Abstract:
A parallel, recursive system for generating and checking a CRC value is disclosed, in which the feedback and forward terms are separated, and the forward terms are reduced. Forward logic, which implements the forward terms, is responsive to a number of bits received from the unit of data, and performs logic operations reflecting the reduced forward logic terms on bits received from the unit of data, to produce a first output. In some cases the forward logic is a direct connection to a number of exclusive-OR logic gates. Feedback logic, responsive to an output of a remainder register, operates to perform feedback logic operations reflecting the feedback terms, on an output of the remainder register to produce a second output. The second output is also coupled to the exclusive-OR logic gates. The exclusive-OR logic gates perform a bit-wise exclusive-OR logic operation on the first output and the second output to produce an input of the remainder register. At the end of processing of the unit of data, the remainder register stores the CRC value, or the inverse of the CRC value.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     Not Applicable 
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not Applicable 
     BACKGROUND OF THE INVENTION 
     Cyclic Redundancy Check (CRC) is a well-known error detection and correction technique used in many transmission and storage systems. A number of redundant bits are added to a message or data block, so that errors occurring during transmission or storage can be detected and, possibly, corrected. The degree of error detection is a function of the size of the message or data block, and the particular CRC. 
     One common CRC used in Local Area Networks (LANs) is defined for the ANSI/IEEE Std. 802 family of LAN standards. In that standard, a 4-octet (32-bit) CRC value is loaded into the Frame Check Sequence (FCS) field of a data unit or packet when it is transmitted. The value is computed as a function of the contents of the data unit. The k bits of data covered by the FCS can be represented as a polynomial f(x) of degree k−1. For example: 
     f(x)=10100100=x 7 +x 5 +x 2    
     f(x)=000 . . . 010100100=x 7 +x 5 +x 2    
     f(x)=101001=x 5 +x 3 +1 
     The specific encoding of the CRC value for ANSI/IEEE 802 is defined by the following generating polynomial: 
     G(x)=x 32 +x 26 +x 23 +x 22 +x 16 +x 12 +x 11 +x 10 +x 8 +x 7 +x 5 +x 4 +x 2 +x+1 
     In existing ASNI/IEEE systems, the CRC value corresponding to a given data unit is formed by the following procedure: 
     (1) The first 32 bits of the data unit are complemented. 
     (2) The k bits of the data unit are then considered to be coefficients of a polynomial f*(x) of degree k=1. 
     (3) f*(x) is multiplied by x 32  and divided by G(x) using modula-2 arithmetic, producing a remainder R(x) of degree less than or equal to 31. 
     (4) The coefficients of R(x) are considered to be a 32-bit sequence. 
     (5) The bit sequence is complemented and the result is the CRC value placed in the FCS field. 
     Steps 1 and 5 allow detection of missing or added zero bits at the beginning of a message. The necessary polynomial division in (3) has a well-known recursive form that processes each bit of the message serially and can be implemented simply in hardware using a linear feedback shift register (LFSR) formed by exclusive-OR gates to perform divisions and registers to hold intermediate results. However, such a serial implementation becomes impractical as data rates increase because only one bit of the message is processed at a time. When the data rate becomes sufficiently high, the serial form cannot generate or check the CRC value of a message within the time it takes to transmit the message. Accordingly, CRC related processing becomes an unacceptable bottleneck on message throughput. 
     To address this problem, some existing systems have introduced a level of parallelism into CRC processing. These systems have extended the serial implementation to process several bits of the message in parallel, based on a modified recursive equation. The standard parallel CRC method is described in U.S. Pat. No. 4,593,393 of Mead et al., filed Feb. 6, 1984, and U.S. Pat. No. 5,103,451 of Fossey, filed Jan. 29, 1990. However, this standard parallel approach cannot process many bits in parallel because the number of terms in the logic equations it implements becomes excessively large. As a result, many exclusive-OR gates are required, causing the system to run too slow, and which further occupy too much area on a chip and consume too much power. In addition, the standard parallel approach suffers from limited performance due to a low degree of pipelining in its processing. 
     As the number of bits being processed in parallel increases, the number of message bits covered by the CRC may not always be exactly divisible by the number of bits being processed in parallel. Existing systems for CRC value generation have not addressed this problem, since such existing systems have typically processed 8 bits in parallel, and the messages covered by the CRC value are typically guaranteed to contain an integer number of octets or bytes (8 bit units). 
     Accordingly it would be desirable to have a system for generating and checking a CRC value that processes many bits in parallel without requiring excessive numbers of exclusive-OR gates. The system should be compatible with existing CRC generation and checking standards, and apply to systems for error detection and correction in communications and storage applications. 
     Further, a system is needed to provide additional pipelining in the processing of CRC values. In addition, a system is required that enables CRC checking to be performed on messages that are not equally divisible by the number of bits being processed in parallel. 
     BRIEF SUMMARY OF THE INVENTION 
     A parallel, recursive system for generating a CRC value for a unit of data is disclosed, in which the feedback and forward terms are separated, and the forward terms are reduced. The unit of data may be either a portion of a data unit that is to be transmitted onto a communications network, a portion of a unit of data that has been received from a communications network, or a data block that has been either read or is to be written to a storage device such as a magnetic disk. 
     A forward logic block, which implements the forward terms, is responsive to a number of bits received from the unit of data, and operates to perform logic operations based on the reduced forward logic terms on the bits received from the unit of data, in order to produce a first output. In an illustrative embodiment in which the number of bits being processed in parallel, also referred to as the size of the portion of the unit of data, is less than or equal to the size of the CRC value, then the forward logic block is a direct connection to a number of exclusive-OR logic gates. 
     A feedback logic block, responsive to an output of a remainder register, operates to perform logic operations based on the feedback terms on an output of the remainder register to produce a second output. The second output is also coupled to the exclusive-OR logic gates. 
     The exclusive-OR logic gates perform a bit-wise exclusive-OR logic operation on the first output and the second output to produce a third output. The third output is coupled to an input of the remainder register. 
     In an exemplary embodiment, a first pipeline register receives the first output, and the exclusive-OR logic performs the bit-wise exclusive-OR logic operation on the second output and an output of the first pipeline register, instead of on the first output and.the second output. A second pipeline register, having the bits from the data unit as an input, further has an output coupled to a first input of a multiplexer. The multiplexer has a second input coupled to the output of the remainder register. The multiplexer is controlled to select the output of the remainder register in the event that all bits of the data unit have been processed by the first logic block and the second logic block. Otherwise, the multiplexer is controlled to select the bits from the unit of data. This has the effect of appending the CRC or FCS to the message. 
     In another embodiment, an inverter coupled is coupled to the output of the remainder register, to allow for CRC values with CRC bits inverted. 
     In another embodiment, the forward logic block determines the first output to be the remainder of the division of a polynomial a(x), by a predetermined generating polynomial G(X), where a(x) corresponds to a subsequence of the unit of data, and wherein a(x) is a polynomial of size j−1, where j is equal to a number of bits of the data unit being processed in parallel. The coefficients of a(x) correspond to the bits of the data unit. The feedback logic block determines the second output to be the remainder of the division of a product polynomial by a predetermined generator polynomial G(X), wherein the product polynomial is the result of multiplying the polynomial r(x) by x j , which has the effect of shifting r(x) by j bits in the direction of more significant bits. The coefficients of the polynomial of r(x) correspond to the bits of the remainder register. 
     In another embodiment, the remainder register is initialized to a predetermined value I(x). I(x) is selected such that the output of said second logic block is all ones (is) in the case where I(x) is an input to said second logic block. I(x) is equal to the hexadecimal value 9226F562 if the generator polynomial is equal to the generator polynomial defined for LANs in IEEE 802. This aspect of the invention is distinct over existing systems in which pre-loading of any result or remainder registers uses an initial value of all is 1s or all 0s. 
     Thus, a system is disclosed for generating and checking a CRC value that processes many bits in parallel without requiring excessive numbers of exclusive-OR gates. The disclosed system is compatible with existing CRC generation and checking standards, and applies to systems for error detection and correction in communications and storage applications. Further, the disclosed system provides increased pipelining in the processing of CRC values. The disclosed system also enables CRC checking to be performed on messages that are not equally divisible by the number of bits being processed in parallel. 
    
