Abstract:
A bidding method including: (a) generating a first bidding function value using an initial bidding price of a first bidder; (b) transmitting the first bidding function value to a successful bidding decider; (c) transmitting a second bidding function value of a second bidder to the first bidder; (d) transmitting a final bidding price of the first bidder and a final bidding price of the second bidder to the successful bidding decider; (e) determining a successful bidding price by comparing the final bidding price of the first bidder and the final bidding price of the second bidder; and (f) verifying whether or not the successful bidding price was manipulated

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     This application is based upon and claims the benefit of priority from Korean Patent Application No. 2004-35533, filed on May 19, 2004, in the Korean Intellectual Property Office, the disclosure of which is incorporated herein in its entirety by reference.  
       BACKGROUND OF THE INVENTION  
       [0002]     1. Field of the Invention  
         [0003]     The invention relates to a bidding method, and more particularly, to a bidding method: for preventing a successful bidding decider from manipulating a successful bidding price; and for preventing a bidder&#39;s denial of a bidding price, in an on-line auction.  
         [0004]     2. Description of the Related Art  
         [0005]     Recently, Internet auctions have become popular. The Internet auction is a system where a bidder sends a bidding price to a successful bidding decider, and the successful bidding decider determines bidding qualifications according to a predetermined reference and allocates an auction target object to the bidder that best meets the successful bidding qualifications.  
         [0006]      FIG. 1  is a view illustrating a related art bidding method.  
         [0007]     A successful bidding decider  10  is provided with bidding prices  22 ,  32  and  42  from bidders  20 ,  30  and  40 . The successful bidding decider  10  determines the bidding price  12  on the basis of a predetermined successful bidding reference, and transmits the determined bidding price  12  to each of the bidders  20 ,  30  and  40 , thereby terminating an auction procedure.  
         [0008]     However, the related art bidding method has the following drawbacks.  
         [0009]     First, the successful bidding decider can manipulate the bidding price. Specifically, since the bidders cannot know the bidding prices of the other bidders, the successful bidding decider can fallaciously determine the bidding price to transmit it to the bidders.  
         [0010]     Second, the successful bidding decider can analyze all bidding prices to provide a specific conspiring bidder with a bidding price that best meets the successful bidding reference.  
         [0011]     Third, a bidder can deny a submitted bidding price.  
         [0012]     Fourth, a third party, such as the owner of the object being auctioned, can bid under the disguise of one of the bidders in an attempt to drive up the successful bidding price.  
       BRIEF SUMMARY OF THE INVENTION  
       [0013]     The invention provides a bidding method for preventing a successful bidding decider from manipulating a successful bidding price and for preventing a bidder from denying his/her bidding price.  
         [0014]     According to an aspect of the invention, a bidding method includes: (a) generating a first bidding function value using an initial bidding price of a first bidder; (b) transmitting the first bidding function value to a successful bidding decider; (c) transmitting a second bidding function value of a second bidder to the first bidder; (d) transmitting a final bidding price of the first bidder and a final bidding price of the second bidder to the successful bidding decider; (e) determining a successful bidding price by comparing the final bidding price of the first bidder and the final bidding price of the second bidder; and (f) verifying whether or not the successful bidding price was manipulated.  
         [0015]     According to another aspect of the invention, a bidding method includes: (a) generating a first bidding function value using an initial bidding price of a first bidder; (b) transmitting the first bidding function value to a successful bidding decider; (c) transmitting a second bidding function value of a second bidder to the first bidder; (d) transmitting a final bidding price of the first bidder and a final bidding price of the second bidder to the successful bidding decider; (e) verifying whether or not the final bidding price of the first bidder corresponds to the initial bidding price of the first bidder; and (f) determining a successful bidding price by comparing the final bidding price of the first bidder and the final bidding price of the second bidder.  
         [0016]     According to a further another aspect of the invention, a bidding system includes: a client adapted to generate an initial bidding price of a first bidder and a first bidding function value using the initial bidding price; and a server adapted to: receive the first bidding function value from the client and a second bidding function value from a second client; transmit the first bidding function value and second bidding function value to the client; receive a final bidding price of the first bidder from the client and a final bidding price of a second bidder from the second client; and determine a successful bidding price by comparing the final bidding price of the first bidder and the final bidding price of the second bidder, wherein the initial bidding price of the first bidder cannot be determined from the first bidding function value.  
