High performance CRC calculation method and system with a matrix transformation strategy

[0001] The present invention relates generally to a CRC calculation, and, more particularly, to a high performance CRC calculation method and system with a matrix transformation strategy.

[0002] Cyclic redundancy code (CRC) has been used for a long time to preserve the integrity of digital data in storage and transmission systems. More particularly, CRC is an important error detection tool used for communications and data processing applications. The CRC schemes are often used for checking integrity of data because they are easy to implement and they detect a large class of errors. CRC is a kind of checksum which is transmitted with data between a source node and a target node over a communications medium. The source node calculates the CRC for the data to be transferred using a predetermined polynomial and then transmits the data along with the CRC to the target node where the CRC of the received data is independently generated using the predetermined generator polynomial and compared with the CRC received from the source node to check if errors have occurred during the transmission. Treating the data or message as a binary polynomial, its CRC corresponding to a particular generator polynomial may be generated by raising the message polynomial to a proper power first and then taking the remainder of the message polynomial divided by the generator polynomial. For CRC generation, data bits are typically serially inputted into a CRC generator in order to produce the appropriate CRC code for transmission along with the data. Traditionally, CRC codes are generated with Linear Feedback Shift Register (LFSR) circuits. An LFSR takes the input data and shifts through a series of flip-flops on successive clock cycles. Combinations of the shift register output and data input are fed back to the flip-flops via exclusive-OR gates. An LFSR can be defined in terms of a generator polynomial which relates the input data and the CRC code via a polynomial expression and of which “+” is an exclusive-OR operation. The state of the flip-flops upon completion of the shifting process is the CRC code.

[0003] For example, ATM uses a FCS field derived from CRC error detection codes for error checking. The integrity of the transmitted or processed message in an ATM system is ensured by the addition at the end of the message of the FCS traveling with the message itself so it can be checked on the reception side for proper transmission. The FCS code has been standardized for data integrity checking as described in the ANSI X3.139-1987 document pages 28 and 29. All the CRC codes constitute a finite Galois Field (GF), and the CRC32 codes belong to the GF generated by the following generator polynomial of degree 32:

[0004] $g\left(x\right)=x^{32}+x^{26}+x^{23}+x^{22}+x^{16}+x^{12}+x^{11}+x^{10}+x^8+x^7+x^5+x^4+x^2+x+1.$
This generator polynomial of degree 32 was chosen as a standard for error checking in Ethernet and then chosen by the ATM standard for AAL5 error checking. In the circuitry for calculating the FCS or checking the message, an LFSR carries out a bit by bit multiplication in the GF, i.e., modulo the polynomial on which GF is generated, and by which each bit of the message is inputted into the LFSR in the manner of most significant bit (MSB) first and division is performed by feedbacks. At the end of the process, the FCS, i.e., the remainder of the division, is within the shift registers.

[0005] Hardware implementation for CRC generators in large scale digital systems is preferred because it is faster. The drawback of hardware implementation of CRCs is that more hardware is required with consequent increase in cost, size and complexity and a decrease in reliability. Software implemented CRCs are known although their use is not widespread because of the speed penalty thought to be inevitable. Those skilled in the art understand that choosing a polynomial of a larger degree will result in greater error detection. However, for the applications of current large scale systems, the desired hardware becomes too complicated and costly to be implemented and the required software needs huge computations. Several improvements were made for CRC generators. For example, by using CRC routines to generate tables consisting of all possible combinations of the chosen polynomial, the checksum generation is reduced to a table lookup. These CRC routines are considered to be the fastest software implementations available, but they take up a great deal of dedicated memory. Early CRC implementations use the concept of LFSR in which the polynomial division is processed one bit at a time. However, the serial processing for the generation of the CRC is relatively slow, and as the technology advanced, single-bit CRC generation was not enough to handle high-speed data processing and transmission, and parallel CRC algorithms were then developed to meet this need. The key reason that existing CRC algorithms are limited in their degree of parallelism is deeply rooted in the concept of LFSRs. All existing algorithms try to solve the same problem, i.e., how to parallelize the bit-by-bit operation of LFSRs. As a result, the degree of parallelism never goes beyond the perceived size of LFSRs.