    
     BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING 
     The invention will be more fully understood by reference to the following detailed description of the invention in conjunction with the drawings, of which: 
     FIG. 1 shows a serial CRC value generator and checker; 
     FIG. 2 shows a parallel CRC value generator and checker with feedback and forward equations separated; 
     FIG. 3 shows a parallel CRC value generator and checker for processing numbers of bits in parallel less than or equal to the size of the CRC value; 
     FIG. 4 shows a parallel CRC value generator and checker for processing numbers of bits in parallel greater than the size of the CRC value; 
     FIG. 5 shows a parallel CRC value generator and checker with termination logic; 
     FIG. 6 shows a pipelined implementation of a parallel CRC value generator and checker; 
     FIG. 7 shows a 32-bit parallel circuit implementation of the CRC value generator and checker of FIG. 3; 
     FIG. 8 shows a 64-bit parallel circuit implementation of the CRC value generator and checker of FIG.,  4 ; 
     FIG. 9 shows a 32-bit parallel circuit implementation of the CRC value generator and checker with pipelining as in FIG. 6; 
     FIG. 10 shows a generalized j-bit CRC value generator and checker circuit; 
     FIG. 11 shows a generalized j-bit CRC value generator and checker circuit with pipelining; 
     FIGS. 12 a  and  12   b  show logic equations of a 32-bit CRC value generator-checker logic module for processing 8 bits in parallel; 
     FIGS. 13 a  and  13   b  show logic equations of a 32-bit CRC value generator-checker logic module for processing 16 bits in parallel; 
     FIGS. 14 a  and  14   b  show logic equations of a 32-bit CRC value generator-checker logic module for processing 24 bits in parallel; 
     FIGS. 15 a ,  15   b , and  15   c  show logic equations of a 32-bit CRC value generator-checker logic module for processing 32 bits in parallel; 
     FIGS. 16 a ,  16   b , and  16   c  show logic equations of a 32-bit CRC value generator-checker logic module for processing 40 bits in parallel; 
     FIGS. 17 a ,  17   b , and  17   c  show logic equations of a 32-bit CRC value generator-checker logic module for processing 48 bits in parallel; 
     FIGS. 18 a ,  18   b ,  18   c , and  18   d  show logic equations of a 32-bit CRC value generator-checker logic module for processing 56 bits in parallel; 
     FIGS. 19 a ,  19   b ,  19   c , and  19   d  show logic equations of a 32-bit CRC value generator-checker logic module for processing 64 bits in parallel; 
     FIGS. 20 a ,  20   b ,  20   c , and  20   d  show logic equations of a 32-bit CRC value generator-checker logic module for processing 72 bits in parallel; 
     FIGS. 21 a ,  21   b ,  21   c , and  21   d  show logic equations of a 32-bit CRC value generator-checker logic module for processing 80 bits in parallel; 
     FIGS. 22 a ,  22   b ,  22   c ,  22   d , and  22   e  show logic equations of a 32-bit CRC value generator-checker logic module for processing 88 bits in parallel; 
     FIGS. 23 a ,  23   b ,  23   c ,  23   d , and  23   e  show logic equations of a 32-bit CRC value generator-checker logic module for processing 96 bits in parallel; 
     FIGS. 24 a ,  24   b ,  24   c ,  24   d , and  24   e  show logic equations of a 32-bit CRC value generator-checker logic module for processing 104 bits in parallel; 
     FIGS. 25 a ,  25   b ,  25   c ,  25   d , and  25   e  show logic equations of a 32-bit CRC value generator-checker logic module for processing 112 bits in parallel; 
     FIGS. 26 a ,  26   b ,  26   c ,  26   d ,  26   e , and  26   f  show logic equations of a 32-bit CRC value generator-checker logic module for processing 120 bits in parallel; 
     FIGS. 27 a ,  27   b ,  27   c ,  27   d ,  27   e , and  27   f  show logic equations of a 32-bit CRC value generator-checker logic module for processing 128 bits in parallel; 
     FIGS. 28 a  and  28   b  show logic equations of a first embodiment of a pipelined 32-bit CRC generator-checker logic module for processing 40 bits in parallel; 
     FIGS. 29 a ,  29   b , and  29   c  show logic equations for a second embodiment of a pipelined 32-bit CRC value generator-checker logic module for processing 40 bits in parallel; 
     FIGS. 30 a  and  30   b  show logic equations of a first embodiment of a pipelined 32-bit CRC value generator-checker logic module for processing 48 bits in parallel; 
     FIGS. 31 a ,  31   b , and  31   c  show logic equations of a second embodiment of a pipelined 32-bit CRC value generator-checker logic module for processing 48 bits in parallel; 
     FIGS. 32 a  and  32   b  show logic equations for a first embodiment of a pipelined 32-bit CRC value generator-checker logic module for processing 56 bits in parallel; 
     FIGS. 33 a ,  33   b , and  33   c  show logic equations of a second embodiment of a pipelined 32-bit CRC value generator-checker logic module for processing 56 bits in parallel; 
     FIGS. 34 a ,  34   b , and  34   c  show logic equations of a first embodiment of a pipelined 32-bit CRC value generator-checker logic module for processing 64 bits in parallel; 
     FIGS. 35 a ,  35   b , and  35   c  show logic equations of a second embodiment of a pipelined 32-bit CRC value generator-checker logic module for processing 64 bits in parallel; and 
     FIGS. 36 a  and  36   b  show an executable model of a 64-bit pipelined CRC value generator-checker circuit. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     1) CRC Fundamentals Using Modula-2 Polynomial Division 
     A message to be covered by CRC protection forms a list of k bits, b 0 ,b 1 ,b k−2 , . . . , b k−1 . By convention, b 0  is the first bit to be transmitted in a serial transmission system. The bits of the message can be represented as a polynomial, f(x), which has degree k−1 and is written as: 
       f ( x )=b 0   x   k−1   +b   1   x   k−2   + . . . +b   k−2   x+b   k−1   
     A generator polynomial, G(x), is chosen, with degree n, such that polynomial division of f(x) by G(x) using modula-2 arithmetic will produce a remainder with degree less than n. When the remainder is converted back into bits, the length of the resulting CRC value is n bits long. Typically, n is chosen to be an integer number of bytes, as in many well known CRC standards such as the ANSI/IEEE 802 LAN CRC (32 bits), the Consultative Committee for International Telegraph and Telephony (CCITT) CRC (16 bits), Asynchronous Transfer Mode (ATM) Header CRC (8 bits), etc. The polynomial division is shown below in equation 1:                    x   n          f        (   x   )           G        (   x   )         =       q        (   x   )       +       r        (   x   )         G        (   x   )                 1                              
     Where q(x) is the quotient of the division that is discarded and r(x) is the remainder. 
     Multiplication (shift) by x n  allows the remainder to be appended to the message. The transmitted message, m(x), is then the concatenation of the message with the remainder, or CRC value, as shown in equation 2 as follows: 
     
