WO2006057948A2

WO2006057948A2 - Condensed galois field computing system

Info

Publication number: WO2006057948A2
Application number: PCT/US2005/042106
Authority: WO
Inventors: James Wilson; Yosef Stein; Joshua Kablotsky
Original assignee: Analog Devices, Inc.
Priority date: 2004-11-22
Filing date: 2005-11-21
Publication date: 2006-06-01
Also published as: EP1825354A4; JP2008521144A; WO2006057948A3; US20060123325A1; EP1825354A2; US7512647B2; CN101095102B; CN101095102A; JP4733143B2

Abstract

A condensed Galois field computing system including a multiplier circuit for multiplying first and second polynomials with coefficients over a Galois field to obtain their product; and a Galois field linear transformer circuit for applying an irreducible polynomial of power n to the product including a partial result generator responsive to terms of power n and greater in the product for providing a folded partial result and a Galois field adder for condensing the folded partial result and the terms less than power n in the product to obtain Galois field transformer of power n of the product.

Description

CONDENSED GALOIS FIELD COMPUTING SYSTEM

FIELD OF THE INVENTION This invention relates to a condensed Galois field computing system.

BACKGROUND OF THE INVENTION

Multiplication of polynomials with coefficients in Galois fields (GF) is widely used in communication systems for Reed Solomon (RS) coding and in advanced encryption. Galois field multiplication is difficult and time consuming for traditional digital signal processors (DSP) to perform: DSP's are optimized for finite impulse response (FIR) filtering and other multiply-accumulate (MAC) intensive operations, but do not efficiently process Galois field types of operations. One approach uses straight forward polynomial multiplication and division over the Galois field using linear feedback shift registers (LFSR' s) which process one bit at a time. This is a very slow process. For example, in broadband communication for AES types of applications, where the bit rate is up to 100 megabits per second, there will be up to 12.5 million GF multiplications per second (MPS) and each multiplication may require many e.g. 60-100 operations. Another approach uses look-up tables to perform the Galois field multiplication.

Typically, this approach requires 10-20 or more cycles which for 12.5 mps results in a somewhat lower but still very large number of operations e.g. 20x12.5=250mps or more. Reed-Solomon codes have been widely accepted as the preferred error control coding

scheme for broadband networks. A programmable implementation of a Reed-Solomon encoder and decoder is an attractive solution as it offers the system designer the unique flexibility to trade-off the data bandwidth and the error correcting capability that is desired based on the condition of the channel. The first step in Reed-Solomon decoding is the computing of the syndromes. The syndromes can be formally defined as Si=R mod G where i=(0,l ...15). The received code word may be expressed in polynomial

form as Rj = T₀X^^'1 + T₁X¹*^"2 + ... r_N-1 where the length of the received word is N. It can be seen that computing the syndrome amounts to polynomial evaluation over the Galois field at the roots as defined by the j ^5t power of the i^?t root of the generator polynomial. For each received word in the Reed-Solomon Algorithm there are sixteen syndromes to be calculated which raise the operations by a factor of sixteen to 400 mps-not practical on current microprocessors. Using the straight forward multiplication instead of the look-up tables raises the operation rate to 1.6 gigs per second. The need for Galois field multiplications is increasing dramatically with the expansion of the communications field and the imposition of encryption requirements on the communication data. This further complicates the matter because each domain error checking encryption needs Galois field multiplication over a different Galois field which requires different sets of look-up tables.

BRIEF SUMMARY OF THE INVENTION It is therefore an object of this invention to provide an improved condensed

Galois field computing system.

It is a further object of this invention to provide such an improved condensed

Galois field computing system which requires less power and less area.

It is a further object of this invention to provide such an improved condensed

Galois field computing system which is much faster than current look-up tables and linear feedback shift register (LFSR) implementations.

It is a further object of this invention to provide such an improved condensed Galois field computing system which reduces the amount of storage required.

It is a further object of this invention to provide such an improved condensed Galois field computing system which dramatically reduces the number of required operations per second.