[0006] Accordingly, it is desired a CRC calculation method and system to reduce the processing for generation of CRC codes.

[0007] The present invention is directed to a methodology to simplify the CRC calculation, by which the process for the CRC generation is speeded up and the circuitry for the system is simplified.

[0008] In a method and system to simplify the CRC calculation, according to the present invention, a linear mapping matrix is used for the operation of the LFSR to generate the CRC and the maximum value of the non-zero entries in the mapping matrix is reduced by applying one or more raw operations to the linear mapping matrix in advance before the computation of mapping the input message to a CRC result for the CRC generation. Various transformation matrixes are provided for the reduction of the maximum value of the non-zero entries in the mapping matrix. In addition, the input messages are padded with specific dummies on the transmission side or the CRC outputs on the reception side are compared with specific patterns in accordance with their length types for the doubleword-wise CRC32 case.

Cyclic Code in a Systematic Form

[0026] As is well-known, an (n, k) linear code C is called a cyclic code if every cyclic shift of a code vector in C is also a code vector in C. To figure out a cyclic code in a systematic form on the transmission side, let the message to be encoded is
M=(m_k−1 . . . m₁m₀)^T,  (EQ-1)
and the corresponding message polynomial is
m(x)=m₀x^k−1+m₁x^k−2+ . . . +m_k−2x+m_k−1.  (EQ-2)
After multiplying m(x) by x^n−k, equation EQ-2 becomes
x^n−km(x)=m₀xⁿ⁻¹+m₁xⁿ⁻²+ . . . +m_k−2x^n−k+1+m_k−1x^n−k.  (EQ-3)
Then, x^n−km(x) is divided by the generator polynomial g(x), and it becomes
x^n−km(x)=q(x)g(x)+r(x).  (EQ-4)
By rearranging equation EQ-4 and inversing the sign of the remainder to replace the original one, it will be obtained the codeword polynomial
x^n−km(x)+r(x)=q(x)g(x).  (EQ-5)
Obviously, this codeword polynomial is divisible by the generator polynomial g(x).

[0027] From the above description, it can be summarized that a cyclic encoding in a systematic form includes:

• [0028] Step 1. Multiplying the message m(x) by x^n−k;
• [0029] Step 2. Deriving the remainder r(x) by dividing x^n−km(x) by the generator polynomial g(x); and
• [0030] Step 3. Combining r(x) with x^n−km(x) to obtain the codeword polynomial x^n−km(x)+r(x).

[0031] Likewise, in order to check the integrity of the received codeword on the reception side, it is verified if the received sequence is divisible by the generator polynomial g(x).

Shortened Cyclic Codes

[0032] Given an (n, k) cyclic code C, if the set of the code vectors for which the l leading high-order information digits are identical to zero, then there are 2^k−l such code vectors and they form a linear subcode of C. If the l zero information digits are deleted, it is obtained a set of 2^k−l vectors of length n−l. These shortened vectors form an (n−l, k−l) linear code, and which code is called a shortened cyclic code and is not cyclic.

Implementation of Divisor

[0033] No matter for a cyclic code encoding or decoding, a divisor of Galois Field GF(2) is needed. For example, a simple Linear Feedback Shift Register (LFSR) is employed to implement the divisor. Furthermore, depending on the dividend sequence shifted into LFSR either from MSB side or Least Significant Bit (LSB) side, there are two schemes for implementation of a divisor, i.e.,

• [0034] Scheme 1: Message is shifted into LFSR from MSB side, which is mathematically equivalent to
  m(x)x^n−kmodg(x)  (EQ-6)
• [0035] Scheme 2: Message is shifted into LFSR from LSB side, which is mathematically equivalent to
  m(x)modg(x)  (EQ-7)
  For illustration, two circuitries are shown in FIG. 1 for these two schemes for the generator polynomial g(x)=x³+x²+1.
  Linear Mapping