       
           m ( x )= x   n   f ( x )+ r ( x )  2 
       
     
     The message m(x) is divisible by G(x) so the remainder will be zero in the absence of errors. 
     This basic procedure is modified in the case of the ANSI/IEEE 802.3 CRC algorithm used in LANs, in order to detect framing errors that result in leading or trailing zeros in serial transmission. The first n bits of the message f(x) are inverted and the remainder is also inverted. 
     In modula-2 notation, the first n bits of the message f(x) are inverted by the following step (equation 3): 
     
       
           f *( x )= f ( x )+ x   n−   L ( x )  3 
       
     
     Where the constant L(x) is defined as a polynomial of order n−1 with all coefficients set to one. 
     
       
           L ( x )= x   n−1   +x   n−2   + . . . +x +1 
       
     
     Substituting equation 3 into equation 1, the division becomes                x   n          f        (   x   )         +       x   k          L        (   x   )             G        (   x   )         =       q        (   x   )       +       r        (   x   )         G        (   x   )                                  
     Inversion of the CRC value is done in a similar way and the message polynomial from equation 2 becomes equation 4: 
     
       
           m ′( x )= x   n   f ( x )+ r ( x )+ L ( x )  4 
       
     
     CRC value checking is based on division of the received message m′(x) by G(x). Without bit inversion, the received message is divided by the generating polynomial, G(x), to yield a zero remainder in the absence of errors.              m   ′          (   x   )         G        (   x   )         =       q   ′          (   x   )                              
     If the division yields a remainder, then an error occurred during transmission or storage. 
     The algorithm is slightly modified if the generating CRC algorithm had employed bit inversion. The first n bits of the received message are inverted and the remainder of the division by the generating polynomial will be L(x) (all is) in the absence of errors.                m   ′          (   x   )       +       x     k   -   n            L        (   x   )             G        (   x   )         =         q   ′          (   x   )       +       L        (   x   )         G        (   x   )                                  
     An optimization is possible that allows the checking hardware to exactly match the generating hardware, which is useful in some applications. The received message, m′(x), is substituted for the original message, f(x) in equation 1. The shift by n bits and division by the generating polynomial still results in no remainder in the absence of errors.              x   n            m   ′          (   x   )           G        (   x   )         =       q   ″          (   x   )                              
     If bit inversion had been deployed during generation, the first n bits of the received message are inverted. In this case, if the received message is error free, the division results in a constant remainder, P(x).                x   n            m   ′          (   x   )         +       x   k          L        (   x   )             G        (   x   )         =         q   ″          (   x   )       +       P        (   x   )         G        (   x   )                     Where                   P        (   x   )         =         x   n          L        (   x   )           G        (   x   )                                
     2) Separation of Feedback and Forward Terms in Serial CRC Generation and Checking 
     The power of CRC protection comes from the fact that it is possible to use simple hardware implementations to calculate the polynomial equations described above in Section 1. The best known of these is the linear feedback shift register (LFSR) implementation, which can be used for both generating and checking CRC values, and which employs exclusive-OR gates to implement incremental polynomial division, together with registers to store intermediate remainders. In a serial algorithm, the remainder is calculated using a recursive form of the polynomial division equations. The level of recursion is set by the number of bits in the message, k. 
     FIG. 1 shows a serial CRC value generator and checker  10 . The system of FIG. 1 is initialized by setting a remainder stored in the CRC Register  12  to an initial value  14 , where the initial value  14  is equal to L(x) if the first n bits of the message are to be inverted, or 0 otherwise. 
     
       
           r   0 ( x )= L ( x ) 
       
     
     or 
     
       
           r   0 ( x )=0 
       
     
     The system of FIG. 1 then proceeds recursively for every bit b i  in the input data  16 . An intermediate remainder is calculated each clock cycle and stored in CRC Register  12  for every bit as follows:                  r   1          (   x   )       =     R        [           x   n            b   0          (   x   )         +     x                     r   0          (   x   )             G        (   x   )         ]                       r   2          (   x   )       =     R        [             x   n            b   1          (   x   )         +     x                     r   1          (   x   )                          G        (   x   )         ]                 ⋮                 r   k          (   x   )       =     R        [             x   n            b     k   -   1            (   x   )         +     x                     r     k   -   1            (   x   )                          G        (   x   )         ]                                    
     Where R[*] is the remainder of the polynomial division *. 
     After all the bits covered by the CRC have passed through the recursive equation, the final remainder, r k (x), is the remainder of the whole division, r(x). If the system of FIG. 1 is generating a CRC value then this is appended to the original message, with or without the CRC value being inverted, as shown in equation 5: 
     
       
           m ( x )= x   n   f ( x )+ r   k ( x )[+ L ( x )]  5 
       
     
     The remainder is compared against a constant  21  by comparison logic  18  in the case of checking to see if any errors have occurred. The constant  21  is either 0 if no bit inversion was used or P(x) if bit inversion was used. 
     In accordance with principles of the present invention, the recursive equation above can be separated into forward terms (with the next data bit as input) and feedback terms (with the current remainder as input), as shown below.                  r   i          (   x   )       =     R        [           x   n            b   i          (   x   )         +     x                     r   i          (   x   )             G        (   x   )         ]                   =       R        [         x   n            b   i          (   x   )           G        (   x   )         ]       +     R        [       x                     r   i          (   x   )           G        (   x   )         ]                     =       A        [       b   i          (   x   )       ]       +     B        [       r   i          (   x   )       ]                                      
     Where A[b i (x)] are the forward terms and B[r i (x)] are the feedback terms. The forward terms are embodied in the system of FIG. 1 by logic block  20 , while the feedback terms are shown as logic block  22 . The logic blocks  20  and  22  may be implemented using exclusive-OR gate trees to perform the necessary polynomial divisions shown above. A modula-2 addition of the remainder outputs of the logic blocks  20  and  22  is performed by exclusive-OR gates  23 , and the result passed to a remainder register, for example CRC Register  12 . 
     In the CRC value generator form of the circuit  10 , as in typical existing systems, the initial value  14  of the CRC Register  12  may be either all zeros (without bit inversion) or all ones (with bit inversion), and the message  16  is fed in bit by bit until all the bits have been processed. The multiplexer  17  is used to append the CRC value  24  (remainder) with or without inversion by an optional inverter  19 , per equation 5. 
     In the CRC value checker form of the circuit, the CRC Register  12  similarly has an initial value  14  of either all zeros (without bit inversion) or all ones (with bit inversion), and the received message  16  is fed in bit by bit until all the bits have been processed. The remainder  24  is checked against a constant  21  to detect any errors. The constant  21  is either 0 if no bit inversion or P(x) with bit inversion. 
     3) Separation of Feedback and Forward Terms in Parallel CRC Generation and Checking 
     The serial algorithm of Section 2 can become difficult to implement at high speeds because it is processing a single bit at a time. The serial recursive equation as described above is not limited to iterating on every bit, but can process many bits simultaneously. The message, f(x), can be grouped into smaller sequences, a i (x), of equal length j, giving rise to another recursive equation.                f        (   x   )       =         b   0          x     k   -   1         +       b   1          x     k   -   2         +   …   +       b     k   -   2          x     +     b     k   -   1                     =       ∑     i   =   0         k   /   j     -   1              x     (     k   -     j        (     i   +   1     )         )              a   i          (   x   )                                        
     Where 
     
