

At this time, the two elements (

Figure 112015070935442-pat00168
Wow
Figure 112015070935442-pat00169
) Is a coordinate vector,
Figure 112015070935442-pat00170
Wow
Figure 112015070935442-pat00171
(100) and (200) of Fig. 1, respectively.

The two elements (

Figure 112015070935442-pat00172
Wow
Figure 112015070935442-pat00173
)
Figure 112015070935442-pat00174
In order to calculate the product of the polynomials
Figure 112015070935442-pat00175
And then calculates a polynomial
Figure 112015070935442-pat00176
Is a polynomial
Figure 112015070935442-pat00177
To < RTI ID = 0.0 >
Figure 112015070935442-pat00178
.

Figure 112015070935442-pat00179

The above equation (1)

Figure 112015070935442-pat00180
About
Figure 112015070935442-pat00181
Called Dixon polynomial
Figure 112015070935442-pat00182
. ≪ / RTI >

Here,

Figure 112015070935442-pat00183
Wow
Figure 112015070935442-pat00184
The
Figure 112015070935442-pat00185
The size of the toplex matrices

Figure 112015070935442-pat00186
ego,

procession

Figure 112015070935442-pat00187
silver
Figure 112015070935442-pat00188
The Hankel matrix of size

Figure 112015070935442-pat00189

Figure 112015070935442-pat00190
Vector
Figure 112015070935442-pat00191
to be.

The definitions of the Toffler matrix and the Wankel matrix are as follows.

(Definition 3)

procession

Figure 112015070935442-pat00192
≪ / RTI >
Figure 112015070935442-pat00193
About
Figure 112015070935442-pat00194
The matrix
Figure 112015070935442-pat00195
Is called a Toffler matrix.

(Definition 4)

procession

Figure 112015070935442-pat00196
≪ / RTI >
Figure 112015070935442-pat00197
About
Figure 112015070935442-pat00198
The matrix
Figure 112015070935442-pat00199
Is referred to as an huckle matrix.

Huckel matrix

Figure 112015070935442-pat00200
And vector
Figure 112015070935442-pat00201
Product of
Figure 112015070935442-pat00202
A toplex matrix
Figure 112015070935442-pat00203
And vector
Figure 112015070935442-pat00204
Product of
Figure 112015070935442-pat00205
. In other words, the following equation (2) holds.

Figure 112015070935442-pat00206

1, reference numeral 100 denotes an element

Figure 112015070935442-pat00207
Vector
Figure 112015070935442-pat00208
≪ / RTI > and the < RTI ID =
Figure 112015070935442-pat00209
Wow
Figure 112015070935442-pat00210
.

In this case,

Figure 112015070935442-pat00211
The three Toeplitz matrix-vector multiplications (< RTI ID = 0.0 >
Figure 112015070935442-pat00212
). This process is illustrated in (200) of FIG.

In other words, (200) in FIG.

Figure 112015070935442-pat00213
Vector
Figure 112015070935442-pat00214
And the sum of the products of the Soffler matrix and the vector (
Figure 112015070935442-pat00215
).

FIG. 2 is an exemplary diagram illustrating the process 200 of FIG. 1 in more detail.

As shown in FIG. 2, the product of the Toeflitz matrix and the vector is divided into four independent blocks (CMF, CVF, CM, and R) do.

Here, in particular, the block CMF is a Toffler matrix

Figure 112015070935442-pat00216
Is symmetrical
Figure 112015070935442-pat00217
In the case of a triangular matrix
Figure 112015070935442-pat00218
And using the special property of the matrix,
Figure 112015070935442-pat00219
or
Figure 112015070935442-pat00220
May be performed in a manner having a lower spatial complexity than the block CMF of the general case (see FIGS. 2 and 3).

FIG. 3 is a block diagram of a TOEFLITS matrix using four blocks,

Figure 112015070935442-pat00221
And vector FIG. 3 is a diagram illustrating an example of a calculation process for a product of a product of a product

Referring again to FIG. 2, FIG. 2 illustrates the multiplication of the Tofflerz matrix and the vector using the four blocks

Figure 112015070935442-pat00223
) Is calculated.

Since all the blocks shown in FIG. 2 are calculated by a method having a spatial complexity less than the second order, the process of FIG. 2, i.e., (200) of FIG.

