Patent Publication Number: US-8121285-B2

Title: Data processing for coding

Description:
BACKGROUND 
     1. Field 
     The present invention relates to an efficient data processing and more specifically to generating an output data having a large minimum weight. 
     2. Description of Related Art 
     Typically, there are three main aspects of a cryptographic system. One aspect is to encrypt data using a secret key, another aspect is to digitally sign the data using a secret key, and the third aspect is to obtain a cryptographic fingerprint or a cryptographic hash. The cryptographic hash is used to digitally sign large files, by first obtaining a small fingerprint and then signing the small fingerprint using a secret key. 
     Methods to encrypt data using a secret key exist and are commonly called block ciphers, as these block ciphers encrypt data by dividing them into blocks of data, and processing a block at a time, with a feedback of some sort. The methods to sign data are called authentication schemes, or public key signature schemes, depending on whether a signature is required to have a property verifiable by a public party. The methods to obtain a fingerprint or cryptographic hash are called cryptographic hash functions, or just hash functions. 
     Typical cryptographic primitives use error-correcting codes to serve as linear mixers. Most error-correcting codes are linear mixers, and tend to have mixing properties desired in cryptographic primitives. 
     SUMMARY 
     Embodiments of the present disclosure provide a system and method for coding data in an efficient manner. The present disclosure teaches how to make a data coding system. 
     Briefly described, in architecture, one embodiment of the system, among others, can be implemented as follows. 
     The system may include a first processor configured to provide a plurality of tables having a plurality of elements, wherein each of the plurality of elements includes a first plurality of bits. A second processor may be coupled to the first processor, and the second processor may be configured to provide a first plurality of data having a second plurality of bits. A third processor may be coupled to the second processor, and the third processor may be configured to process the first plurality of data into a plurality of table indices. A data retriever may be coupled to the third processor, and the data retriever may be configured to retrieve the first plurality of data processed into the plurality of table indices. A data organizer may be coupled to the data retriever, and the data organizer may be configured to organize the first plurality of data into a second plurality of data. An output device may be coupled to the data organizer, and the output device may be configured to output in use the second plurality of data. 
     The present disclosure may also be viewed as providing a method for data processing. The method may include providing a first plurality of bytes of data, non-linearly transforming the first plurality of bytes into a second plurality of bytes, multiplying each of the second plurality of bytes of data by a predetermined constant of a plurality of constants to generate a third plurality of bytes, and organizing in use the third plurality of bytes as a plurality of output bytes. 
     Other systems, methods, features, and advantages of the present disclosure will be, or will become apparent, to a person having ordinary skill in the art upon examination of the following drawings and detailed description. It is intended that all such additional systems, methods, features, and advantages included within this description, be within the scope of the present disclosure, and be protected by the accompanying claims. 
    
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
       Many aspects of the disclosure can be better understood with reference to the following drawings. Components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating principles of the present invention. 
         FIG. 1  is an embodiment of a method of the present disclosure. 
         FIG. 2  illustrates a portion of the method of  FIG. 1 . 
         FIG. 3  illustrates an embodiment of a portion of the method of  FIG. 2 . 
         FIG. 4  illustrates an embodiment of a method of the present disclosure. 
         FIG. 5  illustrates another embodiment of a method of the present disclosure. 
         FIG. 6  illustrates a portion of an embodiment of a method of the present disclosure. 
         FIG. 7  illustrates a portion of  FIG. 6 . 
         FIG. 8  illustrates operations on a byte according to an embodiment of a method of the present disclosure. 
         FIG. 9  illustrates additional operations on the byte of  FIG. 8 . 
         FIG. 10  illustrates operations on a byte according to an embodiment of a method of the present disclosure. 
         FIG. 11  illustrates additional operations on the byte of  FIG. 10 . 
         FIG. 12  illustrates an embodiment of a system of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The present disclosure relates to a system and method for data processing for generating an output data having a large minimum weight. 
     A data mixer has preferable mixing properties if for a large proportion of zero bytes in output bytes, except for a case of all of the output bytes being zero, an input has a large proportion of non-zero input bytes. An embodiment of the present disclosure has preferable mixing properties over 16 bytes, while having an efficient implementation as described below. 
