GB2155669A - Galois field multipliers - Google Patents
Galois field multipliers Download PDFInfo
- Publication number
- GB2155669A GB2155669A GB08405854A GB8405854A GB2155669A GB 2155669 A GB2155669 A GB 2155669A GB 08405854 A GB08405854 A GB 08405854A GB 8405854 A GB8405854 A GB 8405854A GB 2155669 A GB2155669 A GB 2155669A
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- Prior art keywords
- multiplier
- galois field
- supplied
- data word
- input
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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
- G06F7/60—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers
- G06F7/72—Methods or arrangements for performing computations using a digital non-denominational number representation, i.e. number representation without radix; Computing devices using combinations of denominational and non-denominational quantity representations, e.g. using difunction pulse trains, STEELE computers, phase computers using residue arithmetic
- G06F7/724—Finite field arithmetic
-
- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
- H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
- H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
- H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
- H03M13/13—Linear codes
- H03M13/15—Cyclic codes, i.e. cyclic shifts of codewords produce other codewords, e.g. codes defined by a generator polynomial, Bose-Chaudhuri-Hocquenghem [BCH] codes
- H03M13/151—Cyclic codes, i.e. cyclic shifts of codewords produce other codewords, e.g. codes defined by a generator polynomial, Bose-Chaudhuri-Hocquenghem [BCH] codes using error location or error correction polynomials
- H03M13/158—Finite field arithmetic processing
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- General Physics & Mathematics (AREA)
- Engineering & Computer Science (AREA)
- Computational Mathematics (AREA)
- Mathematical Analysis (AREA)
- Mathematical Optimization (AREA)
- Pure & Applied Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Computing Systems (AREA)
- Mathematical Physics (AREA)
- General Engineering & Computer Science (AREA)
- Error Detection And Correction (AREA)
Abstract
A Galois field multiplier for multiplying an arbitrary element beta of a Galois field by alpha <n> where alpha is another element of the Galois field and n is an integer, comprises a data path comprising an input (5) to which an input binary data word representing the element beta is supplied, a plurality of programmable read-only memories (1, 2, 3, 4) connected in series in the path and each able to effect Galois field multiplication, an output (6) connected to the output of the final memory (4) and from which the Galois field multiplied data word is derived, and a control input (7) to which a multiplier data word representing n and consisting of a plurality of binary digits is supplied, one or more of the binary digits being supplied to each memory to control the multiplication effected by that memory. <IMAGE>
Description
SPECIFICATION
Galois field multipliers
This invention relates to Galois field multipliers.
Where an error correcting code is used in binary data processing, for example to provide error correction where digital television signals are to be recorded and reproduced using a digital video tape recorder, Galois field arithmetic is commonly used as in some ways it is easier to implement than ordinary arithmetic, because there are no carries.A particular form of code which can use Galois field arithmetic and has been used in such cases is the Reed
Solomon code, which may be produced by a generator polynomial using a function of the extension field of a primitive polynomial, for example, as represented by GF(28): X8+X4+X3+X2+X0=O (1)
The primitive element a and other elements are generated using: a8 = a4 + a3 + a2 + 1 (2)
The H-matrix for this primitive polynomial is shown in Fig. 1 of the accompanying drawings, where MSB and LSB refer to the most and least significant bits, respectively.
Another way of considering the process of
Galois field multiplication will now be described with reference to Fig. 2 of the accompanting drawings. The centre of the circle shown represents the 8-bit word 00000000.
Around the circumference of the circle are 255 steps designated aO, a', a2, ... a254 representing all the different non-zero patterns of an 8-bit code. A Galois field multiplier can be used to step an input data word around the circle. Thus, when an 8-bit data word is supplied to the Galois field multiplier, the input data word may be considered as having been multiplied by aO, that is by 1. Each successive shift or multiplication by ar has the effect of moving the input data word by one step around the circle, up to a maximum of a254. One further shift or multiplication will bring the input data word back to the original value.Because the polynomial is primitive, any input 8-bit combination other than 00000000 supplied to the Galois field multiplier will cycle in a predetermined manner through all the other possible combinations before returning to the original combination.
