while it is believed that the notation of fig1 - 5 would be clear to those of ordinary skill in the art , it is here reviewed for definiteness . the operations performed are collected together into flowchart boxes . the column that such a box is in indicates which party performs the operation defined in that box . the columns are labeled by party name across the top . some operations show how messages are formed on the right of the equal sign with the message number ( shown in square brackets ) on the left of the equal sign . the operation of a party saving a value under a symbolic name is denoted in the same way as that of forming a message , except that the symbolic name appears on the left instead of a message number . another kind of operation is test for equality and inequality ; these are indicated by the symbols &# 34 ;?=?&# 34 ; and &# 34 ;?≠?&# 34 ;, respectively . the party performing one of these tests within a protocol terminates the protocol when the condition is not satisfied ; the protocol is stopped when the two comparands of an ?=? differ or when the comparands of a ?≠? are the same . where the test is at the end of a protocol , the result of the protocol may be thought of as positive when the test would not have caused the protocol to terminate , and negative otherwise . the final kind of operation is that of sending a message . this is shown by a message number on the left ; followed by the recipient party &# 39 ; s name and an arrow ( these appear for readability as either a recipient name then left pointing arrow , when the recipient is on the left ; or right pointing arrow then recipient name , when the recipient is on the right ); followed by a colon ; finally followed by an expression fully denoting the actual value of the message that should be sent . note that the values of some variables in such message expressions may not be known by the sender and others may be unknown to their recipient . several ways to form expressions are used . one is just the word &# 34 ; random &# 34 ;. this is used to mean that a value is preferably chosen substantially uniformly from an appropriate set , defined in the text , and substantially independently of everything else in the protocol . thus a party should preferably employ a physical random number generator for these purposes , but a variety of other techniques may be applied , as already described for boxes 604 , 607 , 610 , 613 , and 615 . in practice , however , well known pseudo - random generator or hybrid techniques may be applied . since the results of these random expression are used as keys which should not be determinable by the other party to the protocol ( at least until the creating party may choose to release them ), the random generation must be substantially unpredictable to an adversary . the function f is preferably a publicly - agreed one - way function , such functions being well know in the art . when no operation is shown explicitly , the group operation referred to here as multiplication is assumed . another kind of expression involves exponents which denote raising to powers in the group . the well known convention is adopted here that operations in the base are group operations and that arithmetic in the exponent is modulo the order of the group . but parties need not actually know the order of the group , in all but one optional case mentioned later , since parties can simply use natural number arithmetic in the exponent . also , when a random value , as mentioned above , is to be created for use in the exponent , its distribution can be made very close to uniform , even when the order of the group is not known : the exponent is chosen say uniformly from 1 to say the square of an upper bound on the order of the group . for clarity in exposition and concreteness , however , the preferred embodiments will be presented here in terms of the multiplicative group of order prime p . as has already been mentioned , the scope of the present invention should not be considered to be limited to any particular group , and the present detailed description could readily be translated by someone of ordinary skill in the art to any suitable group . turning now to fig1 the first flowchart for part of the preferred embodiment will now be described in detail . this part shows public key creating and issuing , which need only be carried out once by the signer party s , and also the forming of a single undeniable signature for party v . flowchart box 101 shows s choosing x uniformly and at random from the interval 1 to p - 1 , such random selection as already mentioned . then s raises g to the x &# 39 ; th powder modulo p , such exponentiation already having been described and well known in the art . the resulting residue is then called message [ 10 ]. as per the definition of the notation already described , message [ 10 ] is then shown as being sent from s to v . this completes the creating and issuing of a public key by s . box 102 indicates that , after receiving the public key as message [ 10 ], v sends an original message m for signing to s as message [ 11 ]. for the purposes of the present invention the nature or source of m is not essential and it may be regarded as any suitable message ( or blinded message , as has already been mentioned and will be mentioned in detail later ). box 103 shows how , after receiving message [ 11 ], s first forms a signature from it by raising it to the secret power x . the exponentiation is done in this