       
           a   0 ( x )= b   0   x   j−1   +b   1   x   j−2   + . . . + b   j−2 x+b j−1   
       
     
     
       
           a   i ( x )= b   ji   x   j−1   b   ji+1   x   j−2   + . . . +b   j(i+1)−2   x+b   j(i+1)−1   
       
     
     
       
         · 
       
     
     
       
         · 
       
     
     
       
         · 
       
     
     
       
           a   k/j−1 ( x )= b   k−j   x   j−1   +b   k−j+1   x   j−2   + . . . +b   k−2   x+b   k−1   
       
     
     With the restrictions I≦j≦k and k/j is an integer. 
     FIG. 2 shows a parallel CRC value generator and checker with feedback and forward equations separated. The system of FIG. 2 operates in a similar way to the serial system shown in FIG.  1 . Once again, an initial value may be set, as in existing systems, to either L(x) (all 1s) if the first n bits of the message are to be inverted or 0 otherwise. 
     
       
           r   0 ( x )= L ( x ) 
       
     
     or 
     
       
           r   0 ( x )=0 
       
     
     For every j bits in the message the intermediate CRC is calculated as follows:                  r   1          (   x   )       =     R        [           x   n            a   0          (   x   )         +       x   j                       r   0          (   x   )             G        (   x   )         ]                       r   2          (   x   )       =     R        [             x   n            a   1          (   x   )         +       x   j                       r   1          (   x   )                          G        (   x   )         ]                 ⋮                 r     k   /   j            (   x   )       =     R        [             x   n            a       k   /   j     -   1            (   x   )         +         x                j            r       k   /   j     -   1            (   x   )                          G        (   x   )         ]                                    
     After all the bits have passed through the recursive equation, the final remainder is the remainder of the whole message, r(x), and this is appended to the original message as before, with or without inversion. 
     
       
           m ( x )= x   n   f ( x )+ r   k/j ( x )[+ L ( x )]  6 
       
     
     The remainder is compared against a constant in the case of checking, to see if any errors have occurred; either 0, if no inversion was used, or P(x) if inversion was used. 
     Once again, the recursive equation can be separated into forward terms (with the next data bits as input) and feedback terms (with the current intermediate remainder as input).                  r   i          (   x   )       =     R        [           x   n            a   i          (   x   )         +       x   j            r   i          (   x   )             G        (   x   )         ]                   =       R        [         x   n            a   i          (   x   )           G        (   x   )         ]       +     R        [         x   j            r   i          (   x   )           G        (   x   )         ]                     =       A        [       a   i          (   x   )       ]       +     B        [       r   i          (   x   )       ]                                      
     Where A[a i (x)] are the forward terms and B[r i (x)] are the feedback terms. The forward terms are embodied in forward logic block  31 , while the feedback terms are embodied in feedback logic block  33  in FIG.  2 . The logic blocks  31  and  33  may be implemented using exclusive-OR gate trees to perform the necessary polynomial divisions shown above. A modula-2 addition of the resulting remainder outputs of the logic blocks  31  and  33  is performed by exclusive-OR gates  43 , and the result passed to a remainder register, for example CRC Register  32 . 
     In the generator form, the CRC Register  32  is loaded with an initial value  34  equal to either all zeros without bit inversion or all ones with inversion and the input data  36  is fed j bits at a time until all the bits have been processed. A multiplexer  38  is used to append the CRC value  40  (remainder) to the input data  36  with or without inversion by an optional inverter  42 , per equation 6. 
     The input data  36  can be grouped into sub-sequences in the checker similarly as is done in the generator. The CRC Register  32  is initialized to an initial value  34  equal to either all zeros without bit inversion or all ones with inversion and the received message  36  is fed j bits at a time until all the bits have been processed. The CRC value  40  (remainder) is checked against a constant  44  using comparison logic  45  to detect any errors. The constant  44  is either 0 if no bit inversion or P(x) with bit inversion by optional inverter  42 . 
     4) Parallel CRC Value Generation and Checking with Reduced Forward Terms 
     The parallel CRC implementation discussed above in Section 3 runs into difficulty when it is used to process large numbers of bits in parallel. The terms in the equations become unwieldy which results in slower logic, more area and more power. FIG.  3  and FIG. 4 show parallel CRC value generator and checker circuit embodiments which advantageously separate the forward and feedback terms of the recursive equations, and which further reduce the forward terms substantially over those shown in FIG. 2, resulting in an efficient CRC circuit having significantly better performance than existing systems. In embodiments where the number of bits processed is less than or equal to the CRC value bit-width, the forward terms reduce to a simple bit shift that requires no exclusive-OR gates, as shown by the circuit embodiment of FIG.  3 . For cases where the number of bits being processed is greater than the CRC value bit-width, the forward terms are minimized as shown by the circuit embodiment of FIG.  4 . 
     The systems of FIGS. 3 and 4 generate a remainder, or CRC value, in two steps. Firstly, all the data are processed to generate an intermediate remainder, r a (x). Secondly, the final remainder, r(x), is calculated by multiplying (shifting) the intermediate remainder by n bits and dividing by G(x), as shown below.                  f        (   x   )         G        (   x   )         =         q   a          (   x   )       +         r   a          (   x   )         G        (   x   )                   [   Step1   ]                     x   n            r   a          (   x   )           G        (   x   )         =         q   b          (   x   )       +       r        (   x   )         G        (   x   )                   [   Step2   ]                                
     Step 2 can be omitted in the checker case but is included to make the hardware common between generator and checker. 
     The recursive solution to Step  1  proceeds as follows. The input data  50 , f(x), is grouped into smaller sequences, a i (x), of equal length j.                f        (   x   )       =         b   0          x     k   -   1         +       b   1          x     k   -   2         +   …   +       b     k   -   2          x     +     b     k   -   1                     =       ∑     i   =   0         k   /   j     -   1              x     (     k   -     j        (     i   +   1     )         )              a   i          (   x   )                                        
     Where 
     
       
           a   0 ( x ) =b   0   x   j−1   +b   1 x j−2   + . . . +b   j−2   x+b   j−1   
       
     
     
       
           a   i ( x ) =b   ji   x   j−1   +b   ji+1   x   j−2   + . . . +b   j(i+1)−2   x+b   k−1   
       
