US 20070271323 A1 Abstract A Galois field divider engine and method inputs a 1 and a first Galois field element to a Galois field reciprocal generator to obtain an output, multiplies in a Galois field reciprocal generator a first Galois field element by a first element of the Galois field reciprocal generator for predicting the modulo remainder of the square of the polynomial product of an irreducible polynomial m−2 times where m is the degree of the Galois field to obtain the reciprocal of the first Galois field element, and multiplying in the Galois field reciprocal engine the reciprocal of the first Galois field element by a second Galois field element for predicting the modulo remainder of the polynomial product for an irreducible polynomial to obtain the quotient of the two Galois field elements in m cycles; in a broader sense the invention includes a compound Galois field engine for performing a succession of Galois field linear transforms on a succession of polynomial inputs to obtain an ultimate output where each input except the first is the output of the previous Galois field linear transform; Galois field square root is achieved by inputting a Galois field element to a Galois field square root generator to obtain an output which is squared in the Galois field square root generator to predict the modulo remainder of the square of the polynomial product of an irreducible polynomial m−1 times where m is the degree of the Galois field to obtain the square root of the Galois field to obtain the square root of the Galois field element in (m−1) cycles.
Claims(3) 1. A Galois field square root method comprising:
inputting a Galois field element to a Galois field square root generator to obtain an output; and squaring in the Galois field square root generator the output of the Galois field square root generator for predicting the modulo remainder of the square of the polynomial product of an irreducible polynomial m−1 times where m is the degree of the Galois field to obtain the square root of the Galois field element in (m−1) cycles. 2. A Galois field square root engine comprising:
a Galois field square root generator; an input circuit for inputting a Galois field element to the Galois field square root generator to obtain the square root of the Galois field elements in one cycle. 3. The Galois field square root engine of Description This application is a divisional application of U.S. patent application Ser. No. 10/440,330 filed May 16, 2003 which is hereby incorporated herein by reference and to which this application claims priority, and claims priority of U.S. Provisional application to Stein et al. entitled A COMPACT GALOIS FIELD MULTIPLIER, filed Oct. 9, 2002 (AD-337J), U.S. Provisional application Ser. No. 60/334,662, filed Nov. 30, 2001 to Stein et al., entitled GF2-ALU (AD-239J); U.S. Provisional application Ser. No. 60/334,510 filed Nov. 20, 2001 to Stein et al., entitled PARALLEL GALOIS FIELD MULTIPLIER (AD-240J); U.S. Provisional application Ser. No. 60/341,635, filed Dec. 18, 2001 to Stein et al., entitled GALOIS FIELD MULTIPLY ADD (MPA) USING GF2-ALU (AD-299J); U.S. Provisional application Ser. No. 60/341,737, filed Dec. 18, 2001, to Stein et al., entitled PROGRAMMABLE GF2-ALU LINEAR FEEDBACK SHIFT REGISTER—INCOMING DATA SELECTION (AD-300J). This application further claims priority of U.S. patent application Ser. No. 10/395,620 filed Mar. 24, 2003 to Stein et al., entitled COMPACT GALOIS FIELD MULTIPLIER ENGINE (AD-337J); U.S. patent application Ser. No. 10/051,533 filed Jan. 18, 2002 to Stein et al., entitled GALOIS FIELD LINEAR TRANSFORMERER (AD-239J); U.S. patent application Ser. No. 10/060,699 filed Jan. 30, 2002 to Stein et al., entitled GALOIS FIELD MULTIPLIER SYSTEM (AD-240J); U.S. patent application Ser. No. 10/228,526 filed Aug. 26, 2002 to Stein et al., entitled GALOIS FIELD MULTIPLY/MULTIPLY—ADD/MULTIPLY ACCUMULATE (AD-299J); and U.S. patent application Ser. No. 10/136,170, filed May 1, 2002 to Stein et al., entitled RECONFIGURABLE INPUT GALOIS FIELD LINEAR TRANSFORMERER SYSTEM (AD-300J). This invention relates to a Galois field divider engine and method and more generally to a compound Galois field engine for performing a succession of Galois field transforms in one transform operation. In certain applications such as encryption and error control coding, it is necessary to perform arithmetic operations, e.g., add, subtract, square root, multiply, and divide over Galois fields. Any such operation between any two members in a Galois field will result in an output (sum, difference, square root, product, quotient) which is another value in the same Galois field. The number of elements in a Galois field is 2 Division over a Galois field is done by multiplying the dividend by the reciprocal of the divisor. This divisor reciprocal can be generated in a number of ways. One way is to have a stored look-up table of reciprocals where the divisor is the address for the table. One problem with this approach is that for each field of each irreducible polynomial there must be stored a separate table. In