         [0017]     According to a further another aspect of the invention, the methods described above may be implemented in a program stored on a computer recordable recording medium. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0018]     The above and other objects, features and advantages of the invention will become more apparent by describing in detail exemplary embodiments thereof with reference to the attached drawings in which:  
         [0019]      FIG. 1  is a view illustrating a related art bidding method;  
         [0020]      FIG. 2  is a view illustrating a bidding method according to the first exemplary embodiment of the invention;  
         [0021]      FIG. 3  is a view illustrating a method of generating a bidding function value;  
         [0022]      FIG. 4  is a view illustrating a bidding method according to the second exemplary embodiment of the invention;  
         [0023]      FIG. 5  is a view illustrating a bidding method according to the third exemplary embodiment of the invention;  
         [0024]      FIG. 6  is a view illustrating a bidding method according to the fourth exemplary embodiment of the invention;  
         [0025]      FIG. 7  is a view illustrating a bidding method according to the fifth exemplary embodiment of the invention;  
         [0026]      FIG. 8  is a view illustrating a bidding method according to the sixth exemplary embodiment of the invention;  
         [0027]      FIG. 9  is a view illustrating a bidding method according to the seventh exemplary embodiment of the invention; and  
         [0028]      FIG. 10  is a view illustrating a bidding method according to the eighth exemplary embodiment of the invention. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0029]     Exemplary embodiments of the invention will now be described below by reference to the attached drawings. The described exemplary embodiments are intended to assist the understanding of the invention, and are not intended to limit the scope of the invention in any way. Like reference numerals in the drawings denote like elements.  
         [0030]      FIG. 2  is a view illustrating a bidding method according to a first exemplary embodiment of the invention.  
         [0031]     In operation  210 , a client (i.e., a bidder (u 1 )) selects his or her bidding price (v 1 ).  
         [0032]     In operation  220 , the bidder (u 1 ) inputs the bidding price (v 1 ) to a bidding function h(x) to generate a bidding function value (PP 1 ). The bidding function h(x) should provide the bidding function value from the bidding price, but not provide the bidding price from the bidding function value, for the reasons discussed below.  
         [0033]     In operation  230 , the bidder (u 1 ) transmits the generated bidding function value (PP 1 ) to the successful bidding decider. Additionally, any other bidders (u 2 , u 3  . . . ) transmit their respective bidding function values (PP 2 , PP 3  . . . ) to the successful bidding decider so that the successful bidding decider has the bidding function values from each of the bidders. When all bidding function values have been transmitted, the transmission of further bidding function values is no longer allowed.  
         [0034]     In operation  240 , the successful bidding decider transmits all the received bidding function values PP 1 , PP 2 , PP 3 , . . . to each of the bidders. As a result, each bidder knows the bidding function values for the bidding prices made by the other bidders.  
         [0035]     In operation  250 , all bidders transmit their bidding prices v (v 1 , v 2 , v 3  . . . ) to the successful bidding decider. Bidders u 1 , u 2 , u 3  . . . respectively set the bidding prices v 1 , v 2 , v 3  . . .  
         [0036]     In operation  260 , the successful bidding decider compares the bidding prices v 1 , v 2 , v 3  . . . of the bidders u 1 , u 2 , u 3  . . . with one another to determine the successful bidding price V ref .  
         [0037]     In operation  270 , the successful bidding decider transmits the determined bidding price V ref  and all bidding prices v 1 ′, v 2 ′, v 3 ′ . . . to each of the bidders u 1 , u 2 , u 3  . . . The transmitted bidding prices v 1 ′, v 2 ′, v 3 ′ . . . refer to bidding prices that the successful bidding decider indicates are transmitted by each of the bidders.  
         [0038]     In operation  280 , each of the bidders compares the received bidding function values PP 1 , PP 2 , PP 3 , . . . (received in operation  240 ) with input function values h(v 1 ′), h(v 2 ′), h(v 3 ′) . . . generated by inputting the received bidding prices v 1 ′, v 2 ′, v 3 ′ . . . (received in operation  270 ) into the bidding function discussed above, to determine whether or not the successful bidding decider fallaciously reported the bidding price to the bidder.  
         [0039]     More specifically, if the successful bidding decider attempts to manipulate the bidding (i.e., if each of the bidders does not bid with reference to the bidding price transmitted from each bidder), the bidding function values PP 1 , PP 2 , PP 3 , . . . are not identical to the bidding function values F(v 1 ′), F(v 2 ′), F(v 3 ′) . . . since v 1 ′ v 1 , v 2 ′ v 2 , and/or v 3 ′ v 3  . . . Only in the case where the bidding function values PP 1 , PP 2 , PP 3 , . . . are identical to the bidding function values F(v 1 ′), F(v 2 ′), F(v 3 ′) is the bidding determined to be successful. Otherwise, the bidding is determined to be a failure and thus the procedure is ended in a state where the bidding method was not in a secured transparency.  