[0036] Further, the linear feedback shift registers shown in FIG. 1 can be regarded as a linear mapping mathematically, as shown in FIG. 2. For the same generator polynomial g(x)=x³+x²+1, it can be derived the G mapping:
g₀(2)=g_i(2)⊕g_i(1),  (EQ-8a)
g₀(1)=g_i(0), and  (EQ-8b)
g₀(0)=g_i(2),  (EQ-8c)
and this linear mapping can be represented in a matrix form as

[0037] $\begin{array}{cc}\left[\begin{array}{c}{g}_0\left(2\right)\\ g_0\left(1\right)\\ g_0\left(0\right)\end{array}\right]=G\left[\begin{array}{c}{g}_i\left(2\right)\\ g_i\left(1\right)\\ g_i\left(0\right)\end{array}\right]\mathrm{where}& \left(\mathrm{EQ}\text{-}9\right)\\ G=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ 1& 0& 0\end{array}\right]& \left(\mathrm{EQ}\text{-}10\right)\end{array}$
and trivially, the matrix G is invertible and its inverse matrix is

[0038] $\begin{array}{cc}{G}^{-1}=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 1\end{array}\right]& \left(\mathrm{EQ}\text{-}11\right)\end{array}$

[0039] Based on the Scheme 1 and 2, there exist recursive equations between the output of the D-type flip-flops of the polynomial generator g(x) and the input of the encoded message, respectively, as

• [0040] Scheme 1:
  R(k)=G(R(k−1)+M(k−1)), and  (EQ-12)
• [0041] Scheme 2:
  R(k)=GR(k−1)+M(k−1).  (EQ-13)
  Further tracing the output of the D-type flip-flops, i.e., the remainder of a division, in Scheme 1, it results in
  R(0)=I,  (EQ-14a)
  R(1)=G(R(0)+M(0))=GI+GM,  (EQ-14b)

[0042] $\begin{array}{cc}\begin{array}{c}R\left(2\right)=G\left(R\left(1\right)+M\left(1\right)\right)\\ =G^{2}I+G^{2}M\left(0\right)+\mathrm{GM}\left(1\right),\end{array}\dots & \left(\mathrm{EQ}\text{-}14c\right)\\ \begin{array}{c}R\left(k\right)=G\left(R\left(k-1\right)+M\left(k-1\right)\right)\\ =G^{k}I+G^{k}M\left(0\right)+G^{k-1}M\left(1\right)+\dots +\mathrm{GM}\left(k-1\right)\end{array}& \left(\mathrm{EQ}\text{-}14d\right)\end{array}$
Generation of FCS

[0043] In Standard 802.3, the CRC32 is employed to generate FCS and the generator polynomial is
g(x)=x³²+x²⁶+x²³+x²²+x¹⁶+x¹²+x¹¹+x¹⁰+x⁸+x⁷+x⁵+x⁴+x²+1.  (EQ-15)
Mathematically, the CRC value corresponding to a given frame is defined by the following procedures:

• [0044] a.) The first32 bits of the frame are complemented;
• [0045] b.) The k bits of the frame are then considered to be the coefficients of a polynomial m(x) of degree k−1;
• [0046] c.) m(x) is multiplied by X³² and divided by g(x), producing a remainder r(x) of degree less than or equal to 31;
• [0047] d.) The coefficients of r(x) are considered to be a 32-bit sequence; and
• [0048] e.) The bit sequence is complemented and the result is the FCS f(x).
  In the procedure a, disclosed are two implementation methods:
• [0049] Method 1: complementing the first 32 bits of the message directly; and
• [0050] Method 2: initiating the D-type flip-flop with 1 specific value, e.g., 0xffffffff for the Scheme 1 and 0x46af6449 for the Scheme 2.
  The mapping matrix G and its inverse matrix G⁻¹ are shown in FIG. 3.