       FIG. 1  is a flowchart of an embodiment of a method  100  of the present disclosure. The method  100  may include providing a plurality of tables having a plurality of elements, wherein each of the plurality of elements comprises a first plurality of bits (block  102 ), providing a first plurality of data having a second plurality of bits (block  104 ), and processing the first plurality of data into a plurality of table indices (block  106 ). The second plurality of bits may be an input byte. The method  100  may also include retrieving the first plurality of data that has been processed into the plurality of table indices (block  108 ), organizing the first plurality of data into a second plurality of data (block  110 ), and outputting in use the second plurality of data (block  112 ). 
       FIG. 2  illustrates a portion of the method  100  of  FIG. 1 . A matrix  202  is multiplied with four columns of input bytes  206  a 1  through d 4 . A symbol  204  indicates a multiplication over a Galois field (GF). An output column  208  of 4 columns is generated. The plurality of tables in the block  102  of the method  100  may be a minimum distance separable matrix. The arrows shown in the output column  208  indicate operation in a row and a column as described below. 
     M 11 , M 12 , . . . , M 43 , and M 44  may be 8-bit constants, possibly positive or negative. The output column  208  shows “T 3 [c 2 ]” as an entry. A table T 3  may be a table indicating a multiplication of a third column of the matrix  202  by a byte c 2  of input bytes  206  wherein the multiplication may be stored in the table T 3 , wherein the entries in the table T 3  may be indexed by the byte c 2 . Accordingly, in the method  100 , the retrieving the first plurality of data, such as input bytes  206 , processed into the plurality of table indices (block  108 ) may include associating the plurality of table indices, such as the byte c 2 , with the plurality of elements, such as M 11 , M 12 , . . . , M 43 , and M 44 . In an exemplary method, M 11 , M 12 , . . . , M 43 , and M 44  may be bytes which may possibly be viewed as elements of a finite field GF(2^8), i.e., the finite field having 256 elements. Similarly, entries in input bytes  206 , and the output column  208  may be bytes, possibly viewed as elements in the same finite field. A matrix multiplication may be considered over the finite field. It is known in the art that an addition of two elements in the finite field is same as performing a bit-wise exclusive-OR (XOR) of the two elements, each element possibly being a byte or an 8-bit value. 
     In one embodiment, the constants M 11 , M 21 , M 31  and M 41  may be set to 00000001, 00000001, 00000010, and 00000111 respectively. As a person having ordinary skill in the art would appreciate, in a field GF(2^8), elements of the fields which are 8-bit quantities may also be viewed as degree 7 binary polynomials. Thus, 00000111 is the polynomial x^2+x+1, and 00000010 is the polynomial x, and 00000001 is 1. Further, the field GF(2^8) may be defined by an underlying irreducible polynomial, which in the present embodiment may be x^8+x^4+x^3+x+1. 
     In another embodiment, the constants M 11 , M 21 , M 31  and M 41  may be set to 00000001, 00000001, 00000111, and 00000100 respectively, or in terms of GF(2^8), the constants may be set to the polynomials 1, 1, x^2+x+1, and x^2 respectively. 
     In the method  100 , the plurality of elements, such as in the block  102  of  FIG. 1 , may be a plurality of predetermined constants M 11 , M 21 , M 31  and M 41 . Each of the plurality of predetermined constants M 11 , M 21 , M 31  and M 41  may be multiplied by a predetermined byte generated from the second plurality of bits, such as the bits of the input bytes  206 . 
     In the method  100 , the retrieving the first plurality of data, such as input bytes  206 , processed into the plurality of table indices (block  108 ) may include loading the first plurality of data into at least one data storage unit. Such storage unit may be a temporary storage unit, such as a computer memory. 
     In the method  100 , the organizing the first plurality of data, such as the input bytes  206 , into the second plurality of data (block  110 ), such as the output column  208 , may include-performing at least one logic operation on at least one of the first plurality of data, such as the input bytes  206  and the second plurality of data, such as the output column  208 . The performing the at least one logic operation may include performing at least one of an AND operation and an exclusive-OR operation. Further, the performing the at least one logic operation may include exclusive-ORing a byte of the at least one of the first plurality of data, such as the input bytes  206 , into a column having the byte c 2  (vertical arrows shown in the output column  208 ) and transposing the column and exclusive-ORing the column into a row having the byte c 2  (vertical arrows shown in the output column  208 ). 
