these and other objects are attained with encryption / decryption methods and apparatus of the present invention . an embodiment of a method comprises the steps of encrypting a plain - text message by dividing the plain - text message into a multitude of plain - text blocks and encrypting the plain - text blocks to form a multitude of cipher - text blocks . a single pass technique is used in this process to embed a message integrity check in the cipher - text block . the method further comprises the step of decrypting the cipher - text blocks to reform the plain - text blocks , and testing the message integrity check in the cipher - text blocks to test the integrity of the reformed plain - text blocks . it is often advantageous to employ embodiments in which the message integrity check is embedded in the cipher - text blocks by generating a random number , expanding this random number to generate a set of pseudo random numbers , which may be dependent , but are pair - wise differentially uniform in the sense to be defined below , and using the random number and the set of pseudo random numbers to embed the message integrity check in the cipher - text blocks as the cipher - text blocks are being formed . in these embodiments , during the decryption process , the random number and the set of pseudo random numbers are obtained from the cipher - text blocks , and as the cipher - text blocks are decrypted , these set of pseudo random numbers are used to reform the plain - text blocks from the cipher - text blocks . also , the testing step advantageously includes the step of applying a predetermined test to the re - formed plain - text blocks to test the integrity of the re - formed plain - text blocks . it is often advantageous to employ embodiments in which the pairwise differentially uniform pseudo random numbers are embedded in both the plain - text and the cipher - text blocks , leading to encryption schemes which are highly parallelizable . a sequence of n - bit uniformly distributed random numbers s 1 , s 2 , . . . s m is called “ pair - wise differentially - uniform ” if for every n - bit constant number c , and for every pair of indices i , j , i different from j , the probability that the exclusive or function applied to s i and s j results in c is 1 in 2 n . of course , each individual variable s i is random and uniformly distributed if and only if for each n - bit constant number c , the probability that s i equals c is 1 in 2 n . a variable which only approximates this probability in a computational sense is well known in literature as a pseudo random number . the sequence of variables s 1 , s 2 , . . . , s m will be called pair - wise differentially - uniform pseudo random numbers if all or some of the probabilities are approximate in the above sense . it should be noted that a pair - wise independent sequence is a pair - wise differentially - uniform sequence ; hence the latter is a weaker property . however the latter can be computationally cheaper to generate as the following detailed description shows . surprisingly , it suffices to embed a pairwise differentially uniform sequence instead of a pairwise independent sequence to assure message integrity . we now define sequences which are pair - wise differentially - uniform in galois field modulo a prime number p , i . e . gfp . a sequence of n - bit random numbers uniformly distributed in gfp , s 1 , s 2 , . . . s m is herein called “ pair - wise differentially - uniform in gfp ” if for every constant number c in gfp , and for every pair of indices i , j , i different from j , the probability that ( s i – s j ) modulo p results in c is 1 in p . further benefits and advantages of the invention will become apparent from a consideration of the following detailed description , given with reference to the accompanying drawings , which specify and show example embodiments of the invention . in this specification and the drawings , a plain - text message is denoted by “ p ”, and a cipher - text message is denoted by “ c ”. the length of the plain - text and cipher - text are measured in blocks , where a block is the number of bits of input / output of a block cipher used in this construction . in the rest of the document the number n will be used to refer to the number of bits in a block . thus , if the plain - text p is of length m blocks , then the individual blocks of this message are designated p 1 , p 2 , . . . , p m . similarly , the cipher - text blocks are designated c 1 , c 2 , . . . , c m . fig1 – 3 generally illustrate an encryption / decryption procedure of the type referred to as cipher block chaining ( cbc ). in this process , a sending party encrypts a plain - text message using encryption mechanism 101 , and the encrypted message is sent over an insecure or non - secure communication medium 102 to a receiving party . this latter party uses a decryption mechanism 103 to decipher the message . note that both parties have access to the same secret key k . fig2 and 3 respectively show the encryption and decryption mechanisms 101 and 103 in greater detail . with reference to fig2 , mechanism 101 includes a series of cipher blocks 201 . each cipher block is provided with the key k . the first block 201 a is also provided with a random number r that is n bits long . each of the successive cipher blocks in the chain is provided with a respective one block of the plain - text and with the output of the previous block in the chain . advantagously , for each of the cipher blocks after the first one , an exclusive or function is applied to the two text blocks applied to the cipher block . each of the cipher blocks 201 a – 201 m outputs a respective one block of the cipher - text . with reference to fig3 , mechanism 103 includes a series , or chain , of cipher blocks 301 . each block is provided with the key k and with a respective one of the cipher - text blocks . for the first block 301 a in chain 301 , an exclusive or operation is applied to the output of the block and the first cipher - text block . for the other blocks in chain 301 , an exclusive or operation is applied to the output of the block and the input of the previous block . the exclusive or operations performed on the outputs of the cipher blocks produce the original plain - text blocks . fig4 – 7 illustrate an encryption / decryption process embodying an example of the present invention . generally , the encryption process includes three steps . the first step is the randomness generation and its expansion . the second step is the further expansion of the randomness , and the third step is the actual encryption of the plain - text using the above generated randomness to produce the cipher - text . more specifically , in the first step , a random number r is generated . the randomness r may be generated by any of the well known techniques to generate randomness . this number r is applied to the block cipher 401 with key k 2 to produce a pseudo random number r . the number r could also be obtained by any other well known means of expanding randomness . this number r is fed into a pair - wise differentially uniform sequence generator 402 . this generator , using a process discussed below , outputs a series of s values , s 0 , s 1 , s 2 , . . . s m , each of which is also n bits long . fig5 is a flow chart 500 showing how generator 402 works . this generator