The invention results from the realization that an improved condensed Galois field computing system which can operate as a multiplier, multiplier-adder, and multiplier-accumulator requiring less power and area can be achieved by multiplying first and second polynomials with coefficients over a Galois field to obtain their product and then using a Galois field linear transformer circuit to apply an irreducible polynomial of power n to the product including a partial result generator responsive to terms of power n and greater in the product for providing a folded partial result and a Galois field adder for condensing the folded partial result and the terms less than power n in the product to obtain the Galois field transform of power n of the product.

The subject invention, however, in other embodiments, need not achieve all these objectives and the claims hereof should not be limited to structures or methods capable of achieving these objectives.

This invention features condensed Galois field computing system, including a

multiplier circuit for multiplying first and second polynomials with coefficients over a Galois field to obtain their product. There is a Galois field linear transformer circuit for applying an irreducible polynomial of power n to the product including a partial

result generator responsive to terms of power n in greater in the product for providing a folded partial result. And a Galois field adder for combining the folded partial result and the terms less than power n in a product to obtain the Galois field transformer power n of the product.

In preferred embodiments, the partial result generated may include a look-up table. The look-up table may include the folded partial results for the combinations of n or greater. The look-up table may include an address generator. The address generator may provide statistically independent address access. The Galois field adder may include a three input adder. The third input to the three input adder may be a 0 and the Galois field computing system may operate as a Galois field multiplier. The Galois field in or output may be fed back to the Galois field computing system and the Galois field computing system may operate as a multiplier - accumulator. The Galois field adder output may be fed back to the multiplier circuit with one of the first and second polynomials and the other of the first and second polynomials may constitute the third input to the Galois field adder, so that the Galois field computing system operates as a multiplier - adder. The Galois field adder may include an exclusive OR circuit.

The invention also features a condensed Galois field multiplier system including a multiplier circuit for multiplying first and second polynomials with coefficients over a Galois field to obtain their product, and a Galois field linear transformer circuit for applying an irreducible polynomial of power n to the product including a partial result

generator responsive to terms of power n and greater in the product for providing a folded partial result and a Galois field adder for combining the folded partial results and the terms less than power n in the product to perform a Galois field multiply operation.

The invention also features a condensed Galois field multiply-accumulate system including a multiplier circuit for multiplying first and second polynomials with coefficients over a Galois field to obtain their product, and a Galois field linear transformer circuit for applying an irreducible polynomial of power n to the product including a partial result generator responsive to terms of power n and greater in the product for providing a folded partial result and a Galois field adder for combining the folded partial results, the terms less than power n in the product and the Galois field adder fed back to perform a Galois field multiply-accumulate operation.

The invention also features a condensed Galois field multiplier-adder system including a multiplier circuit for multiplying first and second polynomials with coefficients over a Galois field to obtain their product, a Galois field linear transformer circuit for applying an irreducible polynomial of power n to the product including a partial result generator responsive to terms of power n and greater in the product for providing a folded partial result and a Galois field adder whose output is fed back to the multiplier circuit with one of the first and second polynomials and the other of the polynomials is a third input to the Galois field adder along with the folded partial results and the terms less than power n in the product to perform a Galois field multiply-add operation.

BRIEF DESCRIPTION OF THE DRAWINGS Other objects, features and advantages will occur to those skilled in the art from the following description of a preferred embodiment and the accompanying drawings,

in which:

Fig. 1 is a schematic block diagram of a condensed Galois field computing

system according to this invention;

Fig. 2 is a schematic block diagram of the folded partial results look-up table of Fig. 1;

Fig. 3 is a schematic block diagram similar to Fig. 1 with the Galois adder having a third input set for operation as a Galois field multiplier;

Fig. 4 is a schematic block diagram similar to Fig. 1 with the Galois adder having a third input connected for operation as a Galois field multiplier-accumulator; and

Fig. 5 is a schematic block diagram similar to Fig. 1 with the Galois adder having a third input connected for operation as a Galois field multiplier-adder. DISCLOSURE OF THE PREFERRED EMBODIMENT Aside from the preferred embodiment or embodiments disclosed below, this invention is capable of other embodiments and of being practiced or being carried out in various ways. Thus, it is to be understood that the invention is not limited in its application to the details of construction and the arrangements of components set forth in the following description or illustrated in the drawings. If only one embodiment is described herein, the claims hereof are not to be limited to that embodiment. Moreover, the claims hereof are not to be read restrictively unless there is clear and convincing evidence manifesting a certain exclusion, restriction, or disclaimer.