         [0040]     In the bidding method of  FIG. 2 , the bidder u 1  possesses the bidding function values PP 2 , PP 3 , . . . that may be used to confirm the bidding prices of other bidders u 2 , u 3 , . . . that were submitted prior to the determination of the successful bidding price. Thus, this bidding method can prevent the successful bidding decider from manipulating the bidding, as described in detail below.  
         [0041]     First, since each of the bidders u 1 , u 2 , u 3  . . . secures the bidding function values PP 1 , PP 2 , PP 3 , . . . that can confirm the bidding prices of other bidders u 1 , u 2 , u 3 . . . before the successful bidding decider determines the successful bidding price, the successful bidding decider cannot determine the successful bidding price using bidding prices different from those transmitted by the bidders u 1 , u 2 , u 3  . . . Rather, if bidding prices different from those transmitted by the bidders u 1 , u 2 , u 3  . . . are used, this manipulation is discovered in operation  280 .  
         [0042]     Second, if all bidding function values PP 1 , PP 2 , PP 3 , . . . are transmitted to the successful bidding decider, the bidding prices v 1 , v 2 , v 3  . . . of the bidders u 1 , u 2 , u 3  . . . cannot be substantially changed thereafter.  
         [0043]     Third, since the successful bidding decider cannot acknowledge the individual bidding prices v 1 , v 2 , v 3  . . . of the bidders u 1 , u 2 , u 3  . . . from the bidding functions PP (since it is impossible to determine the bidding price from the bidding function value), the successful bidding decider can be prevented from informing a specific bidder of the bidding price of other bidders, which is advantageous for competitiveness of the bidding.  
         [0044]      FIG. 3  is a view illustrating an exemplary method of generating the bidding function value. Other methods are also available.  
         [0045]     As briefly discussed above, a bidding function (h) generates a bidding function value PP from a bidding price V. That is, PP=h(V).  
         [0046]     In operation  310 , the bidder selects the bidding price V.  
         [0047]     In operation  320 , the bidding price is binarized to obtain binary constants V 0 , V 1 , V 2 , . . . , V k−2 , V k−1 , which respectively corresponding to digits. The binary constant V k−1  corresponds to a Most Significant Bit (MSB) of a binary number, and the binary constant V 0  corresponds to a Least Significant Bit (LSB). The binary constant V i  has 0 or 1 due to the binarization.  
         [0048]     In operation  330 , two predetermined prime numbers are selected for binary constants V 0 , V 1 , V 2 , . . . , V i , . . . , V k−1  according to the following three rules to allocate a first prime number p i  and a second prime number q i .  
         [0049]     First, if V i =0, a modular 4 operation of the first prime number has a result value of 1, and a modular 4 operation of the second prime number has a result value of 3.  
         [0050]     Second, if V i =0, the modular 4 operation of the first prime number has a result value of 3, and the modular 4 operation of the second prime number has a result value of 1.  
         [0051]     Third, the second prime number is larger than the first prime number.  
         [0052]     In operation  340 , a pair of the first prime number p i  and the second prime number q i  is defined as a bidding coefficient C i .  
         [0053]     In operation  350 , after the multiplication of the first prime number p i  and the second prime number q i , the multiplication result is arranged in a position of the binary constant V i  corresponding to the first prime number p i  and the second prime number q i  to generate the bidding function value PP.  
         [0054]     For example, in case where the bidding price (V) is 5, the binarized bidding price VB in  is 0101 and accordingly, the binary constants are obtained as V 3 =0, V 2 =1, V 1 =0, V 0 =1. According to the aforementioned rules, the first prime number and the second prime number are selected as P 3 =5, Q 3 =7, Q 2 =13, P 1 =5, Q 1 =7, P 0 =7, Q 0 =13 for the binary constant. The prime number can be arbitrarily selected within the above three rules. If the selected prime number is multiplied and concatenated, the bidding function value is obtained as PP=5*7//7*13//5*7//7*13=35//91//35//91=00100011//01011011//00100011//01011011. That is, the bidding price (V=5) corresponds to the bidding function value (PP=00100011/01011011//00100011//01011011).  