may work as a well known linear feedback shift register ( lfsr ). first a primitive polynomial g ( x ) representing the galois field gf ( 2 n ) is chosen . such polynomials are well known and published in literature . let g stand for the n - bit number representing g ( x )− x n . at step 501 , a variable i is set equal to 0 , and s 0 is set equal to r ; and then at step 502 , i is compared to m . if i is greater than m , then the routine exits . otherwise , the routine proceeds to 503 , 504 , 505 . at step 503 , the variable s i is copied into a temporary variable t . at step 504 , t is left shifted by 1 bit . at step 505 , it is checked if there was a carry produced from the previous shift operation . if so , the quantity g and t are combined by a bit by bit exclusive or operation to produce the new t . if there was no carry produced , then t is left intact . at step 507 , variable s i + 1 is set to t and i is incremented by 1 , and the routine returns to step 502 . an important advantage of this process is that the expansion does not require any cryptographic operations , like block ciphers . it is well known that to generate a pair - wise independent sequence at least two pseudo random numbers are required , as opposed to just one ( i . e . r ) that is required to generate a pair - wise differntially uniform sequence . with reference again to fig4 , after r and the s values are generated , the blocks of plain - text p 1 – p m are encrypted to obtain the cipher - text blocks c 0 – c m + 1 . a series of m + 2 cipher blocks 403 are used to do this . each of the cipher blocks is provided with the key k 1 . the first block 403 a is also provided with the random number r . each of the following cipher blocks , except the last one 403 n , is provided with a combination of a respective one of the plain - text blocks and the output of the preceding cipher block . in particular , this combination is the result of the exclusive or operation performed on the two inputs , on a bit location by bit location basis . the last cipher block 403 n in the series is provided with the combination of ( i ) the output of the previous block , and ( ii ) the result of a series of exclusive or operations performed on the sequence of plain - text blocks p 1 , p 2 , . . . p m . this combination is the result of the exclusive or operation performed on the two inputs . the output of the first cipher block 403 a is the first block of cipher - text c 0 . the other blocks of cipher - text , c 1 – c m , are obtained by performing the exclusive or operation , on a bit location by bit location basis , on the output of each cipher block and a respective one of the s values . specifically , s 1 – s m , are applied to the outputs of blocks 403 b through 403 m respectively , while s 0 is applied to the output of the last block 403 n . there are other known ways of combining two values of same size . thus , the above operation of performing the exculsive - or operation on the output of each cipher block and a respective one of the s values may be generalized to other ways of combining two values of size one block each . in particular , any group structure well known in algebra , whose elements are of size upto one block , supports addition of two elements of that group . such an addition of two elements of a group may also be used to combine the output of each cipher block and a respective one of the s values . as an even more specific example , if an s value is an element of a galois field gfp , then the s value may be added modulo p to the output of a cipher block , where p is a prime number . even though the output of a block cipher is of n bits , and p may be less than 2 n , the output of the block cipher can still be viewed as an element of gfp . known techniques may be employed to perform the first and second steps of the encryption process . the third step is unique in the way pair - wise differentially uniform randomness is used in the encryption process so as to ensure message integrity . the pseudo code for the third step is listed below . block — encrypt is a block cipher which encrypts one block using a key . it takes tow arguments . the first argument is the block to be encrypted , and the second argument is the key . a01 c 0 = block — encrypt ( r , k 1 ) a02 n 0 = c 0 a03 for i = 1 to m − 1 do a04 n i = block — encrypt ( p i xor n ( i − 1 )) a05 c i = n i xor s i a06 endfor a07 checksum = 0 a08 for i = 1 to m − 1 do a09 checksum = checksum xor p i a10 endfor a11 c m = s 0 xor block — encrypt ( n m − 1 xor checksum , k 1 ) fig6 generally illustrates an example of a decryption process . in this process , the cipher - text blocks are applied to decryption 601 , which outputs the plain - text blocks . then these plain - text blocks are used to determined if p m + 1 is equal to the result obtained by applying the exclusive or function , on a bit location by bit location basis , to the sequence of the plain - text blocks p 1 , . . . p m . the message passes or fails the integrity test if p m + 1 is , respectively , equal or not equal to this result . fig7 illustrates the operation of an example of a decryption 601 in greater detail . as shown in this figure , the decryption includes a series of cipher blocks 602 . each of the cipher blocks is provided with a key k , and with a respective one of the cipher - text blocks c 0 , . . . c m + 1 . each of these blocks , except the first one 602 a , is also provided with a respective one of the s values . in particular , blocks 602 b through 602 m are provided with s 1 through s m respectively , and the last cipher block 602 n is provided with s 0 . the exclusive or operation is performed on the c and s values provided to each cipher block . the output of the first cipher block 602 a is the random number r . for each of the cipher blocks 602 b – 602 n , the exclusive or function is applied to the output of the block and the input to the previous block to obtain a respective one of the plain - text blocks p 1 – p m . the pseudo code for decryption process is given below . in this pseudo code , block — decrypt refers to a block cipher which decrypts one block using a key . it takes two arguments . the first argument is the block to be decrypted , and the second argument is the key . b01 r = block — decrypt ( c 0 , k 1 ) b02 expand r into s 0 , s 1 , . . . s m as in 401 and 402 b03 n 0 = c 0 b04 for i = 1 to m − 1 do b05 n i = c i xor s i b06 p i = n i − 1 xor block — decrypt ( n i , k 1 ) b07 end for b08 p m = n m − 1 xor block — decrypt ( c m xor s 0 , k 1 ) b08 checksum = 0 b09 for i = 1 to m − 1 b10 checksum = checksum xor p i b11 endfor b12 if p m = checksum accept decrypted message p as integral b13 else reject p as not integral the pseudo code below is an alternative embodiment using an alternative way to generate the pair - wise differentially uniform random sequence s 0 , s 1 , . . . s m . again , this alternative way uses a primitive polynomial g ( x ), and arithmetic in a galois field . once again let g stand for the n - bit number representing g ( x )− x n . this technique is again well known in literature , and the uniqueness of this invention lies in applying this sequence to the aforementioned third step of the encryption process . c01 w ( 0 )= r ; s ( 0 )= w ( 0 ); c02 for i = 1 to m − 1 do c03 j = i + 1 ; c04 if ( j is a power of 2 ) c05 w ( i )= w ( i − 