particular exemplary embodiment , as already mentioned , in the group of order p . finally , the signature denoted as message [ 12 ] is shown being sent by s to v , who would ordinarily receive it and retain it for possible later use in one of the other protocol parts . it may be pointed out here that if an ordinary digital signature is formed by s on the pair comprising message [ 11 ] and message [ 12 ], sig ( f ([ 11 ], [ 12 ])), and this is later shown to a third party who trusts s , then the third party is able to determine that [ 12 ] is a valid undeniable signature of [ 11 ]. turning now to fig2 the second flowchart for part of a preferred embodiment will now be described in detail . this part shows a first exemplary arrangement for the checking of an undeniable signature , the issuing of which has just been shown in detail in fig1 . box 201 shows how v prepares the initial challenge and sends it to s . first a and b are chosen substantially independently and uniformly at random from 1 to p ( or in somme other suitable way when the order of the group is not known to v , as has already been mentioned ). then message [ 21 ] is formed as the product ( in the group , as already mentioned ) of message [ 12 ] raised to the power a and message [ 10 ] raised to the power b . this message is then sent by v to s , and should have the form shown in the last line of this box . ( but since v does not know x , this is an example of the comment made earlier that neither party acting alone need be able to determine the value of all variables of such expressions .) box 202 is the formation and return of s &# 39 ; s response to the challenge received from v . the multiplicative inverse of x modulo the order of the group is shown in the usual way in the exponent of message [ 21 ], to produce message [ 22 ]. thus , message [ 22 ] is shown as being obtained by applying the inverse of the signing function to the message [ 21 ]. ( a protocol not requiring that the order be known is shown in fig3 to be described in detail .) the resulting message [ 22 ] should be of the form shown , m a g b , and is shown as being supplied to v . box 203 shows the checking of the response [ 22 ] received from s by v . first v uses the values of m , g , a , and b known to v to construct the value that should have been returned by s in case the signature was valid . this is done by raising m to the power a and multiplying the result by g raised to the power b . then v simply compares the value constructed with that received from s in message [ 22 ]. if they are equal , then v stops the protocol , as called for by the definition of the symbol ?≠? given above . in this case , v knows that [ 12 ] is with high certainty the signature of m corresponding to public key [ 10 ]. in the remaining case , that the inequality holds , v continues the protocol with the knowledge that either ( a ) [ 12 ] is not the proper signature or ( b ) s is trying to improperly deny the signature . the rest of this flowchart allows v to distinguish between these two subcases . box 204 is similar to 201 except that c and d are used instead of a and b . first c and d are created by the random expression already described so that they are suitable secret exponents . then message [ 24 ] is formed as the product of message [ 12 ] raised to the c power times message [ 10 ] raised to the d power . finally , message [ 24 ] is sent by v to s . box 205 is again similar to its predecessor , box 202 , and in fact the operations performed by s are the same . the only difference is that the input is message [ 24 ] instead of [ 21 ] and the output is [ 25 ] instead of [ 22 ]. one consequence of this is that s need not know which of these two steps in the protocol is being performed . box 206 shows the final test made by v based on the messages [ 22 ] and [ 25 ] received from s . the test shown is made by comparing the equality of two essentially similarly constructed quantities . the first is the product of message [ 22 ] and g raised to the - b power , all raised to the c power ; the second is message [ 25 ] times g to the - d power all to the a power . notice that the negative exponents on g need not mean that v must compute multiplicative inverses , since the multiplicative inverse of g could have been made public by some other party . as should be obvious to those of skill in the art , however , the comparison can be made in practice without needing multiplicative inverses . there are two cases : if bc & gt ; da , test [ 22 ] c ?=? [ 25 ] a g bc - da or if bc ≦ da , test [ 22 ] c g da - bc ?=? [ 25 ] a . regardless of how the test is made , if the equality holds , then s is with high probability behaving honestly and [ 12 ] is not a valid signature ; if the equality does not hold , then s is believed to be behaving improperly . again the possibility of an ordinary digital signature on the transaction by s is considered . it might in this case contain message [ 21 ] and message [ 22 ] and cold be denoted : sig ( f ([ 21 ], [ 22 ])). the third party would be supplied this digital signature , m , [ 12 ], a , and b by v , and would check the validity of the undeniable signature by checking that the digital signature is valid , [ 21 ]?=? [ 12 ] a [ 10 ] b , and [ 22 ]?