     In the method  100 , the providing the first plurality of data, such as the input bytes  206 , having a second plurality of bits may include arranging the first plurality of data into a p by q matrix where p and q are positive numbers. The arranging the first plurality of data into a p by q matrix may include identifying the first plurality of data by a byte including a first number and a second number, the first number being associated with one of a row number and a column number and the second number being associated with one of a row number and a column number. The first number and the second number may be subscripts as known in matrix algebra. The row number and the column number may be of a matrix. Further, the identifying the first plurality of data by a byte including a first number and a second number may include associating the row number to a row of a location of the byte, possibly in a matrix, and associating the column number to a column of the location of the byte, possibly in a matrix. 
     In the method  100 , the processing the first plurality of data, such as the input bytes  206 , into the plurality of table indices, such as the byte c 2  shown in the output column  208  of  FIG. 2 , may include setting the first plurality of data as one of the plurality of table indices. 
     In the method  100 , the providing the plurality of tables may include providing at least one of a matrix, a circulant matrix, and a minimum distance separable matrix. Circulant matrices have a property that each column is a rotation of entries of other columns. For example, a second column is a rotation of the first column by one byte. Thus, the entire matrix may be specified by specifying the first column and rotating the first column appropriately to generate the other columns. 
       FIG. 3  illustrates an embodiment of a portion of the method  100  of  FIG. 2 . In the method  100 , the transposing the column and exclusive-ORing the column into the row having the byte c 2  may include omitting transposing the column having the byte c 2  and omitting exclusive-ORing the column for a byte located on a diagonal  302  as shown in  FIG. 3 . Vertical arrows indicate an exclusive-OR operation and horizontal arrows indicate a transposing and an exclusive-OR operation. 
       FIG. 4  illustrates an embodiment of a method  400  of the present disclosure. The method  400  may include providing 16 bytes, possibly arranged in 4 columns (block  402 ), applying the method  100  as described above (block  404 ), performing a row rotation (block  406 ), and outputting 16 bytes, possibly arranged in 4 columns (block  408 ). In the method  400 , the outputting in use the second plurality of data from the method  100  may include outputting a permutation of the second plurality of data, such as the output column  208  as described above. In the method  400 , the outputting the permutation of the second plurality of data may include generating a row rotation (block  406 ) of the second plurality of data, such as the output column  208 . 
       FIG. 5  illustrates another embodiment, being a method  500 , of the method  400  of the present disclosure. In the method  500 , an exemplary array of bytes may undergo a row rotation where a row may be left intact. Accordingly, in the method  500 , the generating the row rotation of the second plurality of data, such as the output column  208 , may encompass excluding generating the row rotation of a row of the second plurality of data, such as the output column  208 . The exemplary array of bytes g 1 , g 2 , . . . , q 3 , and q 4  (block  502 ) may be row rotated with a first row left intact (block  504 ), and an output column may be g 1 , g 2 , . . . , q 2 , and q 3  (block  506 ). A first row left intact is merely an illustration that other row(s) may instead be left intact. 
       FIG. 6  illustrates a portion, being a method  600 , of an embodiment of a method of the present disclosure. In the method  600 , an embodiment of a method of the present disclosure transforms input bytes (a 11 , . . . , a 44  at block  602 ) into output bytes (x 11 , . . . , x 44  at block  604 ). 
       FIG. 7  illustrates a portion of  FIG. 6 . In a method  700 , input bytes, such as the input bytes  206  described above, may be arranged in a four by four matrix, with the first column of input bytes being a 11 , a 21 , a 31  and a 41 , the second column being a 12 , a 22 , a 32  and a 42  and so on (block  702 ). Since a table may have entries which may be four bytes long, individual bytes in an intermediate output may be referenced by an additional subscript. Thus, the second byte in the intermediate table lookup T 1 [a 13 ] (block  704 ) may be called T 12 [a 13 ], and the third byte in the same intermediate table lookup may be called T 13 [a 13 ] and so on as in  FIG. 7  (block  706 ). 
       FIG. 8  illustrates operations on a byte according to an embodiment of a method of the present disclosure.  FIG. 8  shows a method  800  where taking a first column and a first row of a matrix (block  802 ), such as the input bytes  206  described above, various tables and table indices are generated (blocks  804 ,  806 ,  808 ,  810 ,  812 ,  814 , and  816 ) for the bytes in question of block  802 . An output is shown in block  818 , such as the output column  208  discussed above. 