1 )& lt ;& lt ; 1 /* left shift w ( i − 1 ) by 1 bit )*/ c06 if ( most significant bit of w ( i − 1 ) is 1 ) then w ( i )= w ( i ) xor g endif c07 endif c08 s ( i )= 0 c09 for k = 0 to n − 1 do c10 if ( least significant bit of j is 1 ) then s ( i )= s ( i ) xor w ( k ) endif c 11 j = j & gt ;& gt ; 1 /* right shift j by 1 bit */ c12 end for c13 end for there is yet another well known technique for generating the pair - wise differentially uniform random sequence s 0 , s 1 , . . . , s m . once again , it uses a primitive polynomial g ( x ). here is the pseudo code for generating the sequence using the well know gray code : d01 w ( 0 )= r ; s ( 0 )= w ( 0 ) d02 for i = 1 to m − 1 do d03 j = i + 1 d04 k = 0 d05 while (( j & amp ; 1 )== 0 ) do d06 k = k + 1 ; j = j & gt ;& gt ; 1 ; /* increment k and right shift j */ d07 end while /* finds the index of the least significant on bit in ( i + 1 ) */ d08 if ( j xor 1 )== 0 )/* ( i + 1 ) is a power of 2 */ d09 w ( k )= w ( k − 1 )& lt ;& lt ; 1 /* left shift w ( k − 1 ) by 1 bit */ d11 if ( most significant bit of w ( k − 1 ) is 1 ) then w ( k )= w ( k ) xor g endif d12 end if d13 s ( i )= s ( i − 1 ) xor w ( k ) d14 end for one well versed with the art can come up with many other ways to generate such pair - wise differentially uniform sequence . as long as the sequence satisfies the pair - wise differentially - uniform property it can be embedded as explained earlier to guarantee an encryption scheme with message integrity . we now describe an alternative embodiment of an encryption scheme in which the pair - wise differentially - uniform sequence is embedded resulting in a secure encryption scheme with message integrity . this encryption scheme is of interest because the same pair - wise differentially - uniform sequence is embedded not just in the output of the cipher blocks , but also in the input blocks . this results in an encryption scheme which is different from the one in fig4 , in that the new scheme is highly parallelizable . fig8 illustrates this alternative encryption mechanism 800 embodying this invention . in this mechanism , each of the blocks is provided with the key k 1 . the first block 403 a is also provided with the random number r . each of the following cipher blocks , except the last one , is provided with a combination of a respective one of the plain - text blocks and a respective one of the s values . specifically , this combination is the result of the exclusive or operation performed on the two inputs , on a bit location by bit location basis . the last cipher block 403 n in the series is provided with the combination of ( i ) s m + 1 and ( ii ) the result of a series of exclusive or operations performed on the sequence of plain - text blocks p 1 , p 2 , . . . , p m . this combination is the result of exclusive or operation performed on the two inputs . as with the system in fig4 , the output of the first cipher block 403 a of mechanism 800 is the first blocks of cipher - text c 0 . the other blocks of cipher - text c 1 – c m are obtained by performing the exclusive or operation , on a bit location by bit location basis , on the output of each cipher block and a respective one of the s values . specifically s 1 – s m are applied to the outputs of blocks 403 b – 403 m respectively , while s 0 is applied to the output of the last cipher block in the series . the invention may also be implemented as an apparatus to encrypt a plain - text message 904 . an example apparatus is shown in fig9 . the apparatus shown includes a randomness generator 901 to generate a first random number ; a randomness transformer 902 to transform said first random number into a first pseudo random number ; a pairwise differentially uniform sequence generator 903 to further expand a randomness of said first random number and / or said first pseudo random number into a set of pair - wise differentially - uniform pseudo random numbers ; an encryptor 905 to divide said plain - text message into a plurality of plain - text blocks , and to encrypt said plain - text blocks to form a plurality of cipher - text blocks ; a checksum generator 908 to combine said plurality of plain - text blocks into at least one check sum ; and an integrity extractor and checker 907 to employ said set of pair - wise differentially - uniform pseudo random numbers , together with said first random number and / or said first pseudo random number , to embed a message integrity check in said cipher - text blocks . in an alternative implementation the said pair - wise differentially - uniform pseudo random numbers may also be employed by the encryptor 905 . other apparatus implementations may be used employing modules known to those skilled in the art to provide the functionality to perform the present invention . the invention may also be implemented as an apparatus to decrypt a cipher - text message 1004 . an example apparatus is shown in fig1 . the apparatus shown includes a decryptor 1005 to divide said cipher - text message 1004 into a plurality of cipher - text blocks , and to decrypt said cipher - text blocks in forming a plurality of plain - text blocks ; a randomness transformer 1002 to transform at least one of said plain - text blocks into a first pseudo random number ; a pairwise differentially uniform sequence generator 1003 to further expand at least one of said plain - text blocks and / or said first pseudo random number into a set of pair - wise differentially - uniform pseudo random numbers ; a checksum generator 1008 to combine said first pseudo random number , and / or said set of pair - wise differentially - uniform pseudo random numbers , and / or said at least one plain - text block to form at least two check sums and to form a plurality of output blocks ; and an integrity extractor and checker 1007 to compare said at least two check sums in declaring success of a message integrity check . in an alternative implementation the said pair - wise differentially - uniform pseudo random numbers may also be employed by the decryptor 1005 . other apparatus implementations may be used employing modules known to those skilled in the art to provide the functionality to perform the present invention . the present invention can be realized in hardware , software , or a combination of hardware and software . the present invention can be realized in a centralized fashion in one computer system , or in a distributed fashion where different elements are spread across several interconnected computer systems . any kind of computer system — or other apparatus adapted for carrying out the methods described herein — is suitable . a typical combination of hardware and software could be a general purpose computer system with a computer program that , when being loaded and executed , controls the computer system such that it carries out the methods described herein . the present invention can also be embedded in a computer program product , which comprises all the features enabling the implementation of the methods described herein , and which — when loaded in a computer system — is able to carry out these methods . computer program means or computer program in the present context include any expression , in any language , code or notation , of a set of instructions intended to cause a system having an information processing capability to perform a particular function