addition, the tables can only be accessed in serial: if parallel operations are required a copy of each table must be provided for each parallel operation. Another approach is to multiply each of the stored Galois field elements by the particular divisor. The value that produces a product of one is then the reciprocal of the particular divisor. Once again all of the values have to be stored and in multiple copies if parallel operation is contemplated. And, a Galois field multiplier is required just to accomplish the retrieval. A third approach uses two linear feedback shift registers (LFSR) each configured to generate a selected Galois field of a particular irreducible polynomial. The first is initialized to the divisor; the second is initialized to “1”. Starting from the divisor value the two are clocked synchronously. When the product of the first LFSR equals “1” the divisor has been multiplied by its reciprocal. The product of the second LFSR at that moment is the Galois field element that is the reciprocal of the divisor. One problem with this approach is that for each Galois field of each irreducible polynomial for each degree a different pair of LFSRs is required. In both, the second look-up table approach, above, and the LFSR approach the search for the reciprocal requires up to It is therefore an object of this invention to provide an improved Galois field divider engine and method. It is a further object of this invention to provide such an improved Galois field divider engine which can complete the search for the divisor reciprocal in m−1 iterations. It is a further object of this invention to provide such an improved Galois field divider engine which can be easily reconfigured to accommodate different irreducible polynomial Galois fields of different degrees. It is a further object of this invention to provide such an improved Galois field divider engine which can function to generate both the divisor reciprocal and multiply it by the dividend. It is a further object of this invention to provide such an improved Galois field divider engine which requires less power and less area. It is a further object of this invention to provide more generally an improved, compound Galois field engine for performing a succession of Galois field transforms in one transform operation. The invention results from the realization that such an improved Galois field division engine and method which is smaller, faster, and more efficient can be achieved with a Galois field reciprocal generator and an input selection circuit for initially inputting a 1 and a first Galois field element to the Galois field reciprocal generator to obtain an output, subsequently multiplying in the Galois field reciprocal generator a first Galois field element by the output of the Galois field reciprocal generator for predicting the modulo remainder of the square of the polynomial product of an irreducible polynomial m−2 times where m is the degree of the Galois field, to obtain the reciprocal of the first Galois field element and multiplying in the Galois field reciprocal engine the reciprocal of the first Galois field element by a second Galois field element for predicting the modulo reminder of the polynomial product for an irreducible polynomial to obtain the quotient of the two Galois field elements in m cycles. It was also realized, more generally, that an improved compound Galois field engine for performing a succession of Galois field linear transforms on a succession of polynomial inputs to obtain an ultimate output where each input, except the first, is the output of the previous Galois field linear transform can be accomplished with an input circuit for providing a first input and a Galois field linear transformer having a matrix of cells responsive to the first input and configured to, in one transform, immediately predict the modulo remainder of the succession of Galois field linear transforms of an irreducible Galois field polynomial to obtain the ultimate output of the Galois field linear transform directly from the first input. This invention features a Galois field divider engine including a Galois field reciprocal generator and an input selection circuit for initially inputting a 1 and a first Galois field element to the Galois field reciprocal generator to obtain an output, subsequently multiplying in the Galois field reciprocal generator a first Galois field element by the output of the Galois field reciprocal generator for predicting the modulo remainder of the square of the polynomial product of an irreducible polynomial m−2 times, where m is the degree of the Galois field, to obtain the reciprocal of the first Galois field element and multiplying in the Galois field reciprocal engine the reciprocal of the first Galois field element by a second Galois field element for predicting the modulo remainder of the polynomial product, for an irreducible polynomial to obtain the quotient of the two Galois field elements in m cycles. In a preferred embodiment, the reciprocal generator may include first and second Galois field