       FIG. 9  illustrates additional operations on the bytes of  FIG. 8 . In a method  900 , bytes of a first column and a first row of a matrix (block  902 ), such as the input bytes  206  described above, are shown. A byte x 11  is shown to be generated (block  918 ) by an addition of bytes at predetermined table indices shown in blocks  904 ,  906 ,  908 ,  910 ,  912 ,  914 , and  916 . 
       FIG. 10  illustrates operations on a byte according to an embodiment, being a method  1000 , of a method of the present disclosure.  FIG. 11  illustrates additional operations, in a method  1100 , on the byte of  FIG. 10 .  FIGS. 10 and 11  accomplish operations to generate a byte x 12 . In  FIG. 10 , input bytes at block  1002  are transformed to output bytes at block  1018  wherein tables and table indices are generated at blocks  1004 ,  1006 ,  1008 ,  1010 ,  1012 ,  1014 , and  1016 . In  FIG. 11 , in a method  1100 , bytes of a second column and a first row of a matrix (block  1102 ), such as the input bytes  206  described above, are shown. A byte x 12  is shown to be generated (block  1118 ) by an addition of bytes in tables at predetermined table indices shown in blocks  1104 ,  1106 ,  1108 ,  1110 ,  1112 ,  1114 , and  1116 . The operations of  FIGS. 8-11  are discussed in more detail below. 
     The output bytes, such as the output column  208  described above, may be arranged in a four by four matrix as well, and individual bytes in the output matrix may be named x-ij, for the byte in the i-th row and the j-th column. Then in one embodiment of the present invention, x- 1   j  may be given by T 11 [a 1   j ] plus T 21 [a 2   j ] plus T 31 [a 3   j ] plus T 41 [a 4   j ] plus T 1   j [a 12 ] plus T 1   j [a 13 ] plus T 1   j [a 14 ], for values  1 ,  2 ,  3 , and  4  for j. This is illustrated in  FIGS. 8 and 9  for x 11 , and in  FIGS. 10 and 11  for x 12 . An addition in the field GF(2^8) is a bit-wise exclusive-OR. It may be noted that the term T 1   j [a 11 ] is missing from the above expression. Possible reasons may be an exclusive-OR operation between two identical values at a corner of a matrix, or omitting from transposing a byte on a diagonal. Similarly, x 2   j  is given by T 12 [a 1   j ] plus T 22 [a 2   j ] plus T 32 [a 3   j ] plus T 42 [a 4   j ] plus T 2   j [a 21 ] plus T 2   j [a 23 ] plus T 2   j [a 24 ], for values  1 ,  2 ,  3 , and  4  for j. 
     Further, x 3   j  may be given by T 13 [a 1   j ] plus T 23 [a 2   j ] plus T 33 [a 3   j ] plus T 43 [a 4   j ] plus T 3   j [a 31 ] plus T 3   j [a 32 ] plus T 3   j [a 34 ], for values  1 ,  2 ,  3 , and  4  for j. And x 4   j  may be given by T 14 [a 1   j ] plus T 24 [a 2   j ] plus T 34 [a 3   j ] plus T 44 [a 4   j ] plus T 4   j [a 41 ] plus T 4   j [a 42 ] plus T 4   j [a 43 ], for values  1 ,  2 ,  3 , and  4  for j. 
     In another embodiment of the present invention, the bytes in a transpose of a column may be first reversed before exclusive-ORing into the output rows. Thus, for j taking values  1 ,  2 ,  3 , and  4 , as described earlier, j may be set to values  4 ,  3 ,  2 , and  1  in that order. Consequently, x- 1   j  may be given by T 11 [a 1   j ] plus T 21 [a 2   j ] plus T 31 [a 3   j ] plus T 41 [a 4   j ] plus T 1   s [a 12 ] plus T 1   s [a 13 ] plus T 1   s [a 14 ]. 
     Another embodiment of a method of the present disclosure may include, providing a first plurality of bytes of data, such as the input bytes  206  described above, non-linearly transforming the first plurality of bytes into a second plurality of bytes, multiplying each of the second plurality of bytes of data by a predetermined constant of a plurality of constants to generate a third plurality of bytes, and organizing in use the third plurality of bytes as a plurality of output bytes. In the aforementioned method, the non-linearly transforming the first plurality of bytes may be an operation to obtain a reciprocal, a square, a cube or similar. The predetermined constant of a plurality of constants may be M 11 , M 21 , M 31 , and M 41  as described above. The organizing in use the third plurality of bytes may include ANDing, exclusive-ORing, and transposing an element on a row or a column of a matrix. 