     
     
       
         · 
       
     
     
       
         · 
       
     
     
       
         · 
       
     
     
       
           a   k/j−1 ( x ) =b   k−j   x   j−1   +b   k−j+1   x   j−2   + . . . +b   k−2   x+b   k−1   
       
     
     With the restrictions 1≦j≦k and k/j is an integer. 
     To implement step  1 , the result register shown as CRC Register  52  is set to an initial value  54  equal to either I(x) if the first n bits of the message  50  are to be inverted or 0 otherwise. 
     
       
           r   0 ( x )= I ( x ) 
       
     
     or 
     
       
           r   0 ( x )=0 
       
     
     For every j bits in the message  50  the intermediate CRC value is calculated below.                              r   1          (   x   )       =     R        [           a   0          (   x   )       +       x   j            r   0          (   x   )             G        (   x   )         ]                       r   2          (   x   )       =     R        [           a   1          (   x   )       +       x   j            r   1          (   x   )             G        (   x   )         ]                       ⋮                       r     k   /   j            (   x   )       =     R        [           a       k   /   j     -   1            (   x   )       +       x   j            r       k   /   j     -   1            (   x   )             G        (   x   )         ]                                    
     After all the bits have passed through the recursive equation the output is the intermediate remainder, 
     
       
           r   a ( x )= r   k/j ( x ). 
       
     
     Step  2  can proceed using the same recursive equation as step  1  if the number of bits being processed in parallel is less than the CRC value length, or j&lt;n.                              r   b1          (   x   )       =     R        [         x   j            r   a          (   x   )           G        (   x   )         ]                       r   b2          (   x   )       =     R        [         x   j            r   b1          (   x   )           G        (   x   )         ]                       ⋮                     r        (   x   )       =         r     n   /   j            (   x   )       =     R        [         x   j            r       n   /   j     -   1            (   x   )           G        (   x   )         ]                                      
     With the restrictions 1≦j≦n and n/j is an integer. 
     If the number of bits being processed in parallel is greater than or equal to the CRC value length, or j≧n, as shown in FIG. 4, then          r        (   x   )       =     R        [         x   n            r     k   /   j            (   x   )           G        (   x   )         ]                              
     After all the bits have passed through the two step process, the final remainder is the remainder of the whole message, r(x) and this is appended to the original message as before, with or without inversion. 
     
       
           m ( x )= x   n   f ( x )+ r ( x )[+ L ( x )] 
       
     
     In the case of checking, the remainder is compared against a constant. The constant is 0 with no bit inversion or P(x) with bit inversion. 
     Once again, the recursive equation can be separated into forward terms with the next j bits of data bit as input and feedback terms with the current remainder as input, which may be implemented as forward logic  64  and feedback logic  66 , respectively, using state machine structures in hardware.                  r   i          (   x   )       =     R        [           a   i          (   x   )       +       x   j            r   i          (   x   )             G        (   x   )         ]                   =       R        [         a   i          (   x   )         G        (   x   )         ]       +     R        [         x   j            r   i          (   x   )           G        (   x   )         ]                     =       A        [       a   i          (   x   )       ]       +     B        [       r   i          (   x   )       ]                                      
     Where A[a i (x)] are the forward terms and B[r i (x)] are the reverse terms. 
     If the number of bits being processed at a time, j, is less than or equal to the number of bits in the CRC value, n, (or j≦n) then a further simplification is possible. The remainder of a polynomial division is equal to the numerator if the order of the numerator is less than or equal to the denominator. 
     
       
           r   i ( x )= A[a   i ( x )]+ B[r   i ( x )] 
       
     
     
       
          = a   i ( x )+ B[r   i ( x )] 
       
     
     So the forward equations reduce down to through-connections, as shown in FIG. 3, resulting in a reduced gate count in hardware. 
     Accordingly, the CRC Register  52  is initialized to an initial value  54  equal to either all zeros without bit inversion by the optional inverter  62  or I(x) with inversion and the message  50  is fed j bits at a time until all the bits have been processed. A multiplexer  58  is used to append the CRC value (remainder) to the message, with or without inversion, for generation and the comparator  60  is used for checking. 
     5) Remainder (CRC) Register Initialization to Invert the Start of a Message 
     As described above, some CRC checking protocols invert the first n bits of the message to detect any leading zeros that might get added to a message during bad framing in serial transmission. Existing systems provide this inversion by initializing the CRC register to all ones. The disclosed system of FIGS. 3 and 4 obtains the same effect, albeit through use of a very different initial value, referred to herein as I(x). 
     An intermediate remainder from processing the first n bits after initialization, r 1 (x) (shown as r i+1    56 ), is calculated from the outputs of the forward and feedback logic blocks  64  and  66  discussed above with reference to FIGS. 3 and 4. The first n bits of the input data  50  produce a polynomial, a 0 (x), that must be added (Modula two) to the constant, L(x), to get the one&#39;s compliment of the data, as shown by the following equations:                  r   i          (   x   )       =     R        [           a   0          (   x   )       +       x   n          I        (   x   )             G        (   x   )         ]                   =       R        [         a   0          (   x   )         G        (   x   )         ]       +     R        [         x   n          I        (   x   )           G        (   x   )         ]                     =         a   0          (   x   )       +     B        [     I        (   x   )       ]                     =         a   0          (   x   )       +     L        (   x   )                                      
     Where the feedback terms, B[I(x)], are defined as in the following equation 7: 
     
       
           B[I ( x )]= L ( x )  7 
       
     
     Thus the value I(x) is equal to a bit sequence, which when input to the feedback logic block  66 , causes the feedback logic block  66  to output L(x) (all is). This fixes the initial value, I(x), to a constant which can be derived by matrix manipulation. Changing to the matrix form of equation 7, 
     
       
         
           B·I=L 
         
       
     
     Where B is an n by n matrix defining the feedback terms, I is a column matrix of n terms defining the initial value and L is a column matrix of n ones defining the one&#39;s complement matrix. 
     To calculate the initial value, I, requires inverting the B matrix. 
       B·I=L   
     
       
         
           B 
           −1 
           ·B·I=B 
           −1 
           ·L 
         
       
     
     
       
         
           I=B 
           −1 
           ·L 
         
       
     