[0051] On the reception side, when the whole of frame is acquired, the output of the Scheme 1 CRC checker is compared with the value of 0xc704dd7b to examine the integrity of the received frame. The reason is explained herewith.

Let the transmitted message (except for FCS) is represented in a polynomial form
m(x)=m₀x^k−1+m₁x^k−2+ . . . +m_k−2x+m_k−1,  (EQ-16)
and defining a polynomial c(x) of degree 31 with all of its coefficients to be
c(x)=1x³¹+1x³⁰+ . . . +1x²+1x+1,  (EQ-17)
then the remainder r(x) generated by the procedure c will be
r(x)=(m(x)+c(x)x^k−32)x³²modg(x).  (EQ-18)
After performing procedures d to e, it will generate FCS and the transmitted sequences
f(x)= r(x), and  (EQ-19)
n(x)=m(x)x³²+f(x)  (EQ-20)
On the reception side, if the integrity of this frame sequence is maintained, then the remainder of a division will be

[0052] $\begin{array}{cc}\begin{array}{c}s\left(x\right)=\left(n\left(x\right)+c\left(x\right)x^k\right)x^{32}\mathrm{modg}\left(x\right)\\ =\left(m\left(x\right)x^{32}+f\left(x\right)+c\left(x\right)x^k\right)x^{32}\mathrm{modg}\left(x\right)\\ =\left(m\left(x\right)x^{32}+c\left(x\right)x^k+f\left(x\right)\right)x^{32}\mathrm{modg}\left(x\right)\end{array}& \left(\mathrm{EQ}\text{-}21\right)\end{array}$
From equation EQ-18, equation EQ-21 can be further modified to be

[0053] $\begin{array}{cc}\begin{array}{c}s\left(x\right)=\left(q\left(x\right)g\left(x\right)+r\left(x\right)+f\left(x\right)\right)x^{32}\mathrm{modg}\left(x\right)\\ =\left(r\left(x\right)+f\left(x\right)\right)x^{32}\mathrm{modg}\left(x\right)\\ =c\left(x\right)x^{32}\mathrm{modg}\left(x\right)\\ =\left[0\mathrm{xc704dd7b}\right]\left[x^{31}\dots x^{1}1\right]\end{array}& \left(\mathrm{EQ}\text{-}22\right)\end{array}$

[0054] Based on a similar derivation, it can be further obtained, if the Scheme 2 is adopted, that the checking pattern will be the value of 0xffffffff.

Parallelized CRC Calculation

[0055] So far the encoding message is sequentially inputted to the CRC calculation with one bit each time, however, for high-speed applications, CRC calculation is desired for the capability of multiple message bits inputted, e.g., byte-wise, at a time to increase the throughput. Consequently, the principal architecture of the previous proposed two schemes is maintained and there is only somewhat difference at the mapping matrix.

[0056] Let the input message and the status of the flip-flops be represented, respectively, with a vector form as
M(k)=[m_k 0 . . . 0]^T, and  (EQ-23)
R(k)=[r_k³¹ r_k³⁰ . . . r_k⁰]^T.  (EQ-24)
Tracing the R(k) influenced by the values of M(k),
initially, R(0)=0,  (EQ-25a)
then, R(1)=G(R(0)+M(0))=GM(0), and  (EQ-25b)
R(2)=G(R(1)+M(1))=G²M(0)+GM(1).  (EQ-25c)
It can be further verified
G²M(0)=m₀×the 1st column of the G² matrix, and  (EQ-26a)
GM(1)=m₁×the 1st column of the G matrix.  (EQ-26b)
Defining a new vector with l(≦32) non-zero entries as