There is shown in Fig. 1 a condensed Galois field computing system 10 according to this invention including a polynomial multiplier 12, a Galois field transformer 13 including a look-up table 14, and a Galois field adder such as exclusive OR gate 16

having two inputs 18 and 20.

In operation an 8-bit polynomial with coefficient over GF(2⁸) X (x₇ - xo) is multiplied by an 8-bit polynomial Y (y₇ - yo) in polynomial multiplier 12. The polynomial product is a 14-bit number. Instead of dividing this 14-bit polynomial product by the field irreducible polynomial of power n to get the modulo reminder as would normally be the procedure, in accordance with this invention, those terms of the polynomial product which are less than the power n of the irreducible polynomial are

delivered directly to the Galois Field adder input 18 of exclusive OR gate 16. Those terms in the product which have powers of n or greater are used to access a partial result stored in look-up table 14 which when provided at input 20 to exclusive OR gate 16 and exclusively ORed or Galois field added by exclusive OR gate 16, produce the Galois field transform Z of 8-bits z₇ -z₀. The realization of this invention is that the terms that are below a power n are unaffected by the reduction process (division of the product by an irreducible polynomial where n is the power of that irreducible polynomial), and so they can be delivered directly without any further operation or manipulation to input 18 of Galois field adder 16. Those of the power n or above must go through the reduction process where they are replaced by a folded partial result using a look-up table, which is then added in adder 16 to the lower power terms from input 18 to produce the proper Galois field output. (A partial result look-up table value is generated by expressing all the product terms of power of n or above in terms of power of n -1 or less and accumulating all the terms). For example, if the irreducible polynomial chosen has a power of eight, all those terms in the product

from polynomial multiplier 12 which have power seven or below would be delivered directly to input 18 while those that had a power of eight or above would be used on line 22 to access the folded partial result in look-up table 14 which when added to the lower power terms on input 18 produce the proper Galois field output.

Galois field polynomial multiplication can be implemented in two basic steps. The first is a calculation of the polynomial product. c(x) = a(x)*b(x) is algebraically expanded, and like powers are collected (addition corresponds to an X-OR operation

between the corresponding terms) to give c(x).

For example c(x) = (a₇x⁷+a₆x⁶+a5X⁵+a₄x⁴+a₃x³+a₂x²+a₁x¹+ao)*

( b₇x⁷+b6x⁶+b5X⁵+b₄x⁴+b3X³+b2x³+bix¹+bo) C(x) = c₁₄x¹⁴+c₁₃x¹³+c₁₂x¹²+c₁₁x¹¹+c₁ox^lo+c₉x⁹+C₈X⁸+c₇x⁷+

c₆x⁶+C5X⁵+c₄x⁴+c₃x³+c₂x²+c₁x¹+c₀

where: Chart I cl4 = a7*b7 cl3 = a7*b6 θ a6*b7 cl2 = a7*b5 θ a6*b6 θ a5*b7 ell = a7*b4 θ a6*b5 θ a5*b6 θ a4*b7 clO = a7*b3 θ a6*b4 θ a5*b5 θ a4*b6 θ a3*b7 c 9 = a7*b2 θ a6*b3 θ a5*b4 θ a4*b5 θ a3*b6 θ a2*b7 c 8 = a7*bl θ a6*b2 θ a5*b3 θ a4*b4 θ a3*b5 θ a2*b6 θ al*b7 c 7 = a7*bθ θ a6*bl θ a5*b2 θ a4*b3 θ a3*b4 θ a2*b5 θ al*b6 θ aθ*b7 c 6 = a6*bθ θ a5*bl θ a4*b2 θ a3*b3 θ a2*b4 ® al*b5 θ aθ*b6 c 5 = a5*bθ θ a4*bl θ a3*b2 θ a2*b3 θ al*b4 θ aθ*b5 c 4 = a4*bθ θ a3*bl θ a2*b2 θ al*b3 θ aθ*b4 c 3 = a3*bθ θ a2*bl θ al*b2 θ aθ*b3 c 2 = a2*bθ θ al*bl θ aθ*b2 c l = al*bθ ® aθ*bl c θ = aθ*bθ

The second is the calculation of d(x) = c(x) modulo p(x).