=? m a g b . such testing may be considered to be shown in fig2 since essentially the same operations are performed by v . turning now to fig3 the third flowchart for part of a preferred embodiment will now be described in detail . this part shows a second alternate arrangement for the checking of an undeniable signature , the issuing of which has already been shown in detail in fig1 . box 301 is similar to box 201 in that a first challenge is created based on two randomly generated exponents , called again here a and b . thus , v chooses these two exponents substantially independently and uniformly , and keeps them secret . what v sends to s in message [ 31 ] is the product of m raised to the a and g raised to the b . notice that since all of these values are known to v , the explicit construction of the message is omitted from the flowchart and its value is shown in the line for the sending of the message only . box 302 entails s raising the received message [ 31 ] to the x power and then applying the one - way function f to the result . this image under the one - way function is what is returned to v by s in message [ 32 ]. box 303 merely indicates that after receiving message [ 32 ] from s , v forwards m , a , and b individually to s in messages [ 33 . 1 ], [ 33 . 2 ], and [ 33 . 3 ], respectively . box 304 first shows how s tests that all the messages received from v during this part of the protocol --[ 31 ], [ 33 . 1 ], [ 33 . 2 ], and [ 33 . 3 ]-- are mutually consistent . this is accomplished by testing the equality of [ 31 ] with the result of reconstructing its value from the others . the reconstruction is accomplished by forming the product of [ 33 . 1 ] raised to the [ 33 . 2 ] with g raised to the [ 33 . 3 ]. if the equality is not satisfied , s stops the protocol , as per the definition of the notation , and knows that v has been supplying improper messages . if the equality is satisfied , s returns to v message [ 31 ] raised to the secret power x in the form of message [ 34 ]. box 305 shows two tests by v . the first checks that [ 34 ] really is the inverse image of [ 32 ] under f . if this test fails , then v stops the protocol knowing that s was supplying improper messages . otherwise v makes a test similar in intention and form to that of box 203 . message [ 34 ] is tested for inequality with the product of message [ 12 ] raised to the power a and message [ 10 ] raised to the power b . if they are equal , then v stops the protocol and knows that with high probability [ 12 ] is indeed the signature of m corresponding to public key [ 10 ]. in case the inequality does hold , v continues the protocol but with the knowledge that either ( a ) [ 12 ] is not the proper signature or ( b ) s has tried to improperly deny the signature . and as with fig2 the remaining part of this flowchart allows v to distinguish between these two subcases . box 306 is similar to box 204 in that a second challenge is created based on two randomly generated exponents , called again here c and d , but they are combined into the challenge in the style of 301 . that is [ 36 ] is formed as the product of m raised to the c times g raised to the d , and it is supplied by v to s . box 307 shows s raising the received message [ 36 ] to the x power and then applying the one - way function f to the result . this image under the one - way function is what is returned to v by s in message [ 37 ]. box 308 denotes that after receiving message [ 37 ] from s , v sends a and b individually to s in message [ 38 . 1 ] and [ 38 . 2 ], respectively . box 309 first shows how s tests the mutual consistency of messages [ 36 ], [ 33 . 1 ], [ 38 . 1 ], and [ 38 . 2 ] received from v . this is accomplished by testing the equality of [ 36 ] and the product of [ 33 . 1 ] raised to the [ 38 . 1 ] times g raised to the [ 38 . 2 ]. if the equality is not satisfied , s stops the protocol knowing that v has been supplying improper messages . if the equality is satisfied , s supplies v with message [ 36 ] raised to the power x called message [ 39 ]. box 310 shows two tests by v . the first checks that [ 39 ] is the inverse image of [ 37 ] under f . if this is not so , then v stops the protocol known that s was supplying improper messages . the second tests messages [ 34 ] and [ 39 ] received from s . the test shown compares the equality of two values . the first value is the product of message [ 34 ] and g raised to the - b power , all raised to the c power ; the second is message [ 39 ] times g to the - d power all to the a power . again , as should be obvious to those of skill in the art , the comparison can be made in practice without computing multiplicative inverses . there are two cases : if bc & gt ; da , test [ 34 ] c ?=? [ 39 ] a g bc - da or if bc ≦ da , test [ 34 ] c g da - bc ?=? [ 39 ] a . no matter how the test is made , if the equality holds , then s is with high probability behaving honestly and [ 12 ] is not a valid signature ; if the equality does not hold , then s is believed certainly to be performing improperly . a digital signature issued for this protocol by s need include only messages [ 31 ] and [ 34 ], and would thus be of the form : sig ( f ([ 31 ], [ 34 ])). the third party would additionally be supplied m , [ 12 ], a , and b by v , and would check the validity of the undeniable signature by checking that the digital signature is valid , [ 31 ]?=? m a g b , and [ 34 ]?