FIG. 1 (300) shows two vectors calculated in (200) of FIG. 1

Figure 112015070935442-pat00224
Wow
Figure 112015070935442-pat00225
)
Figure 112015070935442-pat00226
).

The vectors calculated in (200) and (300)

Figure 112015070935442-pat00227
Wow
Figure 112015070935442-pat00228
From the polynomial
Figure 112015070935442-pat00229
.

1, reference numeral 400 denotes a polynomial

Figure 112015070935442-pat00230
Is a polynomial
Figure 112015070935442-pat00231
To reduce the two elements (
Figure 112015070935442-pat00232
Wow
Figure 112015070935442-pat00233
)
Figure 112015070935442-pat00234
Coordinate vector
Figure 112015070935442-pat00235
.

Next,

Figure 112015070935442-pat00236
The complexity of the multiplier on the output.

1 (100)

Figure 112015070935442-pat00237
Lt; RTI ID = 0.0 >
Figure 112015070935442-pat00238
Wow
Figure 112015070935442-pat00239
), The two matrices (
Figure 112015070935442-pat00240
Wow
Figure 112015070935442-pat00241
) Can be performed in hardware at no cost.

FIG. 1 (200) shows the steps of performing the multiplication of the Toeplitz matrix-vector, and as described above,

Figure 112015070935442-pat00242
, And the actual
Figure 112015070935442-pat00243
When it is in the form of (200) in FIG. 1,
Figure 112015070935442-pat00244
XOR gate,
Figure 112015070935442-pat00245
AND gate,
Figure 112015070935442-pat00246
Time delay is required.

here

Figure 112015070935442-pat00247
Is the time delay required when performing one XOR gate,
Figure 112015070935442-pat00248
Is the time delay required when performing one AND gate.

1 (300) shows that the size

Figure 112015070935442-pat00249
Calculating a sum of two vectors,
Figure 112015070935442-pat00250
With the XOR gate
Figure 112015070935442-pat00251
Time delay is required.

1, reference numeral 400 denotes a polynomial

Figure 112015070935442-pat00252
As a polynomial
Figure 112015070935442-pat00253
As a result,
Figure 112015070935442-pat00254
XOR gates and
Figure 112015070935442-pat00255
Time delay is required.

Thus,

Figure 112015070935442-pat00256
The complexity of the parallel multiplier on
Figure 112015070935442-pat00257
When in shape,
Figure 112015070935442-pat00258
XOR gate,
Figure 112015070935442-pat00259
AND gate,
Figure 112015070935442-pat00260
Time delay.

Fig. 4 is a cross-sectional view of the light-

Figure 112015070935442-pat00261
In the form of
Figure 112015070935442-pat00262
), A finite element
Figure 112015070935442-pat00263
FIG. 4 is a table showing the complexity of the parallel multipliers using the Dixon basis on the table.

Figure 4

Figure 112015070935442-pat00264
In the form of
Figure 112015070935442-pat00265
), The binding Dickson ternary polynomial
Figure 112015070935442-pat00266
The finite element defined by
Figure 112015070935442-pat00267
And the complexity of the parallel multiplier according to the present embodiment as shown in FIG.

As shown in FIG. 4, it can be seen that the parallel multiplier according to the present embodiment has a lower complexity than the conventional parallel multiplier.

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is to be understood that the invention is not limited to the disclosed embodiments, but, on the contrary, I will understand the point. Accordingly, the technical scope of the present invention should be defined by the following claims.

100: symmetric toplex matrix

Figure 112015070935442-pat00268
And the triangle topolitz procession
Figure 112015070935442-pat00269
The formation process of
200: The toplex matrices (
Figure 112015070935442-pat00270
Wow
Figure 112015070935442-pat00271
) And vectors (
Figure 112015070935442-pat00272
Wow
Figure 112015070935442-pat00273
) ≪ / RTI >
Figure 112015070935442-pat00274
,
Figure 112015070935442-pat00275
,
Figure 112015070935442-pat00276
)
300: two vectors (
Figure 112015070935442-pat00277
Wow
Figure 112015070935442-pat00278
)
Figure 112015070935442-pat00279
Calculating and outputting
400: vectors (
Figure 112015070935442-pat00280
Wow
Figure 112015070935442-pat00281
) To receive the two elements (
Figure 112015070935442-pat00282
Wow
Figure 112015070935442-pat00283
)
Figure 112015070935442-pat00284
Coordinate vector
Figure 112015070935442-pat00285
The process of converting to