To illustrate, multiplications are performed with the multiplication of polynomials modulo an irreducible polynomial. For example: (if m(x) = x⁸+x⁴+χ³+χ+l) {57}*{83} = {cl} because,

First Step

(x⁶+x⁴+χ² x+1) + θx⁷

= x¹³θx^πθx⁹θx⁸®x⁶θx⁵θx⁴®x³θl Second Step

For the irreducible polynomial x⁸ +x⁴ + x³ + x + 1

If the polynomial multiplication product = x¹³ Θ x¹¹ Θ x⁹ ® x⁸ ® x ⁶ θ x⁵ Θ x⁴ Θ x³

Θ 1 then each of the terms of power n (8) or above are expressed in terms of n- 1=7 or

less: x⁸ = x⁴ + x³ + x + 1 x⁹ = x-x⁸ = x(x⁴ + x³ + x + 1) = x⁵ + x⁴ + x² + x

X¹¹=x³-x⁸ = x³(x⁴ + x³ + x+l) = x⁷ + x⁶ + χ⁴ + χ³ _χ ⁱ³ _{= χ} ⁵._χ ⁸ _{= χ} ⁵^_χ ⁴ + _χ ³ + _x + i) = _χ ⁹ + _χ ⁸ + _χ ⁶ + _χ ⁵ _which simplifies to x⁶+x³+χ²+l

Assigning a 1 to each term to express the terms of power n and above x x x x X

Terms of power n and greater x° 1 1 1 1 in polynomial product x⁹ 1 1 1 1 x¹¹ 1 1 1 1 x¹³ 1 1 1 1

Folded Partial Result in LUT 14 1 0 1 1 1 0 0 0

By Galois field addition (X-OR 16) of terms of powers less than n in the product with the folded Partial Result for LUT 14 the first Galois field output is obtained.

x⁷ x⁶ x⁵ x⁴ x³ x² x¹ x°

Product terms of power n-1 less 0 1 1 1 1 0 0 1 θ

Folded Partial Result in LUT 14 1 0 1 1 1 0 0 0

Galois Field adder (X-OR)I₆ output 1 1 0 0 0 0 0 1 = X +X +1 = Oxcl

The irreducible polynomial used in this case: x +x +x +x +lis only one of

many that may be used. For example, any one of those shown in Chart II may be

used: Chart π

:GF(2¹) 0x3 (x+1)

:GF(2²)

0x7 (x²+x+l)

:GF(2³)

OxB (x³+x+l)

OxD (x³+x²+l)

:GF(2⁴)

0x13 (x⁴+x+l)

0x19 (x⁴+x³+l)

:GF(2⁵)

0x25 (x⁵+x²+l)

0x29 (x⁵+x³+l)

0x2F (x⁵+x³+χ²+χ+l)

0x37 (x⁵+x⁴+χ²+χ+l)

0x3B (x⁵+x⁴+x³+x+l)

0x3D (x⁵+x⁴+x³+x²+l)

:GF(2⁶)

0x43 (x⁶+x+l)

0x5B (x⁶+x⁴+x³+χ+l)

0x61 (x⁶+x⁵+l)

0x67 (x⁶+x⁵+x²+x+l)

0x6D (x⁶+x⁵+x³+χ²+l)

0x73 (x⁶+x⁵+x⁴+x+l)

:GF(2⁷)