either directly or after conversion to another language , code or notation and / or reproduction in a different material form . thus the present invention includes an embodiment of a method for encrypting a plain - text message . the method includes the steps of : generating a first random number ; transforming the first random number into a first pseudo random number ; further expanding a randomness of the first random number and / or the first pseudo random number into a set of pair - wise differentially - uniform pseudo random numbers ; dividing the plain - text message into a plurality of plain - text blocks ; encrypting the plain - text blocks to form a plurality of cipher - text blocks ; combining the plurality of plain - text blocks into at least one check sum ; and employing the set of pair - wise differentially - uniform pseudo random numbers , together with the first random number and / or the first pseudo random number , to embed a message integrity check in the cipher - text blocks . in some embodiments of the method for encrypting a plain - text message the step of encrypting the plain - text blocks includes employing the first random number , and / or the first pseudo random number , and / or the set of pair - wise differentially - uniform pseudo random numbers ; and / or the step of encrypting includes encrypting the first random number ; and / or the step of encrypting includes encrypting the check sum ; and / or the step of combining includes obtaining the check sum from an exclusive - or of the plurality of plain - text blocks ; and / or the step of transforming the random number includes a non - cryptographic or linear operation ; and / or the step of transforming the random number includes a cryptographic operation ; and / or the said set of pair - wise differentially - uniform numbers are set of pair - wise differentially - uniform numbers in gfp ; and / or the step of employing includes pairing the first random number , and / or the first pseudo random number , and / or the set of pair - wise differentially - uniform pseudo random numbers , with the plurality of cipher - text blocks , and combining each pair to form a plurality of output blocks ; and / or the step of combining each pair includes performing an exclusive - or operation upon components of each pair ; and / or the step of employing includes pairing the first random number , and / or the first pseudo random number , and / or the set of pair - wise differentially - uniform pseudo random numbers , with the plurality of plain - text blocks , and combining each pair to form a plurality of input blocks used in the step of encrypting ; and / or the step of combining each pair includes performing an exclusive - or operation upon components of each pair ; and / or the step of combining each pair includes performing a modulo p addition upon components of each pair , where p is a prime number . furthermore the present invention includes an embodiment of a method for decrypting a cipher - text message . the method includes the steps of : dividing the cipher - text message into a plurality of cipher - text blocks ; decrypting the cipher - text blocks in forming a plurality of plain - text blocks ; transforming at least one of the plain - text blocks into a first pseudo random number ; further expanding at least one of the plain - text blocks and / or the first pseudo random number into a set of pair - wise differentially - uniform pseudo random numbers ; combining the first pseudo random number , and / or the set of pair - wise differentially - uniform pseudo random numbers , and / or the at least one plain - text block to form at least two check sums and to form a plurality of output blocks ; and comparing the at least two check sums in declaring success of a message integrity check . in some embodiments of the method for decrypting a plain - text message the step of decrypting the cipher - text blocks includes employing the first pseudo random number , and / or the set of pair - wise differentially - uniform pseudo random numbers ; and / or the step of combining includes pairing said first pseudo random number , and / or said set of pair - wise differentially - uniform pseudo random numbers , with said plurality of plain - text blocks , and using each pair to form a plurality of output blocks and employing the output blocks to form said at least two check sums ; and / or the step of using each pair includes performing an exclusive - or operation upon components of said each pair ; and / or the step of forming includes dividing the said output blocks into at least two subsets , and obtaining said at least two checksums from an exclusive - or of said subsets of output blocks ; and / or the step of transforming said plain - text blocks includes a non - cryptographic or linear operation ; and / or the step of transforming said plain - text blocks includes a cryptographic operation ; and / or the set of pair - wise differentially - uniform numbers are set of pair - wise differentially - uniform numbers in gfp ; and / or the step of employing includes pairing said first random number , and / or said first pseudo random number , and / or said set of pair - wise differentially - uniform pseudo random numbers , with said plurality of cipher - text blocks , and combining each pair to form a plurality of input blocks used in said step of decrypting ; and / or the step of combining each pair includes performing a modulo p addition upon components of each pair , where p is a prime number . the invention may also be implemented as a method for encryption and decryption of a plain - text message . the method may for example include the steps of : generating a first random number ; transforming said first random number into a first pseudo random number ; further expanding a randomness of said first random number and / or said first pseudo random number into a set of pair - wise differentially - uniform pseudo random numbers ; dividing the plain - text message into a plurality of plain - text blocks ; encrypting said plain - text blocks in forming a plurality of cipher - text blocks ; combining said plurality of plain - text blocks into at least one check sum ; and employing said first random number , said first pseudo random number and said set of pair - wise differentially - uniform pseudo random numbers to embed a message integrity check in said cipher - text blocks to form a cipher - text message ; dividing said cipher - text message into a plurality of cipher - text blocks to form an encryption of said plain - text message ; decrypting said cipher - text blocks in forming a plurality of plain - text blocks ; transforming at least one of said plain - text blocks into a first pseudo random number ; further expanding at least one of said plain - text blocks and / or said first pseudo random number into a set of pair - wise differentially - uniform pseudo random numbers ; combining said first pseudo random number , and / or said set of pair - wise differentially - uniform pseudo random numbers , and / or said at least one plain - text block to form at least two check sums and to re - form the said plain - text message ; and comparing said at least two check sums in declaring success of a message integrity check in decryption of said cipher - text to reform said plain - text message . the invention may also be implemented