=? [ 12 ] a [ 10 ] b . such testing again may be considered to be shown in fig3 since it entails essentially the same operations already shown as performed by v . turning now to fig4 the fourth flowchart for part of the preferred embodiment will now be described in detail . this part shows one kind of blinding , called &# 34 ; exponential blinding ,&# 34 ; of a message by party v , raising the result to a secret power by s , and unblinding of the returned message by v . as will be obvious to those of ordinary skill in the art , and will be described later in detail , these operations are generic : blinding could be performed by v on any message before it is raised to a secret power by s , and the result returned by s could be unblinded . in particular , it could be applied to all three blindings and unblindings shown in fig6 i . e . 606 and 608 ; 609 alone ; or 614 and 616 . box 401 shows how v blinds message u and sends it to s . first v chooses r independently and uniformly from 1 to p - 1 . then v raises u to the power r to form message [ 41 ], which v sends to s . box 402 shows receipt of message [ 41 ] by s and its transformation and subsequent return to v . to make the transformation , s raises message [ 41 ] to the secret power y ; the result is supplied to v as message [ 42 ]. box 403 shows the unblinding of the blinded message received by v . the multiplicative inverse of r modulo the order of the group is applied as an exponent to the message [ 42 ] received from s , and the result is shown as message [ 43 ]. for clarity , the last line of box 403 shows parenthetically that the value of message [ 43 ] should be u raised to the y power . box 404 shows the optional creation of another secret blinding key t , and its use in reblinding the message u . first t is created at random as r was . then message [ 44 ] is formed as u raised to the power t . message [ 45 ] is shown as being created by raising message [ 43 ] to the t power . for clarity , the last line of box 404 again shows parenthetically that the value of message [ 45 ] should be u raised to the power yt . some specific examples will now be presented so that some exemplary embodiments of the generic exponential blinding and unblinding operations just described in detail may be more fully appreciated . in fig1 message [ 11 ] could be blinded by v before being sent to s for signing ( i . e . y = 1 / x ), and the resulting message [ 12 ] could be unblinded by v before it is used in fig2 as is shown by blinder 606 and unblinder 608 already described . message [ 31 ] could also be blinded before being sent to s in the testing of fig3 ( y = x ), and the returned message [ 34 ] could be unblinded before being tested , as shown by blinder 614 and unblinder 616 ; the blinding of message [ 36 ] and the testing of the returned message [ 39 ] would of course be essentially the same . when the same operations are applied for fig2 it will be obvious to those of ordinary skill in the art that the exponent used in boxes 401 and 403 would be exchanged ( with y = 1 / x ) if they are to serve as 606 and 608 , respectively . notice that messages [ 44 ] and [ 45 ] could be regarded as the unsigned and signed form , respectively , of a single blinded message , such as might be used as input to challenger 612 , for either fig2 or fig3 . as would be obvious to those of ordinary skill in the art , the blinding of various messages can be superimposed , to give for example double blinding as disclosed in the application entitled &# 34 ; blind signature systems ,&# 34 ; by the present applicant , already mentioned in the description of the prior art . so called &# 34 ; re - blinding &# 34 ; was disclosed for the unanticipated blind signature system already referenced in the description of the prior art . for the present invention , a kind of re - blinding is also possible . the result of re - blinding is a pair comprising a blinded message and a blinded signature of that message . these could then be used in the protocol of fig2 as just described . some other protocol , such as that disclosed by chaum and evertise in &# 34 ; a secure and privacy - protecting protocol for transmitting personal information between organizations &# 34 ;, proceedings of crypto 86 , a . odlyzko ed ., springer 1987 , might be used to show that these re - blinded messages are related to some other messages in a desired way , and the protocol of fig2 for instance used to show that one member of the pair is in fact a signature on the other member . ordinary digital signatures could be used here again to allow a third party to check a transaction that is blinded in the way shown in fig5 . in addition to the other data already described in detail for fig1 - 3 , the exponent r must also be provided to the third party to allow checking . then the third party performs the checks as already described , except that the expression corresponding to the input to s must be raised to the r power and the multiplicative inverse of r modulo p must be applied to the expression for the output of s , as would be obvious to those of skill in the art . turning now to fig5 the fifth flowchart for part of the preferred embodiment will now be described in detail . this part shows another kind of blinding , related to the &# 34 ; blinding for unanticipated signatures &# 34 ; already referenced in the background of the invention , in which a message is blinded by v , the result is raised by s to a secret power y , and the returned message is unblinded by v . box 501 shows how v blinds message m and sends it to s . first v chooses r independently and uniformly from 1 to p - 1 . then v raises g to the power r and multiplies the result with m to form message [ 51 ], which v sends to s . box 502 shows receipt of