0x83 (x⁷+χ+l) 0x89 (x⁷+x³+l) 0x8F (x⁷+x³+x²+x+l) 0x91 (x⁷+x⁴+l) 0x9D (x⁷+χ⁴+χ³+χ²+l) 0xA7 (x⁷+x⁵+x²+x+l) OxAB (x⁷+χ⁵+x³+χ+l) 0xB9 (x⁷+x⁵+χ⁴+χ³+l) OxBF (x⁷+x⁵+x⁴+x³+x²+x+l) OxCl (x⁷+x⁶+l) OxCB (x⁷+x⁶+χ³+χ+l) 0xD3 (x⁷+x⁶+x⁴+x+l) 0xE5 (x⁷+x⁶+x⁵+x²+l) OxFl (x⁷+x⁶+x⁵+x⁴+l) 0xF7 (x⁷+x⁶+χ⁵+χ⁴+χ²+χ+l) OxFD (x⁷+x⁶+x⁵+χ⁴+χ³+χ²+l)

In summary, applying the irreducible polynomial OxI Ib (x⁸ + x⁴ +χ³ +χ +1) having a power n = 8 to the polynomial product 30, Fig. 2, from the polynomial multiplier 12, that product can be viewed in two sections 32 and 34. Section 32 contains all of those terms of powers less than n, where n = 8 and section 34 represents all of those terms in polynomial product 30 which have powers of 8 or greater. Since the irreducible polynomial that we chose to deal with has a power n = 8, there will be 128 possible combinations or folded partial results that are stored in look-up table 14. Each one is addressed by a different combination of terms which make up the n or greater section 34. More information on Galois field transformers

and multipliers maybe found in U. S. Patent No. 6,587,864 B2 issued July 1, 2003, Stein et al., entitled, GALOIS FIELD LINEAR TRANSFORMER, U.S. Patent No. 6,766,345 B2 issued July 20, 2004, Stein et al. entitled, GALOIS FIELD MULTIPLIER SYSTEM, and U.S. Patent Application No. 10/395,620 filed on March 24, 2003, Stein et al., entitled COMPACT GALOIS FIELD MULTIPLIER ENGINE, each of which is herein incorporated in its entirety by this reference.

Condensed Galois field computing system 10a, according to this invention, may be more generalized using a Galois field adder, exclusive OR gate 16a, that includes three inputs the third input 36 may be a 0 when Galois field computing system 10a is operated as a multiplier. There may also be provided an address generator including adder circuit 38 and base address circuits 40. Then, for example, for advanced encryption standards (AES) look-up table 14a may have a base of 0 added to the addresses coming in on line 22a so that the addresses will be from 0 to 127, and for a Reed-Solomon operation base 40 may add the number 128 to the address coming in on 22a to access addresses from 128 to 255. In that way the different irreducible polynomial based Galois Field multipliers can coexist in the same system. Based on the fact that the base address is a multiple of 2 (the last 7 Lsb's are guaranteed to be "zero") and the address generator adder 38 can be simplified to be a simple OR- circuit. The condensed Galois field computing system according to this invention, 10b, Fig. 4, can be operated as a multiplier-accumulator by applying the output from Galois field adder, exclusive OR gate 16b, back to OR gate 16b.

And it may be operated as multiplier-adder 10c, Fig. 5, by applying one of the polynomials X to polynomial multiplier 12c while the other one Y is applied to input 36c of Galois field adder, exclusive OR gate 16c. The output of Galois field adder 16c is fed back over line 50 to the other input of polynomial multiplier 12c. Further

discussion of Galois field multiply-add and other Galois field operations are discussed in U.S. Patent Application No. 10/228,526, filed August 26, 2002, Stein et al. entitled GALOIS FIELD MULTIPLY/MULTIPLY-ADD/MULTIPLY ACCUMULATE, and U.S. Patent Application No. 10/440,330 filed May 16, 2003, Stein et al. entitled COMPOUND GALOIS FIELD ENGINE AND GALOIS FIELD DIVIDER AND SQUARE ROOT ENGINE AND METHOD, each of which is herein incorporated in

its entirety by this reference.