In decoding an error correcting code, therefore, each input data word can be stepped or multiplied by any desired power of a in a
Galois field multiplier. One prior example of a
Galois field multiplier is shown in Fig. 6.3 on page 162 of "Error Control Coding-Fun- damentals and Applications" by Shu Lin and
Daniel J. Costello, published in 1983. This prior Galois field multiplier comprises a feedback shift register, and pulsing of the shift register effects Galois field multiplication of an arbitrary element ss in the Galois field by a.
Successive shifts of the register will generate vector representations of successive powers of a, in accordance with particular functions of the Galois field of the generator polynomical used, which in this particular case is: GF(24)=X4+X+1 (2)
Likewise, Fig. 6.4 on the same page shows a
Galois field multiplier for multiplying an arbitrary element ss in GF(24) by a3.
In error correction circuits it is commonly necessary, as described above, to multiply ss by an, where ss and an are elements in a
Galois field and n is a variable integer. Moreover, in such error correcting circuits the data rate may be such that the multiplications need to be carried out at a very high frequency; in one particular example one such multiplication was required every 74 nanoseconds. In such a case a Galois field multiplier comprising a shift register is too slow.
According to the present invention there is provided a Galois field multiplier for multiplying an arbitrary element'3 of a Galois field by an where a is another element of the Galois field and n is an integer, the Galois field multiplier comprising: a data path comprising an input to which an input binary data word representing the element ss to be Galois field multiplied is supplied; a plurality of multiplier elements connected in series in said path and each able to effect
Galois field multiplication, and an output connected to the output of the final said multiplier element and from which the Galois field multiplied data word is derived; and a control input to which a multiplier data word representing n and consisting of a plurality of binary digits is supplied, one or more of said binary digits being supplied to each said multiplier element to control the multiplication effected by that multiplier element.
The invention will now be described by way of example with reference to the accompanying drawings, in which:
Figure 1 shows the H-matrix of the primitive polynomial of equation (1);
Figure 2 shows diagrammatically an operation of a Galois field multiplier; and
Figure 3 shows in block form an embodiment of Galois field multiplier according to the present invention.
Referring to Fig. 3, the embodiment of
Galois field multiplier to be described comprises a series of multiplier elements, in the present case four programmable read-only memories (PROMs) 1, 2, 3 and 4. Input 8-bit data words representing arbitrary elements ss of the Galois field and to which the Galois field multiplication is to be applied are supplied successively to an input 5 connected to a data input of the PROM 1. Subsequent to the multiplication the multiplied data word is supplied by way of an output from the PROM 1 to an input of the PROM 2 and so on until the final multiplied output is supplied from an output of the PROM 4 to an output 6.
The number of multiplications, or in the terminology used above, the power n to which a is raised, is determined by an 8-bit multiplier word which is supplied by way of an input 7. Numbering the bits of the multiplier word from the least significant bit to the most significant bit as O to 7, the bits 0 and 1 are supplied to control the PROM 1 the bits 2 and 3 are supplied to control the PROM 2, the bits 4 and 5 are supplied to control the
PROM 3, and the bits 6 and 7 are supplied to control the PROM 4. Thus, the PROM 1 can shift or multiply the input data word by a to the power 0, 1,2 or 3.The PROM 2 can shift or multiply the multiplied input data word received from the PROM 1 by a to the power 0, 4, 8 or 12. The PROM 3 can shift or-multiply the multiplied input data word received from the PROM 2 by a to the power 0, 16, 32 or 48. The PROM 4 can shift or multiply the multiplied input data word received from the PROM 3 by a to the power 0, 63, 128 or 192.
Thus, by suitable selection of the multiplier word supplied to the input 7 any desired number of shifts or multiplications up to a maximum of 255 can be achieved. For example, if 123 multiplications are required, then the multiplier word, reading from the most significant digit, would be 011110111, and the PROMs 1 to 4 would effect shifts of 3, 8, 48 and 64 respectively, giving the required total shift of 123, so the 8-bit data output supplied to the output 6 would represent ss a123, which by definition is another element of the Galois field used.
In the particular embodiment described, the
PROMs 1 to 4 can each be a 1 K x 8 PROM.