[0057] $\begin{array}{cc}{M}_l\left(k\right)={\left[m_k^{1-1}{m}_k^{1-2}\dots m_k^{0}0\dots 0\right]}^T& \left(\mathrm{EQ}\text{-}27a\right)\\ ={\left[m_{k*1}{m}_{k*1+1}\dots m_{k*1+1}0\dots 0\right]}^T& \left(\mathrm{EQ}\text{-}27b\right)\end{array}$
When 1=2 and k=0, it becomes
M₂(0)=[m₀ m₁ . . . 0 . . . 0]^T,  (EQ-28)
and it is further derived
G²M₂(0)=m₀×the 1st column of the G² matrix+m₁×the 2nd column of the G² matrix.  (EQ-29)
Examining the property of the G matrix, it can be found
the 1st column of the G matrix=the 2nd column of the G² matrix.  (EQ-30)
Hence, it is obtained
G²M₂(0)=G²M(0)+GM(1),  (EQ-31)
and it is further included that
G^lM_l(0)=G^lM(0)+G^l−1M(1)+ . . . +GM(l−1), for l≦32.  (EQ-32)

[0058] When the message is inputted in byte-wise form at a time, the input message and calculated remainder vectors are represented as
M₈(k)=[m_k⁷ m_k⁶ . . . m_k⁰ 0 . . . 0]_1×32^T  (EQ-33)
R(k)=[r_k³¹ r_k³⁰ . . . r_k⁰]_1×32^T  (EQ-34)
If the Scheme 1 is adopted, the recursive equation of the input message and calculated remainder is
R(k+1)=T(R(k)+M₈(k)),  (EQ-35)
and the circuitry is shown in FIG. 4.

[0059] Likewise, if the Scheme 2 is adopted, the recursive equation will be
R(k+1)=TR(k)+M₈(k),  (EQ-36)
and its circuitry is also shown in FIG. 4.

[0060] For equation EQ-35 and 36, the mapping matrix T and its inverse are shown in FIG. 5, and of which, the number on the right-hand side of each row indicates how many nonzero entries that row has. For example, in the matrix T, Row 1 has 4 nonzero entries and those rows with maximum nonzero entries, the value of 7, are Row 5, 12 and 13. In the T matrix of FIG. 5, for a specific row, the number of nonzero entries subtracting one is equivalent to how many GF(2) adder (modulo 2) needed. If the maximum value of the non-zero entries in the T matrix is reduced, the operation speed of the CRC circuitry could be promoted further, due to the dramatic reduction of computations in the matrix multiplication or iteration procedure.

Similarity of Matrices

[0061] If there are two matrices T and U with the relationship
U=STS⁻¹,  (EQ-37)
where S is invertible, then the matrix U is similar to the matrix T. Based on the similarity of matrices, there can be obtained two properties

• [0062] a) T^k=S⁻¹U^kS, and
• [0063] b) U is invertible if T is invertible.
  Substitution of the properties into equation EQ-12 and replacement of G with T will result in
  R(k)=S⁻¹US(R(k−1)+M(k−1)),  (EQ-38a)
  SR(k)=U(SR(k−1)+SM(k−1)).  (EQ-38b)
  Let
  {circumflex over (R)}(k)=SR(k), and  (EQ-39a)
  {circumflex over (M)}(k)=SM(k),  (EQ-39b)
  then
  {circumflex over (R)}(k)=U({circumflex over (R)}(k−1)+{circumflex over (M)}(k−1)).  (EQ-40)
  The circuitry corresponding to equation EQ-40 is shown in FIG. 6.

[0064] Further, without effecting the mathematical equality, a 32-bit-wise flip-flops is additionally inserted at the output of the S transform of the circuitry shown in FIG. 6, in order to form a pipeline architecture as shown in FIG. 7.

[0065] On the other hand, if the CRC calculation is applied on the reception side, an alternative method is proposed to omit the practical hardware operation of S⁻¹R(i) for the comparison with P_i, i.e., to compare the output of the flip-flops thereof with the value of SP_i, which can be carried out in advance. The resultant circuitry is shown in FIG. 8.