Although specific features of the invention are shown in some drawings and not in others, this is for convenience only as each feature may be combined with any or all of the other features in accordance with the invention. The words "including", "comprising", "having", and "with" as used herein are to be interpreted broadly and comprehensively and are not limited to any physical interconnection. Moreover, any embodiments disclosed in the subject application are not to be taken as the only possible embodiments.

Other embodiments will occur to those skilled in the art and are within the following claims.

In addition, any amendment presented during the prosecution of the patent application for this patent is not a disclaimer of any claim element presented in the application as filed: those skilled in the art cannot reasonably be expected to draft a claim that would literally encompass all possible equivalents, many equivalents will be unforeseeable at the time of the amendment and are beyond a fair interpretation of what is to be surrendered (if anything), the rationale underlying the amendment may bear no more than a tangential relation to many equivalents, and/or there are many

other reasons the applicant can not be expected to describe certain insubstantial substitutes for any claim element amended.

What is claimed is:

Claims

1. A condensed Galois field computing system comprising: a multiplier circuit for multiplying first and second polynomials with coefficients over a Galois field to obtain their product; and a Galois field linear transformer circuit for applying an irreducible polynomial of power n to said product including a partial result generator responsive to terms of power n and greater in said product for providing a folded partial result and a Galois field adder for combining said folded partial result and the terms less than power n in the product to obtain Galois field transform of power n of said product.

2. The condensed Galois field computing system of claim 1 in which said partial result generator includes a look-up table.

3. The condensed Galois field computing system of claim 2 in which said look¬ up table includes the folded partial results for the combinations of n or greater.

4. The condensed Galois field computing system of claim 2 in which said look¬

up table includes an address generator for said selecting one of the folded partial results tables

5. The condensed Galois field computing system of claim 4 in which said address generator provides statically independent address access.

6. The condensed Galois field computing system of claim 1 in which said Galois field adder includes a three input adder.

7. The condensed Galois field computing system of claim 6 in which said third input to said three input adder is a zero and said Galois field computing system operates as a Galois field multiplier.

8. The condensed Galois field computing system of claim 6 in which said third input to said three input adder is the Galois field adder output fed back and said Galois field computing system operates as a multiplier-accumulator.

9. The condensed Galois field computing system of claim 6 in which the Galois field adder output is fed back to said multiplier circuit with one of said first and second polynomials and the other of said polynomials is the third input to said Galois field adder and said Galois field computing system operates as a multiplier-adder.

10. The condensed Galois field computing system of claim 1 in which the Galois field adder includes an exclusive OR circuit.

11. The condensed Galois field computing system of claim 4 in which said address generator includes an OR circuit for combining said terms of n and greater and said value of base address to generate the look-up table address.

12. The condensed Galois field computing system of claim 2 in which said look¬ up table includes at least one folded partial results table

13. A condensed Galois field multiplier system comprising: a multiplier circuit for multiplying first and second polynomials with coefficients over a Galois field to obtain their product; and a Galois field linear transformer circuit for applying an irreducible polynomial of power n to said product including a partial result generator responsive to terms of power n and greater in said product for providing a folded partial result and a Galois field adder for combining said folded partial results and the terms less than power n in the product to perform a Galois field multiply operation.

14. A condensed Galois field multiply-accumulate system comprising: a multiplier circuit for multiplying first and second polynomials with coefficients over a Galois field to obtain their product; and a Galois field linear transformer circuit for applying an irreducible polynomial of power n to said product including a partial result generator responsive to terms of power n and greater in said product for providing a folded partial result and a Galois field adder for combining said folded partial results, the terms less than power n in the product and the output of the Galois field adder fed back to perform a Galois field multiply-accumulate operation.

15. A condensed Galois field multiplier-adder system comprising: a multiplier circuit for multiplying first and second polynomials with coefficients over a Galois field to obtain their product; and a Galois field linear transformer circuit for applying an irreducible polynomial of power n to said product including a partial result generator responsive to terms of power n and greater in said product for providing a folded partial result and a Galois field adder whose output is fed back to said multiplier circuit with one of said first and second polynomials and the other of said polynomials is a third input to said Galois field adder along with said folded partial results and the terms less than power n in the product to perform a Galois field multiply-add operation.