         [0055]     In the method of  FIG. 3 , the selection and the multiplication of the prime numbers are performed to obtain the bidding coefficient C i  from the bidding price and then obtain the bidding function value from the bidding coefficient C i . However, it is not easy to obtain the selected prime number from the bidding function value and obtain the bidding price from the selected prime number. For example, if a prime number of over 500 bits is selected, it is difficult to perform the aforementioned inverse calculation process using current calculators.  
         [0056]      FIG. 4  is a view illustrating a bidding method according to a second exemplary embodiment of the invention.  
         [0057]     In operation  410 , a client (i.e., a bidder (u 1 )) selects his or her bidding price (v 1 ).  
         [0058]     In operation  420 , the bidder (u 1 ) inputs the bidding price (v 1 ) to a bidding function (h(x)) to generate a bidding function value PP 1 .  
         [0059]     In operation  430 , the bidder (u 1 ) transmits the generated bidding function value (PP 1 ) to a successful bidding decider.  
         [0060]     In operation  440 , the successful bidding decider transmits all of the received bidding function values PP 1 , PP 2 , PP 3 , . . . to each of the bidders.  
         [0061]     In operation  450 , all bidders transmit their bidding prices v″ (v 1 ″, v 2 ″, v 3 ″ . . . ) to the successful bidding decider. The bidding prices v 1 ″, v 2 ″, v 3 ″ . . . refer to bidding prices that, the bidders u 1 , u 2 , u 3  . . . claim are made in operation  410 . Thus, the bidding price (v) of the embodiment of  FIG. 2  is different from the bidding price (v″) of the embodiment of  FIG. 4 .  
         [0062]     In operation  455 , the successful bidding decider checks whether or not a bidding function value h(v″) is identical with the bidding function value PP received from the bidder in operation  430 . The bidding function value h(v″) is generated by inputting the bidding price (v″) received from the bidder in the operation  450 , to the bidding function. That is, it is determined whether or not PP 1  is equal to h(v1″), PP 2  is equal to h(v2″), and PP 3  is equal to h(v3″), . . . If it is determined that PP 1  is equal to h(v1″), PP 2  is equal to h(v2″), and PP 3  is equal to h(v3″) . . . , a next process  460  is performed. Otherwise, the bidding is determined to be a failure and thus the procedure is ended.  
         [0063]     In operation  460 , the successful bidding decider compares the bidding prices (v) of the bidders with one another to determine the successful bidding price V ref . Since it has necessarily been determined that h(v″)=PP (in operation  455 ), it follows that v=v″, and the bidding price (v″) can be determined as an actual bidding price. As a result, the bidding price (v″) can be compared to determine the successful bidding price.  
         [0064]     In operation  470 , the successful bidding decider transmits the determined bidding price V ref  and all bidding prices (v 1 ′, v 2 ′, v 3 ′ . . . ) to each of the bidders. The bidding prices (v 1 ′, v 2 ′, v 3 ′ . . . ) refer to bidding prices that the successful bidding decider indicates are transmitted by the bidders.  
         [0065]     In operation  480 , each of the bidders compares the received bidding function values PP 1 , PP 2 , PP 3 , . . . (received in operation  240 ) with input function values h(v 1 ′), h(v 2 ′), v 3 ′ . . . , generated by inputting the received bidding prices v 1 ′, v 2 ′, v 3 ′. . . (received in operation  270 ) to the bidding function discussed above, to determine whether or not the successful bidding decider fallaciously reported the bidding price to the bidder.  
         [0066]     According to the embodiment of  FIG. 4 , the bidder can be prevented from transmitting a bidding price (v″) in operation  450  that is different from the bidding price (v) transmitted in operation  410 , to the successful bidding decider. Specifically, the bidder cannot deny his/her transmitting bidding price, because if the bidding price (v″) is different from the bidding price (v), it is determined in operation  455  that PP≠H(v″), and thereby that v≠v″.  
         [0067]      FIG. 5  is a view illustrating a bidding method according to the third exemplary embodiment of the invention.  
         [0068]     In operation  510 , a client (i.e., a bidder (u 1 )) selects his or her bidding price (v 1 ).  
         [0069]     In operation  515 , the bidder (u 1 ) inputs the bidding price (v 1 ) to a first lowest bidding function g(v) to generate a bidding coefficient (C). The first lowest bidding function g(v) refers to a function of generating the bidding coefficient (C 1 ) from the bidding price (v), for example, a function comprised of the operations  320  to  340  of  FIG. 3 . According to the first lowest bidding function g(v), more than one bidding coefficient can be generated for each bidding price.  