     In the aforementioned method, the multiplying each of the second plurality of bytes may be performed in a finite field. The finite field may have at least one of a characteristic two and a size 256. 
     In another embodiment, a non-linear transformation may be first applied to the input bytes  206  individually, before the 16 bytes are mixed. Thus, a pre-computed table may already incorporate a non-linear transformation into the pre-computed table. 
       FIG. 12  illustrates an embodiment of a system of the present disclosure. A system  1200  may include a first processor  1210  configured to provide a plurality of tables, such as the matrix  202  of  FIG. 2 , having a plurality of elements M 11  . . . M 44 , wherein each of the plurality of elements M 11  . . . M 44  may include a first plurality of bits, a second processor  1220  coupled to the first processor  1210 , the second processor  1220  being configured to provide a first plurality of data having a second plurality of bits, such as the input bytes  206 , and a third processor  1230  coupled to the second processor  1220 , the third processor  1230  being configured to process the first plurality of data, such as the input bytes  206 , into a plurality of table indices, such as byte c 2  shown in the output column  208  of  FIG. 2 . 
     The system  1200  may include a data retriever  1240  coupled to the third processor  1230 , the data retriever  1240  being configured to retrieve the first plurality of data, such as the input bytes  206 , processed into the plurality of table indices, a data organizer  1250  coupled to the data retriever  1240 , the data organizer  1250  being configured to organize the first plurality of data, such as the input bytes  206 , into a second plurality of data, such as the output column  208 , and an output device  1260  coupled to the data organizer  1250 , the output device  1260  being configured to output in use the second plurality of data. 
     In the system  1200 , the data organizer  1250  may be configured to perform at least one operation selected from the group consisting of: exclusive-OR a byte of the at least one of the first plurality of data into a column having the byte, transpose the column, and exclusive-OR the column into a row having the byte. 
     As a person skilled in the art would appreciate, the system  1200  may include a computer  1270  having the computer program product  1280 . As shown, the computer  1270  also includes the first processor  1210 , the second processor  1220 , the third processor  1230 , the data retriever  1240 , the data organizer  1250 , and the output device  1260  coupled to respective components as described above in a manner known in the art, such as electromagnetically, time-multiplexed, and space-multiplexed. 
     The foregoing method  100  or elements of the method  100  could also be stored on a computer-readable storage medium having computer-executable instructions to implement the method  100  or the elements of the method  100 . A computer program product  1280 , shown in  FIG. 12 , may have computer-executable instructions for: providing a plurality of tables having a plurality of elements, wherein each of the plurality of elements comprises a first plurality of bits, such as in the block  102  of the method  100 , providing a first plurality of data having a second plurality of bits, such as in the block  104  of the method  100 , processing the first plurality of data into a plurality of table indices, such as in the block  106  of the method  100 , retrieving the first plurality of data processed into the plurality of table indices, such as in the block  108  of the method  100 ), and organizing the first plurality of data into a second plurality of data, such as in the block  110  of the method  100 , and outputting in use the second plurality of data, such as in the block  112  of the method  100 . 
     In the computer program product  1280 , the organizing the first plurality of data may include performing at least one operation selected from the group consisting of: exclusive-OR a byte of the at least one of the first plurality of data into a column having the byte, transpose the column, and exclusive-OR the column into a row having the byte. In the computer program product  1280 , the outputting in use the second plurality of data may include generating a row rotation of the second plurality of data. 
     As a person having an ordinary skill in the art would appreciate, an arrow entering a block or a symbol indicates an input and an arrow leaving a block or a symbol indicates an output. Similarly, connections described below may be of any electromagnetic type, such as electrical, optical, radio-frequency, and magnetic. 
     The terminology used herein is for the purpose or describing particular embodiments only and is not intended to be limiting or the invention. As used herein, the singular forms “a,” “an,” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprises” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. 
     The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the invention. An embodiment was chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various embodiments with various modifications as are suited to the particular use contemplated. 
     As a person having ordinary skill in the art would appreciate, the elements or blocks of the methods described above could take place at the same time or in an order different from the described order. 
     It should be emphasized that the above-described embodiments are merely some possible examples of implementation, set forth for a clear understanding of the principles of the disclosure. Many variations and modifications may be made to the above-described embodiments of the invention without departing substantially from the principles of the invention. All such modifications and variations are intended to be included herein within the scope of this disclosure and the present invention and protected by the following claims.