     For example, by using this equation, the initial value I(x) for a generator/checker circuit compatible with the ANSI/IEEE 802 CRC algorithm is calculated as follows: 
     I(x)=x 30 +x 26 +x 25 +x 23 +x 21 +x 19 +x 18 +x 17 +x 16 +x 14 +x 13 +x 10 +x 6 +x 3 +x+1 
     where the generating polynomial, G(x), is: 
     G(x)=x 32 +x 26 +x 23 +x 22 +x 16 +x 12 +x 11 +x 10 +x 8 +x 7 +x 5 +x 4 +x 2 +x+1 
     the resulting initial value in hexadecimal is 9226F562. 
     6) Terminating the CRC Generator/Checker 
     In many systems, the total number of bits in the message covered by the CRC is guaranteed to be divisible by the number of bits being processed in parallel, such that k/j is an integer. However, in certain applications, such as data communications, where packets are not fixed in length and can come in byte increments, such a guarantee is not always feasible. Even where the number of bits covered by the CRC is known to always be divisible by 8, it is undesirable to limit the number of bits being processed in parallel to  8 . Fortunately, as illustrated by the embodiment shown in FIG. 5, a system is disclosed herein which allows wider implementations. As previously described above, the recursive equations for both the standard and present parallel systems process a constant number of bits, j, per iteration. The equations are still valid if j is a variable and the entire message is processed. 
     The message  50  can be split up into a set of sequences, of length j bits, followed by a single sequence of length m bits where m is less than or equal to j. The set of j-bit sequences can be processed using the same recursive forms as the standard and herein disclosed parallel implementations. The m-bit sequence is processed at the end using a separate equation.                f        (   x   )       =         b   0          x     k   -   1         +       b   1          x     k   -   2         +   …   +       b     k   -   2          x     +     b     k   -   1                     =         ∑     i   =   0         k   /   j     -   2              x     (     k   -     j        (     i   +   1     )         )              a   i          (   x   )           +       a   m          (   x   )                                      
     Where 
     
       
           a   0 ( x )= b   )   x   j−1   +b   1   x    j−2   + . . . +b   j−2   x+b   j−1   
       
     
     
       
           a   i ( x )= b   ji   x   j−1   +b   ji+1   x   j−2   + . . . b   j(i+1)−2   x+b   j(i+1)− 1 
       
     
     
       
         · 
       
     
     
       
         · 
       
     
     
       
         · 
       
     
     
       
           a   k/j−2 ( x )= b   k−2j   x   j−1   +b   k−2j+1   x   j−2   + . . . +b   k−j−2   x+b   k−j−1   
       
     
     
       
           a   m ( x )= b   k−m   x   m−1   +b   k−m+1   x    m−2   + . . . +b   k−2   x+b   k−1   
       