         [0070]     In operation  520 , the bidder (u 1 ) inputs the bidding coefficient (C) to a second lowest bidding function f(C) to generate the bidding function value PP 1 . According to the second lowest bidding function f(C), one bidding function value should be generated for each bidding coefficient. In comparison with  FIG. 4 , the bidding function, the first lowest bidding function and the second lowest bidding function have a relation of h(v)=f(g(v)).  
         [0071]     In operation  530 , the bidder (u 1 ) transmits the generated bidding function value (PP 1 ) to the successful bidding decider.  
         [0072]     In operation  540 , the successful bidding decider transmits all of the received bidding function values PP 1 , PP 2 , PP 3 , . . . to each of the bidders.  
         [0073]     In operation  550 , all of the bidders transmit their bidding prices v″(v 1 ″, v 2 ″, v 3 ″ . . . ) and bidding coefficients C″ (C 1 ″, C 2 ″, C 3 ″ . . . ) to the successful bidding decider. The bidding prices v1″, v2″, v 3 ″ . . . refer to bidding prices that the bidders u 1 , u 2 , u 3  . . . claim are made in operation  510 , and the bidding coefficients C 1 ″, C 2 ″, C 3 ″ . . . refer to bidding coefficients that the bidders u 2 , u 3  . . . . claim are generated in operation  515 .  
         [0074]     In operation  555 , the successful bidding decider checks whether or not a bidding function value f(C″), which is generated by inputting the bidding coefficient C″ received from the bidder in operation  550  to the second lowest bidding function f(C), is identical with the bidding function value PP received from the bidder in operation  530 . That is, it is determined whether or not PP 1  is equal to f(C 1 ″), PP 2  is equal to f(C 2 ″), and PP 3  is equal to f(C 3 ″), . . . If it is determined that PP 1  is equal to f(C 1 ″), PP 2  is equal to f(C 2 ″), and PP 3  is equal to f(C 3 ″), . . . a next process  560  is performed. Otherwise, the bidding is determined to be a failure and thus the procedure is ended.  
         [0075]     Processes  560  through  580  are similar to operations  460  through  480  of  FIG. 4 .  
         [0076]     In  560 , the successful bidding decider compares the bidding prices (v) of the bidders with one another to determine the successful bidding price V ref . Since it has necessarily been determined that f(c″)=PP (in operation  555 ), it follows that v=v″, and the bidding price (v″) can be determined as an actual bidding price. As a result, the bidding prices (v″) can be compared to determine the successful bidding price.  
         [0077]     In operation  570 , the successful bidding decider transmits the determined bidding price V ref  and all bidding prices (v 1 ′, v 2 ′, v 3 ′ . . . ) to each of the bidders. The bidding prices (v 1 ′, v 2 ′, v 3 ′ . . . ) refer to bidding prices that the successful bidding decider indicates are transmitted by the bidders.  
         [0078]     In operation  580 , each of the bidders compares the received bidding function values PP 1 , PP 2 , PP 3 , . . . (received in operation  540 ) with input function values h(v 1 ′), h(v 2 ′), v 3 ′ . . . generated by inputting the received bidding prices (v 1 ′, v 2 ′, v 3 ′ . . . ) (received in operation  570 ) to the bidding function discussed above, to determine whether or not the successful bidding decider fallaciously reported the bidding price to the bidder.  
         [0079]     The embodiment of  FIG. 5  is different from that of  FIG. 4  in that a bidding coefficient, not a bidding price, is used for checking the original bidding price submitted by the bidder. Specifically, the embodiment of  FIG. 5  is different from that of  FIG. 4  in that the bidding coefficient, not the bidding price, is used to compare the bidding function value PP with the bidding function value f(C″) in operation  555 . The method of  FIG. 5  has an advantage in that an operation required in the operation  555  is more simplified.  
         [0080]      FIG. 6  is a view illustrating a bidding method according to the fourth exemplary embodiment of the invention.  
         [0081]     In operation  610 , a client (i.e., a bidder (u 1 )) selects his or her bidding price (v 1 ).  
         [0082]     In operation  615 , the bidder (u 1 ) inputs the bidding price (v 1 ) to a first lowest bidding function g(v) to generate a bidding coefficient (C).  
         [0083]     In operation  620 , the bidder (u 1 ) inputs the bidding coefficient (C) to a second lowest bidding function f(C) to generate the bidding function value PP 1 .  