Claims (4)

The matrix generating unit
Figure 112015070935442-pat00286
Element of
Figure 112015070935442-pat00287
Vector
Figure 112015070935442-pat00288
And a symmetric Toffler's matrix
Figure 112015070935442-pat00289
And the triangle topolitz procession
Figure 112015070935442-pat00290
;
A vector output unit
Figure 112015070935442-pat00291
Element of
Figure 112015070935442-pat00292
Vector
Figure 112015070935442-pat00293
And the TOFLitz matrices < RTI ID = 0.0 >
Figure 112015070935442-pat00294
Wow
Figure 112015070935442-pat00295
Multiplication with (
Figure 112015070935442-pat00296
,
Figure 112015070935442-pat00297
,
Figure 112015070935442-pat00298
And outputting the vectors as vectors;
The vector sum output section outputs the two calculated vectors (
Figure 112015070935442-pat00299
Wow
Figure 112015070935442-pat00300
)
Figure 112015070935442-pat00301
); And
Vector conversion unit converts the vectors (
Figure 112015070935442-pat00302
Wow
Figure 112015070935442-pat00303
), And the two elements
Figure 112015070935442-pat00304
Wow
Figure 112015070935442-pat00305
Of the
Figure 112015070935442-pat00306
Coordinate vector
Figure 112015070935442-pat00307
(2 < n > ) of a finite field GF (2 < n > ).
The method according to claim 1,
The two elements (
Figure 112015070935442-pat00308
Wow
Figure 112015070935442-pat00309
)
Figure 112015070935442-pat00310
In order to compute,
Using the following equation (1), the product of the polynomials
Figure 112015070935442-pat00311
And then calculates a polynomial
Figure 112015070935442-pat00312
Is a polynomial
Figure 112015070935442-pat00313
Lt; RTI ID = 0.0 >
Figure 112015070935442-pat00314
( 2n ) by using the Dixon basis.
(1)
Figure 112015070935442-pat00315

Here, the expression (1)
Figure 112015070935442-pat00316
About
Figure 112015070935442-pat00317
Called Dixon polynomial
Figure 112015070935442-pat00318
Lt; / RTI >< RTI ID = 0.0 >
procession
Figure 112015070935442-pat00319
Wow
Figure 112015070935442-pat00320
The
Figure 112015070935442-pat00321
The size of the toplex matrices
Figure 112015070935442-pat00322
ego,
procession
Figure 112015070935442-pat00323
silver
Figure 112015070935442-pat00324
The Hankel matrix of size
Figure 112015070935442-pat00325

Figure 112015070935442-pat00326
Vector
Figure 112015070935442-pat00327
to be.
Finite element
Figure 112015070935442-pat00328
Element of
Figure 112015070935442-pat00329
Vector
Figure 112015070935442-pat00330
And a symmetric Toffler's matrix
Figure 112015070935442-pat00331
And the triangle topolitz procession
Figure 112015070935442-pat00332
A matrix generator for generating a matrix;
The finite element
Figure 112015070935442-pat00333
Element of
Figure 112015070935442-pat00334
Vector
Figure 112015070935442-pat00335
And the TOFLitz matrices < RTI ID = 0.0 >
Figure 112015070935442-pat00336
Wow
Figure 112015070935442-pat00337
Multiplication with (
Figure 112015070935442-pat00338
,
Figure 112015070935442-pat00339
,
Figure 112015070935442-pat00340
And outputting the vectors as vectors;
The calculated two vectors (
Figure 112015070935442-pat00341
Wow
Figure 112015070935442-pat00342
)
Figure 112015070935442-pat00343
And outputs the vector sum output; And
The vectors (
Figure 112015070935442-pat00344
Wow
Figure 112015070935442-pat00345
), And the two elements
Figure 112015070935442-pat00346
Wow
Figure 112015070935442-pat00347
Of the
Figure 112015070935442-pat00348
Coordinate vector
Figure 112015070935442-pat00349
( 2n ) using the Dixon basis. The parallel multiplication apparatus according to claim 1,
The apparatus according to claim 3,
The two elements (
Figure 112015070935442-pat00350
Wow
Figure 112015070935442-pat00351
)
Figure 112015070935442-pat00352
In order to calculate the product of the polynomials by using the following equation (1)
Figure 112015070935442-pat00353
And then calculates a polynomial
Figure 112015070935442-pat00354
Is a polynomial
Figure 112015070935442-pat00355
Lt; RTI ID = 0.0 >
Figure 112015070935442-pat00356
( 2n ) on the finite field GF ( 2n ).
(1)
Figure 112015070935442-pat00357