as an article of manufacture comprising a computer usable medium having computer readable program code means embodied therein for causing encryption of a plain - text message , the computer readable program code means in the article of manufacture comprising computer readable program code means for causing a computer to effect the apparatus and / or steps of methods of the present invention . the invention may also be implemented as a computer program product comprising a computer usable medium having computer readable program code means embodied therein for causing encryption of a plain - text message , the computer readable program code means in the computer program product comprising computer readable program code means for causing a computer to effect the apparatus and / or the steps of methods of the present invention . the invention may also be implemented as a program storage device readable by machine , tangibly embodying a program of instructions executable by the machine to perform method steps for encrypting a plain - text message , said method steps comprising the steps of methods and / or apparatus of the present invention . symmetric key encryption has become an integral part of today &# 39 ; s world of communication . it refers to the schemes and algorithms used to secretly communicate data over an insecure channel between parties sharing a secret key . it is also used in other scenarios like data storage . there are two primary aspects of any security system : confidentiality and authentication . in its most prevalent form , confidentiality is attained by encryption of bulk digital data using block ciphers . the block ciphers ( e . g . des , aes ), which are used to encrypt fixed length data , are used in various chaining modes to encrypt bulk data . one such mode of operation is cipher block chaining ( cbc ). the security of cbc has been well studied ([ 1 ] m . bellare , a . desai , e . jokiph , p . rogaway , “ a concrete security treatment of symmetric encryption : analysis of the des modes of operation ”, 38th ieee focs , 1997 ). cipher block chaining of block ciphers is also used for authentication . the cbc - mac ( cbc message authentication code ) is an international standard . the security of cbc mac was demonstrated in ([ 2 ] m . bellare , j . kilian , p . rogaway , “ the security of cipher block chaining ”, crypto 94 , lncs 839 , 1994 ). authentication in this setting is also called message integrity . despite similar names , the two cbc modes , one for encryption and the other for mac are different , as in the latter the intermediate results of the computation of the mac are kept secret . in fact in most proprietary security systems , two different passes with two different keys , one each of the two modes is used to achieve both confidentiality and authentication . nevertheless , it is enticing to combine the two passes into one , that is in a single cipher block chaining pass , both confidentiality and authentication are assured . many such attempts have been made , which essentially use a simple checksum or manipulation detection code ( mdc ) in the chaining mode ([ 3 ] v . d . gligor , p . donescu , “ integrity aware pcbc encryption schemes ”, 7th intl . workshop on security protocols , cambridge , lncs , 1999 ). unfortunately , all such previous schemes are susceptible to attacks ( see e . g . the section on lower bound ). recently we presented a new variant of cbc mode , which in a single pass achieves both confidentiality and authentication . to encrypt a message of length m blocks , it requires a total of ( m + log m ) block encryptions . all other operations are simple operations , like exclusive - or . to contrast this with the usual cbc mode , the encryption pass requires m block encryptions , and the mac computation requires another m block encryptions . we also show that there is indeed a matching lower bound to our mode of operation , in a reasonable ( linear ) model of computation . this also explains why all previous attempts which tried to attain both features together , without the extra log m cryptographic operations , have failed . a simpler ( though not as efficient ) version of the mode just requires a usual cbc encryption of the plain - text appended with the checksum ( mdc ), with a random initial vector r . as already mentioned , such a scheme is susceptible to message integrity attacks . however , if one “ whitens ” the complete output with a random sequence , the scheme becomes secure against message integrity attacks . whitening just refers to xor - ing the output with a random sequence . the random sequence could be generated by running the block cipher on r + 1 , r + 2 , . . . r + m ( but with a different shared key ). this requires m additional cryptographic operations , and hence is no more efficient than generating a mac . the efficiency of this new mode comes from proving that the output whitening random sequence need only be pair - wise independent . in other words , if the output whitening sequence is s 1 , s 2 , . . . s m , then each s i is required to be random , but only pair - wise - independent of the other entries . such a sequence is generated by performing only log m cryptographic operations like block encryption . we now show that an even weaker condition than pair - wise independence suffices . a sequence of uniformly distributed n - bit random numbers s 1 , s 2 , . . . s m , is called pair - wise differentially - uniform if for every constant c , and every pair i , j , i ≠ j , probability that s i ⊕ s j is c is 2 − 1 . we show that the output whitening sequence need only be pair - wise differentially - uniform . the pair - wise independent sequence generated to assure message integrity can also be used to remove chaining from the encryption mode while still assuring confidentiality . this results in a mode of operation for authenticated encryption which is highly parallelizable . once again , we now show that a pair - wise differentially - uniform sequence suffices to guarantee security of both confidentiality and authentication in this parallelizable version . recently , another scheme was proposed in ([ 4 ] v . g . gligor , p . donescu fast encryption and authentication : xcbc encryption and xecb authentication modes “ http :// www . nist . gov / aes / modes ”), where the output is whitened with the sequence r , 2r , 3r , . . . . however , this scheme is not as secure as whitening the sequence with either the pair - wise independent sequence or the pair - wise differentially - uniform sequence . we first prove the theorems for the variants which employ sequences which are pair - wise independent . these are described in fig1 and fig1 . they are different from the example embodiments ( i . e . fig4 and fig8 ) in that to generate a pair - wise independent sequence at least two new pseudo random numbers need to be generated ( i . e . w 1 , # f : in fig1 ), as opposed to just one pseudo random number r in the example embodiments in fig4 and fig8 . the proof of security of the scheme in fig1 ( i . e . the one using pair - wise independent sequence ) can then be generalized to prove the security of the example embodiment ( i . e . the one using pair - wise additively - uniform sequence ). in fig1 ( and also in fig1 ) a subset construction is employed to produce the seqeunce s 0 , s 1 , . . . s m − 1 from iv 1 , iv 2 , . . . , iv t . the subset construction works as follows ( t =┌ log ( m + 2 )┐): for i = 1 to 2 t − 1 do let & lt ; a 1 , a 2 , . . . a t & gt ; be the binary representation of i s i - 1 = ∑ j = 1 t ⁢ ( a j · iv j ) the scheme in fig1 will be referred to as the iacbc scheme . the scheme in fig1 will be referred to as the iapm scheme . the rest of the paper is organized as follows . the next section gives definitions of random permutations , and formalizes the notions of security , for both confidentiality and message integrity . next we state the theorem for the security of the new modes of operation . we then prove that the new scheme is secure for message integrity . lastly , we describe our model of computation for the lower bound , and prove the lower bound . definition ( random function ) a random function is a function chosen randomly from { 0 , 1 } n →+{ 0 , 1 } l . it could also be viewed as a random sequence ( uniformly chosen ) of length 2 n of 1 bit strings . definition ( random permutation ) a random permutation is a function chosen randomly from class of permutations in { 0 , 1 } n →+{ 0 , 1 } n . it could also be viewed as a random sequence chosen uniformly from the class of all 2 n length sequences of l bit strings , such that each l bit string is represented once in every sequence . the following notion is new ( i . e . non - standard ). the new notion and the following theorem help simplify the proof of message integrity . it essentially separates the approximations in calculating the success probability that result from replacing random permutations by random functions in theorem 3 . definition ( random permutation - like functions ( rpf )) a random permutation - like function with parameter q is a pair of random functions & lt ; f , g & gt ;, with the following restriction for i ∈[ 1 . . . q ] define π ( i )= min { j : j ≦ q and f ( j )= f ( i )} if j = f ( i ) for some i ≦ q , j arbitrary , then g ( j )= π ( i ). a permutation f can be viewed as a pair & lt ; f , f − 1 & gt ;. theorem 1 : let & lt ; f , g & gt ; be a random permutation - like function with parameter q . let p be a random permutation . consider an adversary which is allowed calls to a pair of oracles & lt ; o 1 , o 2 & gt ;, with the restriction that it is only allowed to call o 1 on inputs 1 , 2 , . . . , q , whereas there is no restriction on calls to o 2 . any such adversary a that makes at most q total queries to a pair of oracles has probability at most q 2 / 2 n of distinguishing & lt ; f , g & gt ; from & lt ; p , p − 1 & gt ;. definition ( pair - wise differentially - uniform ): a sequence of uniformly distributed n - bit random numbers s 1 , s 2 , . . . s m , is called pair - wise differentially - uniform if for every n - bit constant c , and every pair i , j , i ≠ j , probability that s i ⊕ s j is c is 2 − 1 . definition a sequence of random numbers s 1 , s 2 , . . . s m uniformly distributed in gfp , is called pair - wise differentially - uniform in gfp if for every constant c in gfp , and every pair i , j , i ≠ j , probability that ( s i – s j ) mod p is c is 1 / p . we give definitions of schemes which explicitly define the notion of secrecy of the input message . of course , pseudorandom permutations can be used to build encryption schemes which guarantee such message secrecy ([ 1 ]). in addition , we also define the notion of message integrity . moreover , we allow arbitrary length input messages ( upto a certain bound ). let coins be the set of infinite binary strings . let l ( n )= 2 o ( n ) , and w ( n )= o ( n ). let n be the natural numbers . definition a ( probabilistic , symmetric , stateless ) encryption scheme with message integrity consists of the following : initialization : all parties exchange information over private lines to establish a private key x ∈ { 0 , 1 } n . all parties store x in their respective private memories , and | x |= n is the security parameter . message sending with integrity awareness : be polynomial - times function ensembles . in e , the third argument is supposed to be the length of the plain - text . similarly , in d the second argument is the length of the cipher - text . we will drop this argument when it is clear from context . the functions e and d have the property that for all x ∈ { 0 , 1 } n , for all m ∈ { 0 , 1 } l ( n ) , c ∈ coins we will usually drop the random argument to e as well , and just think of e as a probabilistic function ensemble . we will also drop n when it is clear from context . thus we will write l for l ( n ) etc . definition ( security under find - then - guess [ 1 ]) consider an adversary a that runs in two stages . during the adversary &# 39 ; s find stage he endeavors to come up with a pair of equal length messages , m 0 , m 1 , whose encryptions he wants to tell apart . he also retains some state information s . in the adversary &# 39 ; s guess stage he is given a random cipher - text y for one of the plain - texts m 0 , m 1 , together with s . the adversary is said to “ win ” if he correctly identifies the plain - text . an encryption scheme is said to be ( t , q , μ , ε )- secure in the find - then - guess sense , if for any adversary a which runs in time at most t and asks at most q queries , these totaling at most μ bits , ad ⁢ ⁢ υ a ⁢ = def ⁢ 2 · p ⁢ ⁢ r [ ( m 0 , m 1 , s ) ← a e x ⁡ ( · ) ⁡ ( find ) ; b ← { 0 , 1 } ; y ← e x ⁡ ( m b ) : a e x ⁡ ( · ) ( guess , y , s ) = b ] - 1 ≤ ε definition ( integrity awareness ): consider an adversary a running in two stages . in the first stage ( find ) a asks r queries of the oracle e x . let the oracle replies be c 1 , . . . c r . subsequently , a produces a cipher - text c , different from each c i , i ∈ [ 1 . . . r ]. since d has length of the cipher - text as a parameter , the breakup of d x ( c ) as m ∥ m ′, where | m ′|= w ( n ), is well defined . the adversary &# 39 ; s success probability is given by succ ⁢ = def ⁢ pr ⁡ [ mdc ⁡ ( m ) = m ′ ] we state the theorem for security under the find - then - guess notion of security . the proof follows standard techniques ([ 1 ]). theorem 2 : let a be an adversary attacking the encryption scheme iacbc in fig1 ( with f being a random function f ) in the find - then - guess sense , making at most q queries , totaling at most μ bits . then , ad ⁢ ⁢ υ a ≤ ( μ 2 n 2 - μ n ) · 1 2 n in this section we show that the mode of operation iacbc in fig1 guarantees message integrity with high probability . we start with some informal observations to aid the reader in the eventual formal proof . first thing to note is that since each encryption has a new random seed r , it does not help the adversary to have more than one pair of plain - text cipher - text messages . thus , essentially the problem of message integrity is the following . given p 1 , and corresponding c 1 , can the adversary generate another c 2 different from c 1 , such that on decryption the plain - text passes the integrity check . we will take the following approach . we first restrict ourselves to the random permutation - like function model . that is , we model the block cipher by a random permutation - like function . using theorem 1 , one can show that the following theorem also holds for the random permutation model . finally , yet another standard reduction shows that the theorem holds for pseudorandom permutations . theorem 3 : let a be an adversary