     
     Where I≦j≦k and 1≦m≦j. 
     These equations translate into modified hardware implementations where the last part of the message is processed in a termination logic block. A parallel implementation processing j bits in parallel and terminating with processing m bits is shown in FIG.  5 . The main recursive block  69  has the forward terms, A[a i (x)], implemented by forward logic  72 , and the feedback terms, B[r i (x)], implemented by feedback logic  74 . The termination logic block  70  h as the forward terms, A[a m (x)] as in terminating forward logic  76 , and the feedback terms, B└r i (x)┘, as in terminating feedback logic  78 . In the embodiment described above in section  4 , for parallel CRC value generation and checking with reduced forward terms, and in the case where in is less than or equal to the length of the CRC value, the data input, a m  80 is set to zero which further simplifies the design. 
     The termination logic block  70  that processes the final m-bit sequence operates similarly to the recursive logic block  69 , with forward and feedback logic, except the termination feedback logic  78  operates only on the last output of the register  52   a , and the forward terms operate only on the last m bits of the message. It is possible to continuously calculate candidate m-bit sequences of different lengths in parallel and select the correct CRC at the end of the message, in order to allow a range of m termination values. In this way the number of bits being processed in parallel can be de-coupled from the length of the message. 
     7) Pipelined Version of Parallel Implementations 
     It has been shown above that the CRC generator and checker circuits can be broken down into forward and reverse terms to produce a state machine. The forward terms can further be pre-calculated using a pipeline because they only have input data as input. A pipeline structure allows a faster implementation in hardware and improves test access to the logic blocks. Multiple stages of pipelining are possible in the forward path so the ultimate speed of the implementation will always be defined by the speed of the feedback path. 
     FIG. 6 shows an illustrative embodiment of a pipelined system. The elements of the circuit shown in FIG. 6 are described above with regard to FIG. 4, with the exception of pipeline registers  90  and  91 . In FIG. 6, the pipeline registers  90  and  92  are pre-loaded with data from the input data  50  during a clock cycle preceding operation of the remaining circuit elements of the circuit. Subsequently, the circuit processes the input data using the output of the pipeline register  90  as input to exclusive-OR gates  23 , and the output of pipeline register  92  as an input to the multiplexer  58 . In this way, the pipeline register  92  synchronizes the input data  50  with the output of the CRC Register  52 . The pipelining shown in FIG. 6 improves circuit performance by increasing the level of parallelism in the circuit. 
     8) CRC Circuit and Module Implementations 
     The disclosed parallel CRC design can be embodied in many applications. FIGS. 7,  8  and  9  show three illustrative generator-checkers: a 32-bit wide CRC generator-checker (FIG.  7 ), a 64-bit CRC generator-checker (FIG. 8) and a pipelined 64-bit CRC generator-checker (FIG.  9 ). The circuits shown in FIGS. 7-9 include CRC logic modules  100 , remainder registers  102 , input data register  104 , CRC Controller State Machine  106 , multiplexer  108 , and inverter  108 . The pipelined implementation of FIG. 9 further includes pipeline registers  120 . During operation of the embodiments shown in FIGS. 7-9, input data  50  is received into the input register  104 , and subsequently input to the CRC logic modules  100 . The input data  50  is passed to the CRC logic modules  100  in 32 bit portions in the embodiment of FIG. 7, and in 64 bit portions in the embodiments of FIGS. 8 and 9. The outputs of the CRC logic modules  100  are then stored in the remainder registers  102 , and fed back as r 31  . . . r 0   135  into the CRC logic modules  100 . This process repeats recursively until all the input data  50  has been processed. At the end of the input data  50 , the multiplexer  108  outputs the final contents of a selected one of the 32-bit remainder registers  102 , based on the assertion of its controls  112 . The inverter  110  then inverts the output of the multiplexer  108 , resulting in the final CRC value. 
     As illustrated in FIGS. 7-11, variable length m-bit termination sequences are handled by duplicating the termination block once for every possible m-bit sequence length. Selection of the correct CRC from the array of CRC remainder registers ( 102   a - 102   d  in FIG. 7) is done with a multiplexer ( 108  in FIG. 7) in hardware. 
     The termination equations exactly match the recursive equations when the length of the m-bit sequence is equal to j, where j is the number of bits being processed in parallel by the recursive equations. In that case, the recursive logic block ( 100   d  in FIG. 7) then processes the m-bit sequence and the correct CRC appears at the output of the recursive CRC remainder register ( 102   d  in FIG.  7 ). 
     Further simplification of the termination logic is possible if the parallel algorithm embodied in FIGS. 7-11 is used. The shift specified in Step  2  of the disclosed algorithm, as described above in section  4 , is implemented by appending n zero bits to the end of the message, where n is the number of bits in the CRC value being generated. The n zero bits occupy part, or all, of the final m bits used in the termination logic. This overlap is dependent on the values of n and m, which are constant, so the n zero bits that fall in the m-bit sequence can be hard-wired to zero. This eliminates some XOR gates in the termination logic as well as simplifying routing in hardware implementations. 
     For example, in FIG. 7, logic blocks  100   a ,  100   b  and  100   c  are termination logic blocks for handling termination sequences having 24, 16 and 8 valid bits respectively. The CRC controller state machine  106  operates to first initialize the circuit using the initial value  54 , and then feeds j bits ( 32  in FIG. 7) at a time into the circuit over data input lines  50 . The CRC controller state machine detects a last word of the message by indication provided over last word signal  109 , and also the number of valid bytes in the last word from valid bytes signal  107 . The CRC controller state machine  106  also appends n zero bits at the end of the message, where n is the size of the CRC value (32 bits in the embodiment of FIG.  7 ), by selecting the hardwired zero input  111  of the multiplexer  104  when the final byte of the last word of the message has been received. 
     The CRC controller state machine  106  then selects which remainder register of remainder registers  102  holds the last value to be passed through the multiplexer  108 . Specifically, for the example embodiment shown in FIG. 7, if there are 8 valid bits in the final word, the CRC controller state machine operates to select the remainder register  102   a  using the controls  112  of the multiplexer  108  as the final CRC to pass through the multiplexer  108 , if there are 16 valid bits the contents of remainder register  102   b  is selected, if there are 24 bits the contents of remainder register  102   c  is selected, and if the last word is aligned to a j-bit boundary, the contents of remainder register  102   d  is selected. 
     FIGS. 10 and 11 show a generalized form of the ordinary and pipelined versions respectively. Illustrative logic equations for the CRC logic modules  100  of FIGS. 7-11 are given in FIGS. 12-35. Similar designs are possible for other CRC equations but are not included here. There is no inherent limitation of the bus width for any of these designs. 
     An exemplary embodiment for a communications system having a line rate of 10 gigabits per second is shown by the 64-bit pipelined design of FIG. 9, running at 156.25 MHz, which offers moderate exclusive-OR tree size and good test access. The design requires approximately 4070 exclusive-OR gates (about 40% less than the standard design in existing systems) and uses a maximum of 20 equation terms in 5 levels. 
     The terms r 31 -0 in the logic equations of FIGS. 12-35 correspond to feedback signals r 31 -r0  135  as shown in FIGS. 7-11, while the terms b 0 , b 1 , b 2 , . . . bn correspond to the non-feedback inputs z 0 , z 1 , z 2 , . . . zn or b 0  . . . b(j−1) as in FIGS. 7-11. The + operator represents an exclusive-OR logic operation. 
     As shown in FIGS. 7-11, each one of the CRC logic modules  100  may have a number of zero bit, non-feedback inputs z 0  . . . zn that is a multiple of 8. Those logic modules that have only zero bits as inputs, such as  100   a - 100   c  in FIG. 7, and  100   a - 100   d  in FIG. 8, are purely termination modules and are not used to process bits from the input data. 
     For example, in embodiments where the number of input data bits processed in parallel is greater than the number of bits in the generated CRC value, as in FIG. 8, the logic modules that have only zero bits for non-feedback inputs are those logic modules processing numbers of bits in parallel less than or equal to the number of bits in the CRC value being generated. Accordingly, in FIG. 8, logic module  100   a  only has 8 zero bit inputs z 7  . . . z 0 , logic module  100   b , has 16 zero bit inputs z 15  . . . z 0 , logic module  100   c  has 24 zero bit inputs z 23  . . . z 0 , and so on, up to the logic module processing a number of bits in parallel equal to the number of bits in the CRC value being generated ( 100   d  in FIG.  8 ). The non-feedback inputs to logic modules processing numbers of bits greater than the number of bits in the CRC value being generated are the outputs of multiplexer  104  (b 63  through b 0 ). 
     However, in the embodiment of FIG. 7, the CRC logic module  10   d , which processes a number of bits in parallel equal to the size of the CRC value being generated, receives the output of multiplexer  104  as a non-feedback input. This is because in the embodiment of FIG. 7, the CRC logic module  100   d  processes the input data selected by the multiplexer  104  during processing of the data block before the last word signal  109  is asserted. When the last word signal  109  is asserted, then the multiplexer  104  appends the zero bits  111 , which become inputs to the logic module  10   d . Accordingly, if the contents of register  102   d  is selected by multiplexer  108  as the CRC value after the complete packet or data block has been processed, that contents will reflect processing of the zero bits  111  appended to the input data by the multiplexer  104 . 