Here, the expression (1)
Figure 112015070935442-pat00358
About
Figure 112015070935442-pat00359
Called Dixon polynomial
Figure 112015070935442-pat00360
Lt; / RTI >< RTI ID = 0.0 >
procession
Figure 112015070935442-pat00361
Wow
Figure 112015070935442-pat00362
The
Figure 112015070935442-pat00363
The size of the toplex matrices
Figure 112015070935442-pat00364
ego,
procession
Figure 112015070935442-pat00365
silver
Figure 112015070935442-pat00366
The Hankel matrix of size
Figure 112015070935442-pat00367

Figure 112015070935442-pat00368
Vector
Figure 112015070935442-pat00369
to be.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR20200023486A (en) * 2017-07-24 2020-03-04 아이오와 스테이트 유니버시티 리서치 파운데이션, 인코퍼레이티드 System and method for inverting chirp Z-transforms to O (n log n) time and O (n) memory
KR20200022844A (en) * 2018-08-24 2020-03-04 공주대학교 산학협력단 A PARALLEL GF(2^m) MULTIPLIER AND MULTIPLICATION METHOD USING GAUSSIAN NORMAL BASIS

Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR101418686B1 (en) 2013-08-02 2014-07-10 공주대학교 산학협력단 Subquadratic Space Complexity Parallel Multiplier and Method using type 4 Gaussian normal basis
KR101533929B1 (en) 2014-06-27 2015-07-09 공주대학교 산학협력단 Subquadratic Space Complexity Parallel Multiplier for using shifted polynomial basis, method thereof, and recording medium using this

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR100670780B1 (en) * 2004-10-29 2007-01-17 한국전자통신연구원 Apparatus for hybrid multiplier in GF2^m and Method for multiplying
KR100950581B1 (en) * 2007-12-06 2010-04-01 고려대학교 산학협력단 Bit-parallel multiplier and multiplying method for finite field using redundant representation

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR101418686B1 (en) 2013-08-02 2014-07-10 공주대학교 산학협력단 Subquadratic Space Complexity Parallel Multiplier and Method using type 4 Gaussian normal basis
KR101533929B1 (en) 2014-06-27 2015-07-09 공주대학교 산학협력단 Subquadratic Space Complexity Parallel Multiplier for using shifted polynomial basis, method thereof, and recording medium using this

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR20200023486A (en) * 2017-07-24 2020-03-04 아이오와 스테이트 유니버시티 리서치 파운데이션, 인코퍼레이티드 System and method for inverting chirp Z-transforms to O (n log n) time and O (n) memory
KR20200133283A (en) * 2017-07-24 2020-11-26 아이오와 스테이트 유니버시티 리서치 파운데이션, 인코퍼레이티드 SYSTEMS AND METHODS FOR INVERTING THE CHIRP Z-TRANSFORM IN O(n log n) TIME AND O(n) MEMORY
KR102183973B1 (en) 2017-07-24 2020-12-03 아이오와 스테이트 유니버시티 리서치 파운데이션, 인코퍼레이티드 System and method for inverting chirp Z-transform into O(n log n) time and O(n) memory
KR20200022844A (en) * 2018-08-24 2020-03-04 공주대학교 산학협력단 A PARALLEL GF(2^m) MULTIPLIER AND MULTIPLICATION METHOD USING GAUSSIAN NORMAL BASIS
KR102372466B1 (en) * 2018-08-24 2022-03-11 공주대학교 산학협력단 A PARALLEL GF(2^m) MULTIPLIER AND MULTIPLICATION METHOD USING GAUSSIAN NORMAL BASIS

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