         [0084]     In operation  630 , the bidder (u 1 ) transmits the generated bidding function value (PP 1 ) to the successful bidding decider.  
         [0085]     In operation  640 , the successful bidding decider transmits all of the received bidding function values PP 1 , PP 2 , PP 3 , . . . to each of the bidders.  
         [0086]     In operation  650 , all of the bidders transmit their bidding prices v″ (v 1 ″, v 2 ″, v 3 ″ . . . ) and C″ (C 1 ″, C 2 ″, C 3 ″ . . . ) to the successful bidding decider.  
         [0087]     In operation  655 , the successful bidding decider determines whether or not a bidding function value f(C″), which is generated by inputting the bidding coefficient C″ received from the bidder in operation  650  to the second lowest bidding function f(C), is identical with the bidding function value PP received from the bidder in operation  630 . That is, it is determined whether or not PP 1  is equal to f(C 1 ″), PP 2  is equal to f(C 2 ″), and PP 3  is equal to f(C 3 ″), . . . If it is determined that PP 1  is equal to f(C 1 ″), PP 2  is equal to f(C 2 ″), and PP 3  is equal to f(C 3 ″), . . . a next process  660  is performed. Otherwise, the bidding is determined to be a failure and thus the procedure is ended.  
         [0088]     In operation  660 , the successful bidding decider compares the bidding prices (v″) of the bidders with one another to determine the successful bidding price V ref . Since it has necessarily been determined that f(c″)=PP (in operation  655 ) it follows that v=v″, and the bidding price (v″) can be determined as an actual bidding price. As a result, the bidding price (v″) can be compared to determine the successful bidding price.  
         [0089]     In operation  670 , the successful bidding decider transmits the determined bidding price V ref  and all bidding prices (v 1 ′, v 2 ′, v 3 ′ . . . ) to each of the bidders.  
         [0090]     In operation  680 , each of the bidders compares the received bidding function values PP 1 , PP 2 , PP 3 , . . . (received in operation  640 ) with input function values f(c 1 ′), f(c 2 ′), f(c 3 ′) . . . generated by inputting the received bidding prices (v 1 ′, v 2 ′, v 3 ′ . . . ) (received in operation  670 ) to the bidding function discussed above, to determine whether or not the successful bidding decider fallaciously reported the bidding price to the bidder.  
         [0091]     The embodiment of  FIG. 6  is different from that of  FIG. 5  in that a bidding function value PP is compared with the bidding function value f(c′) to determine whether or not any of the bidders manipulated the bidding. The method of  FIG. 6  has an advantage in that time taken to determine whether or not the bidding was manipulated is reduced due to the simplification of the operation  680 .  
         [0092]      FIG. 7  is a view illustrating a bidding method according to the fifth exemplary embodiment of the invention.  
         [0093]     In operation  710 , a client, (i.e. a bidder (u 1 )) selects his or her bidding price (v 1 ).  
         [0094]     In operation  720 , the bidder (u 1 ) inputs the bidding price (v 1 ) to a bidding function h(x) to generate a bidding function value PP 1 .  
         [0095]     In operation  730 , the bidder (u 1 ) transmits the generated bidding function value (PP 1 ) to the successful bidding decider.  
         [0096]     In operation  740 , the successful bidding decider transmits bidding function values PP 1 ′, PP 2 ′, PP 3 ′, . . . to each of the bidders, which, the successful bidding decider indicates are received from each of the bidders.  
         [0097]     In operation  745 , all of the bidders determine whether or not the bidding function values PP 1 ′, PP 2 ′, PP 3 ′, . . . include the bidding function values PP 1 , PP 2 , PP 3 , . . . , which the bidder has transmitted to the successful bidding decider in operation  730 . That is, the bidder (u 1 ) checks whether or not a relation of (PP 1 ′, PP 2 ′, PP 3 ′, . . . )           PP 1  is satisfied, the bidder (u 2 ) checks whether or not a relation of (PP 1 ′, PP 2 ′, PP 3 ′, . . . )           PP 2  is satisfied, and the bidder (u k ) checks whether or not a relation of (PP 1 ′, PP 2 ′, PP 3 ′, . . . )           P Pk  is satisfied. If it is determined that the bidding function values PP 1 ′, PP 2 ′, PP 3 ′, . . . include the bidding function values PP 1 , PP 2 , PP 3 , . . . , a next process  750  is performed. Otherwise, the bidding is determined to be a failure, and is ended.  
         [0098]     The next processes are the same as those of the embodiment shown in  FIG. 2 .  