attacking the iacbc encryption scheme with random permutation - like function & lt ; f , g & gt ; making at most r queries in the first stage , totaling at most μ bits ( where μ ≦ qn , q being the parameter of f ). then , succ & lt ; ( μ 2 n 2 ) · 1 2 n + o ⁡ ( μ n ) · 1 2 n for sake of clarity , we assume that the adversary only has one query in the first stage with plain - text p of length m blocks and corresponding cipher - text c ( μ = mn ). in the first stage , we do a modification to the iacbc algorithm . the modified algorithm uses f ( ξ ( i )) instead of f ( m i ) for queries f ( m i ), where ξ ( i )= min { j : j ≦ i and m j = m i } given that f is random , the behavior of the modified algorithm and the original algorithm is identical . its query in the second stage is with cipher - text c ′≠ c . we will used primed variables to denote the variables in the second stage . for example , p ′ m will denote the last decrypted block ( if c ′= c 0 ′, . . . c m ′). first note that , r , iv 1 , iv 2 , . . . , iv t are uniformly random and independent variables . also , they are all independent of p . ∀ i , j ∈[ 0 . . . m ]: m i ≠ m j this happens with high probability as in theorem 2 . this implies that n 0 , n 1 , . . . , n m are uniformly random and independent variables . also , they are all independent of r , iv 1 , iv 2 , . . . , iv t . we first consider the case where the length of c ′ and c is same . let i be the smallest index in which c and c ′ differ . it is easy to see that n i ≠ n i ′. the case i = m is trivial , as m m ′≠ m m with high probability , and hence p m ′ ≠ ∑ i = 1 m - 1 ⁢ p i ′ = ∑ i = 1 m - 1 ⁢ p i . next , we consider the case i ∈ [ 1 . . . m − 1 ]. we first prove the following : now , n i ′= c i ′⊕ s i , as s i ′= s i , i being greater than zero . but , c j = n j ⊕ s j ( for j & gt ; 0 ), and n j is independent of s j . in fact , since n 0 , n 1 , . . . , n m are independent and independent of iv 1 , iv 2 , . . . , iv t , the set of variables c 0 , c 1 , . . . , c m are independent and also independent of iv 1 , iv 2 , . . . , iv t , and hence independent of each s k ( for any k ∈[ 0 . . . m ]). this fact that variables c 0 , c 1 , . . . , c m are independent of iv 1 , iv 2 , . . . , iv t can be seen as follows , where c 0 , . . . c m are any constants : prob [ ⩓ i ⁢ c i = c i  ⁢ iv 1 , … ⁢ ⁢ iv t ] ⁢ = prob [ ⩓ i ⁢ n i = c i ⊕ s i  ⁢ iv 1 , … ⁢ ⁢ iv t ] = ∏ i ⁢ ⁢ prob ⁡ [ n i = c i ⊕ s i ] = ∏ i ⁢ ⁢ prob ⁡ [ c i = c i ] = prob ⁡ [ ⩓ i ⁢ c i = c i ] the second equation above follows because n 0 , n 1 , . . . , n m are independent of each other and independent of iv 1 , . . . iv t . thus , s k ( for each k ) is independent of whole of c . in fact , the previous argument is easily generalized to prove that s k is independent of c and p . also , c ′ is completely determined by c and p ( i . e . c ′ is a function only of c and p ), and hence s k is independent of c and c ′. since i ≠ j ( we already know that n i ≠ n i ), s i ⊕ s j = s k , for some k ∈[ 0 . . . m ]. since , s k is random and independent of c and c ′, the probability that s k = c i ′⊕ c j is 2 − n . the case when j = 0 is proved similarly . for case ( 2 ), for j & lt ; i , n j ′= n j , and hence case ( 1 ) applies . for case ( 2 ) to hold for a particular j & gt ; i would require again , as before , s k ( for any k ∈[ 0 . . . m ]) is independent of c i ′⊕ c j ′. and hence the probability is at most 2 − 1 . thus , the disjunction ( 1 ) or ( 2 ) holds with probability at most 2 ( m + 1 )* 2 − n . now , we consider the case i = 0 , i . e . c 0 ′≢ c 0 = n 0 . we show that with high probability , for all j ∈ [ 1 . . . m ], c 0 ′≢ n j . we consider the individual event n j = c 0 ′, or s j = c 0 ′ γc j . again , s j = c 0 ′ γc j holds with probability 2 − n . thus , with probability at most m * 2 − n , there exists a j ∈ [ 1 . . . m ] such that c 0 ′(= n 0 ′)= n j . thus , m 0 ′= g ( n 0 ′) is a random variable independent of all previous variables . this implies , that with high probability , iv 1 ′, . . . iv t ′ are random and independent variables , independent of all previous variables r , iv 1 , . . . iv t , n 0 , n 1 , . . . n m . thus , with high probability n 1 ′≢ n 1 , and now the previous case applies . thus , we have that with high probability , there is an i ∈ [ 1 . . . m − 1 ] such that thus , m i ′= g ( n i ′) is a random variable independent of all of r , r ′, iv 1 , iv 1 ′, . . . iv t , iv t ′, n 0 , n 0 ′, . . . n m , n m ′, and also independent of p 1 , p 2 , . . . p m - 1 , and all m j ′ ( j ≢ i ). p m ′ = ∑ j = 1 m - 1 ⁢ p j ′ = ∑ j = 1 m - 1 ⁢ ( m j ′ ⊕ n j - 1 ) ⁢ ⁢ and ⁢ ⁢ mdc ⁡ ( p ) = ∑ j = 1 m - 1 ⁢ p j m i ′ = ∑ j = 1 m - 1 ⁢ ( p j ⊕ n j - 1 ) ⊕ ∑ j ≠ 1 ⁢ m j ′ the lhs being independent of rhs , the probability of the event is 2 − n . for the case when the lengths of c and c ′ are different , we just remind the reader that a designated set s 0 is used in the last block . note that the proof of theorem 3 only required that s i γ s j equal a constant with low probability . infact , the theorem holds for an encryption scheme where a pair - wise differentially - uniform sequence of uniformly distributed random numbers is used instead of a pair - wise independent sequence of uniformly distributed random numbers . similalry , theorem 2 holds for an encryption scheme employing a pair - wise differentially - uniform sequence of uniformly distributed random numbers . theorem 2 and 3 also hold for encryption schemes which employ sequences which are pair - wise differentially - uniform in gfp ; the success probabiltites , however are now in terms of 1 / p instead of 1 / 2 n . theorem 4 : let a be an adversary attacking the iapm encryption scheme ( with f being a random function f ) in the find - then - guess sense , making at most q queries , totaling at most μ bits . thsn , ad ⁢ ⁢ υ a ≤ ( μ 2 n 2 - μ n ) · 1 2 n theorem 4 also holds for an encryption scheme variant of iapm employing a pair - wise differentially - uniform sequence of uniformly distributed random numbers . in this section we show that the log m additional cryptographic operations in the iacbc scheme are essentially the least one has to do to assure message integrity along with message secrecy . we consider the following model . we assume a fixed block size n for a block cipher ( or random permutations or length preserving random functions ). any application of one of these will constitute one application of a cryptographic operation . the only other operations allowed are linear operations over ( gf2 ) n , i . e . bit - wise exclusive - or . of course , operations of testing whether an n bit quantity is zero is also allowed . since , the scheme could be probabilistic , as iacbc is , we also allow v blocks of randomness , r 1 , . . . , r v . let , the message to be encrypted be of size m blocks , i . e . mn bits . call the input blocks p 1 , . . . , p m . let there be m + k invocations of random functions , and let the inputs to these functions be m 1 , m 2 , . . . , m m + k . similarly , let the outputs of these random functions be n 1 , n 2 , . . . , n m + k . let , c = c 1 , c 2 , . . . c m + t be a linear function of p &# 39 ; s , r &# 39 ; s , m &# 39 ; s and n &# 39 ; s . here 0 ≦ t ≦ k . our aim is to show that either the scheme is not secrecy secure , or it is not message integrity secure , or it is not