Clearly many modifications are possible. For example, the particular choice of an 8-bit data word, an 8-bit multiplier, and the number of
PROMs in the described embodiment are merely given by way of example. Advantages will be gained in reduced memory capacity in any case where the number of shifts or multiplications is divided between two or more
PROMs.
If the circuit was arranged with eight series connected multiplier elements, so that each multiplier element was controlled by only one bit of the multiplier word n, then each of the multiplier elements could be a random logic circuit. In the case described above, the eight multiplier elements would then multiply by a to the power 0 or 1; a to the power 0 or 2; a to the power 0 or 4; a to the power 0 or 8; a to the power 0 or 16; a to the power 0 or 32; a to the power 0 or 64; and a to the power 0 or 128, respectively. Such an arrangement is advantageous where a special integrated circuit incorporating all the multiplier elements is to be designed for effecting the Galois field multiplication.
Claims (6)
1. A Galois field multiplier for multiplying an arbitrary element ss of a Galois field by an where a is another element of the Galois field and n is an integer, the Galois field multiplier comprising: a data path comprising an input to which an input binary data word representing the element ss to be Galois field multiplied is supplied; a plurality of multiplier elements connected in series in said path and each able to effect
Galois field multiplication, and an output connected to the output of the final said multiplier element and from which the Galois field multiplied data word is derived; and a control input to which a multiplier data word representing n and consisting of a plurality of binary digits is supplied, one or more of said binary digits being supplied to each said multiplier element to control the multiplication effected by that multiplier element.
2. A multiplier according to claim 1 wherein each said multiplier element is a programmable read-only memory.
3. A multiplier according to claim 2 comprising four said memories, and wherein said multiplier data word consists of eight binary digits, respective pairs of which are supplied to respective said memories.
4. A multiplier according to claim 1 wherein each said multiplier element is a logic circuit.
5. A multiplier according to claim 4 comprising eight said logic circuits, and wherein said multiplier data word consists of eight
binary digits, respective digits of which are supplied to respective said logic circuits.
6. A Galois field multiplier substantially as hereinbefore described with reference to Fig.
3 of the accompanying drawings.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| GB08405854A GB2155669A (en) | 1984-03-06 | 1984-03-06 | Galois field multipliers |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| GB08405854A GB2155669A (en) | 1984-03-06 | 1984-03-06 | Galois field multipliers |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| GB8405854D0 GB8405854D0 (en) | 1984-04-11 |
| GB2155669A true GB2155669A (en) | 1985-09-25 |
Family
ID=10557661
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| GB08405854A Withdrawn GB2155669A (en) | 1984-03-06 | 1984-03-06 | Galois field multipliers |
Country Status (1)
| Country | Link |
|---|---|
| GB (1) | GB2155669A (en) |
Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4791600A (en) * | 1986-07-28 | 1988-12-13 | Tektronix, Inc. | Digital pipelined heterodyne circuit |
| WO2005062472A1 (en) * | 2003-12-12 | 2005-07-07 | Analog Devices, Inc. | Encoding and decoding of reed-solomon codes using look-up tables for galois field multiplications |
Citations (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| GB2109135A (en) * | 1981-11-09 | 1983-05-25 | Rca Corp | Digital signal processing |
-
1984
- 1984-03-06 GB GB08405854A patent/GB2155669A/en not_active Withdrawn
Patent Citations (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| GB2109135A (en) * | 1981-11-09 | 1983-05-25 | Rca Corp | Digital signal processing |
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US4791600A (en) * | 1986-07-28 | 1988-12-13 | Tektronix, Inc. | Digital pipelined heterodyne circuit |
| WO2005062472A1 (en) * | 2003-12-12 | 2005-07-07 | Analog Devices, Inc. | Encoding and decoding of reed-solomon codes using look-up tables for galois field multiplications |
| US7162679B2 (en) | 2003-12-12 | 2007-01-09 | Analog Devices, Inc. | Methods and apparatus for coding and decoding data using Reed-Solomon codes |
Also Published As
| Publication number | Publication date |
|---|---|
| GB8405854D0 (en) | 1984-04-11 |
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| Date | Code | Title | Description |
|---|---|---|---|
| WAP | Application withdrawn, taken to be withdrawn or refused ** after publication under section 16(1) |