         [0099]     In operation  750 , all of the bidders transmit their bidding prices v (v 1 , v 2 , v 3  . . . ) to the successful bidding decider.  
         [0100]     In operation  760 , the successful bidding decider compares the bidding prices of the bidders to determine the bidding price V ref .  
         [0101]     In operation  770 , the successful bidding decider transmits the determined bidding price V ref  and all bidding prices v 1 ′, v 2 ′, v 3 ′ . . . to each of the bidders u 1 , u 2 , u 3  . . .  
         [0102]     In operation  780 , each of the bidders compares the received bidding function values PP 1 , PP 2 , PP 3 , . . . (received in operation  740 ) with input function values h(v 1 ′), h(v 2 ′), h(v 3 ′) . . . , which are generated by inputting the received bidding prices v 1 ′, v 2 ′, v 3 ′ . . . (received in operation  770 ) to the bidding function discussed above, to determine whether or not the successful bidding decider fallaciously reported the bidding price to the bidder.  
         [0103]     The method of  FIG. 7  is different from the method of  FIG. 2  in that the operation  745  is further performed.  
         [0104]     Specifically, in the exemplary method of  FIG. 2 , bidders cannot recognize that the successful bidding decider transmits manipulated bidding function values PP 1 ′, PP 2 ′, PP 3 ′, . . . to the bidder in operation  740 . As a result, if the successful bidding decider transmits the manipulated bidding prices PP 1 ′, PP 2 ′, PP 3 ′, . . . , instead of the actual bidding prices PP 1 , PP 2 , PP 3 , . . . , to the bidder, the determination of whether or not the bidding is manipulated (operation  780 ) can be no longer relied upon.  
         [0105]     However, according to the method of  FIG. 7 , since all of the bidders determine whether or not the bidding function values are authentic, it is impossible for manipulated bidding function values to be transmitted to the bidders in operation  750 . In other words, according to the embodiment of  FIG. 7 , the successful bidding decider is prevented from transmitting the manipulated bidding function value to the bidder, thereby increasing the reliability of the results of the operation  780  for preventing manipulation of the bidding price.  
         [0106]      FIG. 8  is a view illustrating a bidding method according to the sixth exemplary embodiment of the invention.  
         [0107]     In operation  810 , a client (i.e. , a bidder (u 1 )) selects his or her bidding price (v 1 ).  
         [0108]     In operation  820 , the bidder (u 1 ) inputs the bidding price (v 1 ) to a bidding function h(x) to establish a bidding function value PP 1 .  
         [0109]     In operation  830 , the bidder uses his or her secret key K Sc  to perform a digital signature, and then transmits the digital-signature bidding function value PP//S(PP, K Sc ) to the successful bidding decider.  
         [0110]     In operation  835 , the successful bidding decider uses a public key K Pc  of the bidder to verify the received digital-signature bidding function value PP//S(PP, K Sc ). The successful bidding decider has already possessed the public key K Pc  of the bidder through a certification organization. The digital signature is widely known, and a description thereof is omitted. If the signature is verified in operation  835 , it is secured that a current successful bidding decider receives the bidding function value PP from a currently signing bidder (u 1 ).  
         [0111]     The operations  840  through  880  are similar to the operations  240  through  280  of  FIG. 2 .  
         [0112]     According to the method of  FIG. 8 , the bidder (u 1 ) cannot deny that the bidder (u 1 ) transmits the bidding function value PP to the successful bidding decider in operation  830 .  
         [0113]      FIG. 9  is a view illustrating a bidding method according to the seventh exemplary embodiment of the invention.  
         [0114]     Each of the embodiments shown in  FIGS. 2 through 8  can be combined with one another. As an example, the embodiment of  FIG. 9  represents one of the combined embodiments of  FIGS. 2 through 8 .  
         [0115]     In operation  910 , a client (i.e. a bidder (u 1 )) selects his or her bidding price (v 1 ), and inputs the bidding price (v 1 ) to a bidding function h(x) to establish a bidding function value PP 1 . The bidding function h(x) has a relation of h(x)=f(g(v)) as shown in  FIG. 5 .  
         [0116]     In operation  915 , the bidder uses the secret key K Sc  to apply his or her digital signature to the bidding function value PP, and then transmits the digital-signature bidding function value PP//S(PP, K Sc ) to the successful bidding decider.  