invertible , or k + v = ω ( log n ). more formally , we would like the scheme to behave as a random function from mn bits to ( m + t ) n bits . the scheme is not secrecy secure if an adversary can distinguish the scheme from such a random function with probability ≧ 1 − 2 − 1 . for message integrity , let there be u & gt ; 0 mdc functions d 1 , d 2 , . . . , d u . without loss of generality ( see below ), assume that these are linear functions of r &# 39 ; s , m &# 39 ; s and n &# 39 ; s , and they are linearly independent . the scheme is not message integrity secure , if given p and c , an adversary can produce a c ′≢ c , such that on inversion , all the mdc functions evaluate to zero with high probability . for invertibility , we assume the scheme has the following structure : there is a subset of n &# 39 ; s which can be written as linear functions of just the c &# 39 ; s . the corresponding m &# 39 ; s then may lead to determination of some more m &# 39 ; s , and hence n &# 39 ; s . using , these new m &# 39 ; s and n &# 39 ; s , a second subset of n &# 39 ; s can be written as a linear combination of previously determined m &# 39 ; s , n &# 39 ; s and c , and so on . we are forced to take this approach , as by just allowing a system of equations with unique inverse is not enough . the unique inverse may exist but may not be efficiently computable . for example , c 1 = m 1 γ n 1 , may have a unique inverse , but may be intractable to compute . due to the fact that c is completely determined by r &# 39 ; s , m &# 39 ; s , n &# 39 ; s and p &# 39 ; s , it follows from the above characterization that c can be expressed as linear expressions in only n &# 39 ; s , m &# 39 ; s and r &# 39 ; s . for otherwise , the scheme is not secrecy secure ( i . e . there is a linear relationship between only c &# 39 ; s and p &# 39 ; s ). similarly , p can be expressed as linear expressions in only n &# 39 ; s , m &# 39 ; s and r &# 39 ; s . this justifies the above restriction on mdcs . d i = ∑ j = 1 m + k ⁢ ( a j i · m j ) ⊕ ∑ j = 1 m + k ⁢ ( b j i · n j ) ⊕ ∑ j = 1 υ ⁢ ( c j i · t j ) we say that n i and n j resolve if n i γ n j can be written as a linear combination of only the c &# 39 ; s and the p &# 39 ; s . similarly , for m i and m j . suppose there exists a pair i , j , i ≢ j , i , j ∈ [ 1 . . . m + k ] such that 1 . n i and n j resolve 2 . m i and m j resolve 3 . for all x ∈ [ 1 . . . u ], a i ′ γa j x = 0 , and b i x γ b j x = 0 then , we show that an adversary can produce a new c ′≢ c , such that all the mdc functions evaluate to zero . note that , if there exists a c ′ such that then , we have a similar set of relations for m , and hence given ( 3 ), all the mdc functions would evaluate to zero . since c can be expressed only in terms of n &# 39 ; s , m &# 39 ; s and r &# 39 ; s it is not difficult to come up with such a c ′. moreover , we have also assumed in our schemes , that a c ′ uniquely determines n ′, and m ′. finally , we show that if k + v is not ω ( log n ), then there exists a pair i , j satisfying ( 1 ), ( 2 ) and ( 3 ). let [ p 1 . . . p m r 1 . . . r v n 1 . . . n m + k ]· b =[ c 1 . . . c m ] the rank of the matrix b is m . for a fixed p , let the resulting matrix be b ′, i . e . [ r 1 . . . r v n 1 . . . n m + k ]· b ′=[ c 1 . . . c m ] the rank of the new matrix b ′ is still m , for otherwise we have a non - trivial linear relationship between c and p , and hence the scheme is not random . this implies that [ r 1 . . . r v n 1 . . . n m + k ]=[ f ( c )]+( gf 2 ) n v 1 + . . . +( gf 2 ) n v k + v where f ( c ) is a set of linear functions of c &# 39 ; s , and v i are linearly - independent binary row - vectors . for a subset of n &# 39 ; s with indices a set j ⊂ [ 1 . . . m + k ] to be pair - wise independent thus requires k + v & gt ; log | j |. in other words , there exists i , j ∈ j , i ≢ j , n i and n j resolve if k + v & lt ; log | j |. stated differently , there is a set of size | j |=( m + k )/ 2 k + v in which all pairs of n &# 39 ; s resolve with each other . a similar statement holds for m &# 39 ; s . thus , there is a set of size | j |=( m + k )/ 2 2 ( k + v ) in which all pairs of n &# 39 ; s resolve with each other , and all pairs of m &# 39 ; s resolve with each other . ∀ k ∈[ 1 . . . u ], ∀ i , j ∈ j : a i k ∈ a j k = 0 combining these arguments , we get that there exists a pair satisfying ( 1 ), ( 2 ) and ( 3 ) if 2u + 2 ( k + v )& lt ; log n . to complete the proof , we show that ( k + v )≧ u . we can write p &# 39 ; s and d &# 39 ; s as linear functions of r &# 39 ; s , m &# 39 ; s and c &# 39 ; s ( as discussed earlier n &# 39 ; s can be replaced by r &# 39 ; s , m &# 39 ; s and c &# 39 ; s ). thus , we have a matrix a such that [ c 1 . . . c m r 1 . . . r v m 1 . . . . m m + k ]· a =[ p 1 . . . p m d 1 . . . . d u ] the matrix a has rank at least m + u , for otherwise one would get a non - trivial linear relationship between d &# 39 ; s and p &# 39 ; s . in fact , for a fixed c , the rank of the resulting matrix a ′ is still at least m + u , for otherwise we would get a non - trivial linear relationship between d &# 39 ; s , p &# 39 ; s and c &# 39 ; s . however , on a valid encryption , d &# 39 ; s evaluate to zero . thus , for valid encryptions we have a non - trivial linear relationship between the p &# 39 ; s and the c &# 39 ; s , which renders the encryption distinguishable from random . thus , m + k + v ≧ m + u . a new mode of operation for combining confidentiality and authentication was recently described in [ 3 ]. the mode of operation is called ia - pcbc ( integrity aware plain - text cipher - text block chaining ). it was however shown by the author that the scheme is not secure for message integrity . we just remark here that the scheme was essentially as described in the model in this section . to encrypt a m blocks , only m + 2 block encryptions are employed in ia - pcbc . the claimed security came from mixing addition over integers modulo 2 n , with exclusive - or operations . however , one can be approximated in terms of others with reasonably high probability , and then the attack follows by the lower bound . it is noted that the foregoing has outlined some of the more pertinent objects and embodiments of the present invention . this invention may be used for many applications . thus , although the description is made for particular arrangements and methods , the intent and concept of the invention is suitable and applicable to other arrangements and applications . it will be clear to those skilled in the art that other modifications to the disclosed embodiments can be effected without departing from the spirit and scope of the invention . the described embodiments ought to be construed to be merely illustrative of some of the more prominent features and applications of the invention . other beneficial results can be realized by applying the disclosed invention in a different manner or modifying the invention in ways known to those familiar with the art . while it is apparent that the invention herein disclosed is well calculated to fulfill the objects stated above , it will be appreciated that numerous modifications and embodiments may be devised by those skilled in the art , and it id intended that the appended claims cover all such modifications and embodiments as all within the true spirit and scope of the present invention .