multipliers. The first Galois field multiplier may include a first polynomial multiplier circuit and a first Galois field linear transformer. The first Galois field linear transformer may include a matrix of cells. The first Galois field linear transform may include a matrix section and a unity matrix section. The second Galois field multiplier may include a second polynomial multiplier circuit and a second Galois field linear transformer. The second Galois field linear transformer may include a matrix of cells. The second Galois field linear transformer matrix of cells may include a matrix section and a unity matrix section. The output of the first Galois field multiplier may be fed to both multiply inputs of the second Galois field linear multiplier to provide the square of that output. The Galois field reciprocal generator may include a Galois field multiplier including a first polynomial multiplier and a first Galois field transformer and a second Galois field transformer for calculating the square of the first Galois field multiplier output. The second Galois field transformer may be approximately one half the size of the first Galois field transformer. The first and second Galois field transformers each may include a matrix of cells and the second Galois field transformer may include approximately one half the number of cells of the first Galois field transformer. The Galois field reciprocal engine may include a Galois field multiplier and a program circuit for programming the Galois field multiplier to perform a compound multiply-square operation for m−2 times followed by a multiply operation. The invention also features in a broader sense a compound Galois field engine for performing a succession of Galois field linear transforms on a succession of polynomial inputs to obtain an ultimate output where each input except the first is the output of the previous Galois field linear transform. There is an input circuit for providing a first input and a Galois field linear transformer having a matrix of cells responsive to the first input and configured to, in one transform, immediately predict the modulo remainder of the succession of Galois field linear transforms of an irreducible Galois field polynomial to obtain the ultimate output of the Galois field linear transform directly from the first input. This invention also features a method of Galois field division including initially inputting a 1 and a first Galois field element to a Galois field reciprocal generator to obtain an output, multiplying in the Galois field reciprocal generator a first Galois field element by the output of the Galois field reciprocal generator for predicting the modulo remainder of the square of the polynomial product of an irreducible polynomial m−2 times where m is the degree of the Galois field to obtain the reciprocal of the first Galois field element, and multiplying in the Galois field reciprocal engine the reciprocal of the first Galois field element by a second Galois field element for predicting the modulo remainder of the polynomial product for an irreducible polynomial to obtain the quotient of the two Galois field elements in m cycles. This invention also features a Galois field square root engine including a Galois field square root generator and an input circuit for inputting a Galois field element to the Galois field square root generator to obtain the square root of the Galois field elements in one cycle. In a preferred embodiment, the Galois field square root engine may include a Galois field multiplier, and a program circuit for programming the Galois field multiplier to perform a compound square operation of m-l times in one cycle. The invention also features a Galois field square root method including inputting a Galois field element to a Galois field square root generator to obtain an output and squaring in the Galois field square root generator the output of the Galois field square root generator for predicting the modulo remainder of the square of the polynomial product of an irreducible polynomial m−1 times where m is the degree of the Galois field to obtain the square root of the Galois field element in (m−1) cycles. 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: 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. Before disclosing the compound Galois field engine and the divisor engine and method of this invention an explanation of Galois field transformers and multipliers is presented for a better understanding. A Galois field GF(n) is a set of elements on which two binary operations can be performed. Addition and multiplication must satisfy the commutative, associative and distributive laws. A field with a finite number of elements is a finite field. An example of a binary field is the set {0, 1} under modulo 2 addition and modulo 2 multiplication and is denoted GF(2). The modulo 2 addition and multiplication operations are defined by the tables shown in the following illustration. The first row and the first column indicate the inputs to the Galois field adder and multiplier. For e.g. 1+1=0 and 1*1=1.