[0066] For convenience, let Ψ(T) represent the maximum value of non-zero entries among all rows of the matrix T. To analyze the circuitry of FIG. 7, if both Ψ(U) and Ψ(S) are less than Ψ(T), an advantage will be acquired that the CRC calculation can operate in a higher clocking. To find such matrices U and S, one strategy is proposed that the matrix S is produced by the multiplication of some basic row operation matrices R_{(i, j)}, whose function is to add Row i with Row j to thereby produce a new Row j. For a 3×3 matrix T, for example, to generate another matrix U whose Row 0 and 1 are the same as those of the matrix T and whose Row 2 is the summation of Row 0 and 2 of the matrix T, the matrix U can be acquired by multiplying the matrix T in the left side by a row operation matrix

[0067] $\begin{array}{cc}{R}_{\left(0,2\right)}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 1& 0& 1\end{array}\right].& \left(\mathrm{EQ}\text{-}41\right)\end{array}$
Moreover, the invertible matrix of R_(0,2) will be itself, i.e.,

[0068] $\begin{array}{cc}{R_{\left(0,2\right)}}^{*}{R}_{\left(0,2\right)}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=I,\mathrm{and}& \left(\mathrm{EQ}\text{-}42\right)\end{array}$
thus
R_(0,2)⁻¹=R_(0,2).  (EQ-43)
Searching Algorithm for S and U Matrices

• [0069] Step 1: Let S(0)=I, S⁻¹(0)=I, U(0)=T and k=0.
• [0070] Step 2: If there exists a row operation matrix R_{(i, j)} so that both Ψ(R(i,j)U(k)R(i,j))<Ψ(U(k)) and Ψ(R(i,j)S(k))≦Ψ(T), then go to Step 3, else go to Step 4.
• [0071] Step 3: S(k+1)=R(i,j)S(k), S⁻¹(k+1)=S⁻¹(k)R(i,j), U(k+1)=R(i,j)U(k)R(i,j), and k=k+1, then go to Step 2.
• [0072] Step 4: Let S=S(k), S⁻¹=S⁻¹(k), and U=U(k), then STS⁻¹=U, Ψ(U)<Ψ(T), and Ψ(S)≦Ψ(T).
  Doubleword-wise CRC 32

[0073] After searching by running the program to implement the above steps, there are total 242 solutions of S and U matrices satisfied the criteria in the case of doubleword-wise CRC32, among them an excellent solution is shown in FIGS. 9 and 10.

[0074] Other than the case of the message inputted in byte-wise form, however, the MAC frame is based on octet format, and the length of the processed message is not always divisible by 4. As a result, some dummies are padded on the message in order to have the doubleword-wise format when the doubleword-wise CRC calculation is employed. Two strategies for dummy padding are further proposed:

• [0075] Strategy 1: padding with some zero-valued octets before the prefix of the processed message for the transmission side; and
• [0076] Strategy 2: padding with some zero-valued octets after the suffix of the processed message for the reception side.

[0077] When the message is inputted in doubleword-wise form at a time, the input message and calculated remainder vectors are
M₃₂(k)=[m_k³¹ m_k³⁰ . . . m_k⁰]_1×32^T, and  (EQ-44a)
R(k)=[r_k³¹ r_k³⁰ . . . r_k⁰]_1×32^T.  (EQ-44b)
Similar to the byte-wise case, the recursive equation R(k) for the Scheme 1 and 2 are
R(k+1)=T(R(k)+M₃₂(k)), and  (EQ-45a)
R(k+1)=T R(k)+M₃₂(k).  (EQ-45b)
In the doubleword-wise case, no matter what the length of a frame is, they can be classified in accordance with their length into four types: 4n, 4n+1, 4n+2 and 4n+3. If the Strategy 1 is adopted, the initial values of the flip-flops will vary with the length type as listed in

[0078]

TABLE 1
Length type	Padding number	C(0)	C(1)
4n + 3	1	00 ff ff ff	ff 00 00 00
4n + 2	2	00 00 ff ff	ff ff 00 00
4n + 1	3	00 00 00 ff	ff ff ff 00
4n	4	00 00 00 00	ff ff ff ff