     The generalized circuit shown in FIG. 10 may include CRC logic modules for processing varying numbers of bits in parallel. With regard to the pipelined circuit shown in FIG. 9, the CRC logic modules  100  include a first stage, shown by logic modules  100 A 40 ,  100 A 56 ,  100 A 64 , and a second stage, shown by logic modules  100 B 40 ,  100 B 56 , and  100 B 64 . These two stages are also shown in the generalized pipelined representation shown in FIG.  11 . The logic modules  100   a  through  100   d  may similarly have numbers of zero input bits in multiples of 8, up to the number of bits in the CRC value being generated, for example 32 as shown in FIG.  9 . 
     FIGS. 12 a  and  12   b  show logic equations for the output signals of CRC logic module  10   a  as shown in FIGS. 7-11. 
     The logic module  100   a  requires 114 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 8, and has a maximum number of levels of exclusive-OR gates equal to 3. 
     FIGS. 13 a  and  13   b  show logic equations for the output signals of CRC logic module  100   b  as shown in FIGS. 7-11. The logic module  100   b  requires 215 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 11, and has a maximum number of levels of exclusive-OR gates equal to 4. 
     FIGS. 14 a  and  14   b  show logic equations for the output signals of CRC logic module  100   c  as shown in FIGS. 7-11. The logic module  100   c  requires 319 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 14, and has a maximum number of levels of exclusive-OR gates equal to 4. 
     FIGS. 15 a ,  15   b , and  15   c  show logic equations for the output signals of CRC logic module  100   d  as shown in FIGS. 7-11. The logic module  100   d  requires 452 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 18, and has a maximum number of levels of exclusive-OR gates equal to 5. 
     FIGS. 16 a ,  16   b , and  16   c  show logic equations for the output signals of CRC logic module  100   e  as shown in FIGS. 7-11. The logic module  100   e  requires 557 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 16, and has a maximum number of levels of exclusive-OR gates equal to 5. 
     FIGS. 17 a ,  17   b , and  17   c  show logic equations for the output signals of CRC logic module  100   f  as shown in FIGS. 8 and 10. The logic module  100   f  requires 669 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 27, and has a maximum number of levels of exclusive-OR gates equal to 5. 
     FIGS. 18 a ,  18   b ,  18   c , and  18   d  show logic equations for the output signals of CRC logic module  10   g  as shown in FIG.  8 . The logic module  100   g  requires 807 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 31, and has a maximum number of levels of exclusive-OR gates equal to 5. 
     FIGS. 19 a ,  19   b ,  19   c , and  19   d  show logic equations for the output signals of CRC logic module  100   h  as shown in FIG.  8 . The logic module  100   h  requires 937 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 35, and has a maximum number of levels of exclusive-OR gates equal to 6. 
     FIGS. 20 a ,  20   b ,  20   c , and  20   d  show logic equations for the output signals of a logic module having 72 zero bit inputs, as would be used in an embodiment of the generalized circuit of FIG.  10 . The logic module shown in FIGS. 20 a ,  20   b ,  20   c , and  20   d  requires 1049 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 41, and has a maximum number of levels of exclusive-OR gates equal to 6. 
     FIGS. 21 a ,  21   b ,  21   c , and  21   d  show logic equations for the output signals of a CRC logic module having 80 zero bit inputs, as would be used in an embodiment of the generalized circuit of FIG.  10 . The logic module of FIGS. 21 a ,  21   b ,  21   c , and  21   d  requires 1169 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 45, and has a maximum number of levels of exclusive-OR gates equal to 6. 
     FIGS. 22 a ,  22   b ,  22   c ,  22   d , and  22   e  show logic equations for the outputs of a CRC logic module having 88 zero bit inputs, as would be used in an embodiment of the generalized circuit of FIG.  10 . The logic module of FIGS. 22 a ,  22   b ,  22   c ,  22   d , and  22   e  requires 1305 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 51, and has a maximum number of levels of exclusive-OR gates equal to 6. 
     FIGS. 23 a ,  23   b ,  23   c ,  23   d , and  23   e  show logic equations for the outputs of a CRC logic module having 96 zero bit inputs, as would be used in an embodiment of the generalized circuit of FIG.  10 . The logic module of FIGS. 23 a ,  23   b ,  23   c ,  23   d , and  23   e  requires 1440 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 53, and has a maximum number of levels of exclusive-OR gates equal to 6. 
     FIGS. 24 a ,  24   b ,  24   c ,  24   d , and  24   e  show logic equations for the outputs of a CRC logic module having 104 zero bit inputs, as would be used in an embodiment of the generalized circuit of FIG.  10 . The logic module of FIGS. 24 a ,  24   b ,  24   c ,  24   d , and  24   e  requires 1572 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 56, and has a maximum number of levels of exclusive-OR gates equal to 6. 
     FIGS. 25 a ,  25   b ,  25   c ,  25   d , and  25   e  show logic equations for the outputs of a CRC logic module having 112 zero bit inputs, as would be used in an embodiment of the generalized circuit of FIG.  10 . The logic module of FIGS. 25 a ,  25   b ,  25   c ,  25   d , and  25   e  requires 1709 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 60, and has a maximum number of levels of exclusive-OR gates equal to 6. 
     FIGS. 26 a ,  26   b ,  26   c ,  26   d ,  26   e , and  26   f  show logic equations for the outputs of a CRC logic module having 120 zero bit inputs, as would be used in an embodiment of the generalized circuit of FIG.  10 . The logic module of FIGS. 26 a ,  26   b ,  26   c ,  26   d ,  26   e , and  26   f  requires 1850 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 65, and has a maximum number of levels of exclusive-OR gates equal to 7. 
     FIGS. 27 a ,  27   b ,  27   c ,  27   d ,  27   e , and  27   f  show logic equations for the outputs of a CRC logic module having 128 zero bit inputs, as would be used in an embodiment of the generalized circuit of FIG.  10 . The logic circuit of FIGS. 27 a ,  27   b ,  27   c ,  27   d ,  27   e , and  27   f  requires 1995 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 70, and has a maximum number of levels of exclusive-OR gates equal to 7. 
     FIGS. 28 a  and  28   b  show logic equations for outputs of CRC logic module  100 A 40 , having 40 zero bit inputs, as shown in FIGS. 9 and 11. The logic module  100 A 40  requires 114 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 8, and has a maximum number of levels of exclusive-OR gates equal to 3. 
     FIGS. 29 a ,  29   b , and  29   c  show logic equations for outputs of CRC logic module  100 B 40 , having 40 zero bit inputs, as shown in FIGS. 9 and 11. The logic module of FIGS. 29 a ,  29   b , and  29   c  requires 443 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 19, and has a maximum number of levels of exclusive-OR gates equal to 5. 
     FIGS. 30 a  and  30   b  show logic equations for outputs of CRC logic module  100 A 48  as shown in FIG. 9, having 48 zero bit inputs. The logic module of FIGS. 30 a  and  30   b  requires 215 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 11, and has a maximum number of levels of exclusive-OR gates equal to 4. 
     FIGS. 31 a ,  31   b , and  31   c  show logic equations for outputs of CRC logic module  100 B 48 , as shown in FIG. 9, having 48 zero bit inputs. The logic module of FIGS. 31 a ,  31   b , and  31   c  requires 454 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 20, and has a maximum number of levels of exclusive-OR gates equal to 5. 
     FIGS. 32 a  and  32   b  show logic equations for outputs of CRC logic module  100 A 56 , as shown in FIG. 9, having 56 zero bit inputs. The logic module of FIGS. 32 a  and  32   b  requires 319 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 14, and has a maximum number of levels of exclusive-OR gates equal to 4. 
     FIGS. 33 a ,  33   b , and  33   c  show logic equations for outputs of CRC logic module  100 B 56 , as shown in FIG. 9, having 56 zero bit inputs. The logic module of FIGS. 33 a ,  33   b , and  33   c  requires 488 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 20, and has a maximum number of levels of exclusive-OR gates equal to 5. 
     FIGS. 34 a ,  34   b , and  34   c  show logic equations for the outputs of CRC logic module  100 A 64 , as shown in FIG. 9, having 64 zero bit inputs. The logic module of FIGS. 34 a ,  34   b , and  34   c  requires 452 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 18, and has a maximum number of levels of exclusive-OR gates equal to 5. 
     FIGS. 35 a ,  35   b , and  35   c  show logic equations for the outputs of CRC logic module  100 B 64 , as shown in FIG. 9, having 64 zero bit inputs. The logic module of FIGS. 35 a ,  35   b , and  35   c  requires 485 2-input exclusive-OR gates, has a maximum number of terms for any one equation of 20, and has a maximum number of levels of exclusive-OR gates equal to 5. 
     FIGS. 36 and 36 b  show an executable model of a 64-bit pipelined CRC value generator-checker circuit. 
     Those skilled in the art should readily appreciate that the functions of the present invention can be implemented in many forms, including using hardware components such as Application Specific Integrated Circuits or other hardware, or some combination of hardware components and software. Where a portion of the functionality is provided using software, that software may be provided to the computer in many ways; including, but not limited to: (a) information permanently stored on non-writable storage media (e.g. read only memory devices within a computer such as ROM or CD-ROM disks readable by a computer I/O attachment); (b) information alterably stored on writable storage media (e.g. floppy disks and hard drives); or (c) information conveyed to a computer through communication media such as computer or telephone networks via a modem. 
     While the invention is described through the above exemplary embodiments, it will be understood by those of ordinary skill in the art that modification to and variation of the illustrated embodiments may be made without departing from the inventive concepts herein disclosed. Accordingly, the invention should not be viewed as limited except by the scope and spirit of the appended claims.