message [ 51 ] by s and its signing and subsequent return to v . to make the signature , s raises message [ 51 ] to the secret power y ; the result is supplied to v as message [ 52 ]. box 503 shows the unblinding of the signed blinded message received by v . the multiplicative inverse of message [ 10 ] raised to the r is first formed . then this is multiplied with message [ 52 ] received from s , and the result is shown as message [ 53 ]. again for clarity , the last line of box 503 shows parenthetically that message [ 53 ] should have the value m raised to the y power . box 504 shows the optional creation of another secret blinding key t , and its use in re - blinding the message m . first t is created at random as r was . then message [ 54 ] is formed as m times g to the power t . message [ 55 ] is shown as being created by raising message [ 10 ] to the t power and multiplying the result by message [ 53 ]. for clarity , the last line of box 504 again shows parenthetically that the value of message [ 55 ] should be m raised to the power t times g raised to the power yt . it should be pointed out that the forming of message [ 55 ] has been shown for clarity only in the case when x = y , but , as would be obvious to those of skill in the art , in the case when y = 1 / x message [ 55 ] would not be formed from message [ 10 ], but would rather be formed from the analog of message [ 10 ] that contains the value g 1 / x . notice that messages [ 54 ] and [ 55 ] could again be regarded as the unsigned and signed form , respectively , of a single blinded message . as an example use of such unanticipated signature techniques adapted to this setting , the signing operation of fig1 might be performed so as to yield v an undeniable signature unlinkable by s . that is , if a plurality of such signatures are obtained with independent r &# 39 ; s , then s should be unable to determine anything about which signature corresponds with which instance of the signing process . the pair comprising a blinded message and a blinded signature of that message used in re - blinding has already been shown in box 504 , and the comments already made for box 404 could apply to this box as well . some specific examples will now be presented so that some exemplary embodiments of the generic unanticipated signature blinding and unblinding operations just described in detail may be more fully appreciated . in fig1 message [ 11 ] could be blinded by v before being sent to s for signing ( i . e . y = x ), and the resulting message [ 12 ] could be unblinded by v before it is used in fig2 as is shown by blinder 606 and unblinder 608 already described . message [ 31 ] could also be blinded before being sent to s in the testing of fig3 ( y = x ), and the returned message [ 34 ] could be unblinded before being tested , as shown by blinder 614 and unblinder 616 ; the blinding of message [ 36 ] and the testing of the returned message [ 39 ] would of course be essentially the same . when the same operations are applied for fig2 it will be obvious to those of skill in the art that the exponent used in boxes 501 and 503 would be exchanged ( with y = 1 / x ) if they are to serve as 606 and 608 , respectively . notice that messages [ 54 ] and [ 55 ] could be regarded as the unsigned and signed form , respectively , of a single blinded message , such as might be used as input to challenger 612 , for either fig2 or fig3 . ordinary digital sigantures could again be used to allow a third party to check a transaction that is blinded in the way shown in fig5 . in addition to the other data already described in detail for fig1 - 3 , the exponent r must also be provided to the third party to allow checking . then the third party performs the checks as already described , except that the blinding factor g r must be included in the expression corresponding to the input to s and [ 10 ] - r must be included in the expression for the output of s . as again would be obvious to those of ordinary skill in the art , the blinding of various messages can be superimposed to give double blinding as already mentioned and re - blinding is also possible as already described during the detailed description for fig4 . another variation that would be obvious to those of ordinary skill in the art would involve plural original message parts in a signature . the signature would consist of the product of each such message part raised to a different power . the challenge would contain a separate message corresponding to each part of a signature . the response would be the product of all such messages of a challenge , each having the exponent corresponding to the corresponding message part . a further and not necessarily mutually exclusive use anticipated would be to include more than two terms in a challenge message . with such an arrangement the mutual consistency of more than two message / signature pairs could be tested while keeping some of the message and processing costs the same . different random exponents could be used on each term , but if there were sufficiently many terms , it is anticipated that various possibly randomly chosen combinations of possibly smaller exponents might be used . while these descriptions of the present invention have been given as examples , it will be appreciated by those of ordinary skill in the art that various modifications , alternate configurations and equivalents may be employed without departing from the spirit and scope of the present invention .