The recursive equations are
R(1)=TC(0), and  (EQ-46a)
R(2)=T(R(1)+C(1))=T²C(0)+TC(1)  (EQ-46b)
Let the initial values of the flip-flops be R(0), so that
T²R(0)=T²C(0)+TC(1), or  (EQ-47a)
R(0)=C(0)+T⁻¹C(1),  (EQ-47b)
and R(0) is listed in

[0079]

	TABLE 2
	Length type	The initial value of R(0)
	4n + 3	0x9bf1a90f
	4n + 2	0x09b93859
	4n + 1	0x816474c5
	4n	0x46af6449

[0080] In the strategy 2, the resultant output of the flip-flops will vary with the length of the processed frame, which result implies, for examining the integrity of a received frame, the output Pi is compared with a specified pattern depending on the length type i, in the following rule
Pi=G⁸ⁱ[0xc704dd7b]^T, for i=1, 2, 3 and 4,  (EQ-48)
and the pattern in

[0081]

TABLE 3
Length type	Padding number	Pattern (Pi)
4n + 3	1	0x4710bb9c
4n + 2	2	0x3a7abc72
4n + 1	3	0x8104c946
4n	4	0x69044bb59

If the method 1 is adopted, from Table 2 and the relationship of {circumflex over (R)}(k)=SR(k), it is obtained the initial value of the flip-flops in this transform-type CRC calculation as

[0082]

	TABLE 4
	Length type	Initial value of R(0)
	4n + 3	0x9bf1a10b
	4n + 2	0x09b9385d
	4n + 1	0x816474c5
	4n	0x46af744d

Likewise, if the method 2 is adopted, the checking pattern, based on the transformation and Table 3, will be

[0083]

TABLE 5
Length type	Padding member	Pattern
4n + 3	1	0x4710a398
4n + 2	2	0x3a7abc72
4n + 1	3	0x8104d146
4n	4	0x6904b35d

[0084] The comparison of normal and transform-type CRC calculations is summarized in

[0085]

	TABLE 6
			2XOR		2XOR
			needed in	2XOR	needed in
	Ψ(T)/Ψ(U)	Ψ(S)	T/U	needed in S	S−1
Normal	17	NA	420	NA	NA
Transform	15	3	394	4	4

Byte-wise CRC32

[0086] After searching by running the program, there are more than 10 million solutions of S and U matrices satisfied the criteria in the case of byte-wise CRC32, among them two excellent solutions are provided, one for the transmission side and the other for the reception side. For the excellent solution of the transmission CRC generator, FIGS. 11 and 12 show the matrices U_tx-08 and S_tx-08, and FIG. 13 depicts the corresponding circuitry. For the excellent solution of the reception CRC checker, FIGS. 14 and 15 show the matrices U_rx-08 and S_rx-08, and FIG. 16 depicts the corresponding circuitry.

[0087] From equation EQ-40, the initial value of the flip-flops can be calculated out. Let
{circumflex over (R)}(4)=(U⁴+U³+U³+U){circumflex over (M)}(0).  (EQ-49)
Assume
{circumflex over (R)}(4)=U⁴{circumflex over (R)}(0),  (EQ-50)
then
{circumflex over (R)}(0)=(I+U⁻¹+U⁻²+U⁻³){circumflex over (M)}(0).  (EQ-51)
The comparison of normal and transform-type CRC calculations is summarized in

[0088]

	TABLE 7
			2XOR		2XOR
			needed in	2XOR	needed in
	Ψ(T)/Ψ(U)	Ψ(S)	T/U	needed in S	S−1
Normal	7	NA	106	NA	NA
Tx	5	3	80	0	11
Transform
Rx	5	4	78	0	0
Transform

[0089] While the present invention has been described in conjunction with preferred embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications and variations that fall within the spirit and scope thereof as set forth in the appended claims.