In general, if p is any prime number then it can be shown that GF(p) is a finite field with p elements and that GF(p In addition, polynomials whose coefficients are binary belong to GF(2). A polynomial over GF(2) of degree m is said to be irreducible if it is not divisible by any polynomial over GF(2) of degree less than m but greater than zero. The polynomial F(X)=X Galois field addition is easy to implement in software, as it is the same as modulo addition. For example, if 29 and 16 are two elements in GF(2 Galois field multiplication on the other hand is a bit more complicated as shown by the following example, which computes all the elements of GF(2 It can be seen that 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) which is algebraically expanded, and like powers are collected (addition corresponds to an XOR operation between the corresponding terms) to give c(x). For example c(x)=(a 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) =x {57}*{83}={c1} because,
An improved Galois field multiplier system Each of the fifteen polynomial c(x) term includes an AND function as represented by an * and each pair of terms are combined with a logical exclusive OR as indicated by a ⊕. This product as represented in Chart II is submitted to a Galois field linear transformer circuit which may include a number of Galois field linear transformer units each composed of 15×8 cells, which respond to the product produced by the multiplier circuit to predict the modulo remainder of the polynomial product for a predetermined irreducible polynomial. The A The Galois field multiplier presented here GF(2 An example of the GF multiplication occurs as follows:
There is shown in Conventional Galois field multiplier engine The number of cells Each cell The efficacy of engine The reduction in the number of required cells is not limited to only polynomials having the same power as the irreducible polynomial. It also applies to any of those having the power of one half or less of the power of the irreducible polynomial. For example, the eight by fifteen matrix If it is desirable to service the intermediate polynomials of power five, six and seven the unity matrix section can be replaced with a sparse matrix section The number of input registers can be reduced from three to two and the number of external buses relied upon to communicate with the digital signal processor (DSP) Another feature is the reconfigurability of Galois field linear transformer circuit The operation of a reconfigurable input Galois field linear transformer circuit is explained in U.S. patent application Ser. No. 10/136,170, filed May 1, 2002 to Stein et al., entitled RECONFIGURABLE INPUT GALOIS FIELD LINEAR TRANSFORMERER SYSTEM (AD-300J) and all its priority applications and documents which are incorporated herein in their entirety by this reference. Although thus far for the sake of simplicity the explanation has been with respect to only one engine, a number of the engines may be employed together as shown in A polynomial multiplier circuit There is shown in The fact that
Replacing q with 2 According to (3) for n=7 we need to calculate β The circuit of
As can be seen, the final value of β ^{−1 }is obtained in (n−1) cycles. The same circuit is generating β^{−1 }for all intermediate powers of m GF(2^{m}) {m=3.7}, for example if m=4, β^{2} ^{ m } ^{−2}=14 is generated at n=3.
In one embodiment, Galois field reciprocal generator The values at inputs Galois field transformers When the Galois field divider engine has been reduced as shown in A A A all with the exclusive OR functions between them. This results in the output c In operation, initially the GFLT is programmed as a compound multiplier performing (GF_MPY(α,β) Thus far the invention has focused on a Galois field divider engine and method and to the ability to reduce that engine in size by first reducing the size of one of the Galois field linear transformers and eliminating one of the polynomial multipliers and then by combining the functions of the two linear transformers so that a succession of Galois field linear transforms on a succession of polynomial inputs is performed to obtain the ultimate output (quotient) as shown in Another example of this fact can be seen in the square root operation of a Galois field member β. There is shown in The fact that √{square root over (β)}=β Replacing q with 2 The Galois field square root method of this invention is shown in In summary, generally a compound Galois field engine 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: Referenced by
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