         [0117]     In operation  920 , the successful bidding decider uses the public key K Pc  of the bidder to verify the received digital-signature bidding function value PP//S(PP, K Sc ). If so, it is secured that a current successful bidding decider receives the bidding function value PP from a current signing bidder (u 1 ).  
         [0118]     In operation  925 , the successful bidding decider transmits bidding function values PP 1 ′, PP 2 ′, PP 3 ′, . . . that the successful bidding decider indicates are received from each of the bidders, to each of the bidders.  
         [0119]     In operation  930 , all of the bidders receive the bidding function values PP 1 ′, PP 2 ′, PP 3 ′, . . . any verify whether or not the bidding function values PP 1 , PP 2 ′, PP 3 ′, . . . include the bidding function values PP 1 , PP 2 , PP 3 , . . . which the bidder has transmitted to the successful bidding decider in operation  915 . If it is determined that the bidding function values PP 1 ′, PP 2 ′, PP 3 ′, . . . include the bidding function values PP 1 , PP 2 , PP 3 , . . . , it is confirmed that the successful bidding decider is not transmitting manipulated bidding function values to the bidders.  
         [0120]     In operation  935 , all bidders transmit their bidding prices v″ (v1″, v2″, . . . ) and C″ (C1′, C2″, . . . ) to the successful bidding decider.  
         [0121]     In operation  940 , the successful bidding decider verifies whether or not the bidding function value f(C″), which is generated by inputting the bidding coefficient C″ received from the bidder in operation  935  to the second lowest bidding function f(C), is identical with the bidding function value PP received from the bidder in the operation  915 . That is, it is determined whether or not PP 1  is equal to f(C1″), PP 2  is equal to f(C2″), and PP 3  is equal to f(C3″), . . . Thus, the bidder can be prevented from transmitting a bidding price (v″) different from the original bidding price (v) of the operation  910  to the successful bidding decider. That is, the bidder cannot change his or her originally transmitted bidding price.  
         [0122]     In operation  950 , the successful bidding decider compares the bidding prices of the bidders to determine the bidding price V ref .  
         [0123]     In operation  955 , the successful bidding decider transmits the determined bidding price V ref  and all bidding prices c 1 ′, c 2 ′, c 3  ′ . . . to each of the bidders.  
         [0124]     In operation  960 , each of the bidders compares the received bidding function values PP 1 , PP 2 , PP 3 , . . . from operation  925  with the input function values f(c 1 ′), f(c 2 ′), f(c 3 ′) . . . , which are generated by inputting the received bidding coefficients c 1 ′, c 2 ′, c 3 ′ . . . of the operation  955  to the second lowest bidding function f(C), to verify whether or not the successful bidding decider fallaciously reported the bidding price to the bidder. That is, in operation  960 , it is verified whether or not the successful bidding decider manipulated the bidding price.  
         [0125]      FIG. 10  is a view illustrating a bidding method according to the eighth exemplary embodiment of the invention.  
         [0126]     In the ninth exemplary embodiment, since data transmissions between a successful bidding decider and a bidder in the other exemplary embodiments are not all in an encrypted state, it is possible that a third party could copy the transmitted data. In order to overcome this drawback, a message may be transmitted between the bidder and the successful bidding decider in an encrypted state, as shown in  FIG. 10 .  
         [0127]     For message encryption and decryption, a public key infrastructure is used. According to a concept of the public key infrastructure, the successful bidding decider possesses a public key K Pc  of the bidder, and the bidder possesses the public key K Ps  of the successful bidding decider.  
         [0128]     In operation  1010 , the bidder (u 1 ) selects his or her bidding price (v 1 ), and establishes the bidding function value PP 1 .  
         [0129]     In operation  1015 , the bidder uses the secret key K Sc  of the bidder to add his or her digital signature to the bidding function value PP, and then establishes the digital-signature bidding function value PP//S(PP, K Sc ), and then encrypts the generated bidding function value PP//S(PP, K Sc ) by using the public key K Ps  of the successful bidding decider. Finally, the bidder transmits the encrypted bidding function value PP//S(PP, K Sc ) to the successful bidding decider.  
         [0130]     In operation  1017 , the successful bidding decider decrypts the encrypted bidding function value E 1  by using the secret key of the successful bidding decider to generate the digital signed bidding function value PP//S(PP, K Sc ).  
         [0131]     In operation  1020 , the digital signed bidding function value PP//S(PP, K Sc ) established in the operation  1017  is verified using the public key K Pc  of the bidder.  
         [0132]     In operation  1025 , the successful bidding decider transmits the bidding function values PP 1 ′, PP 2 ′, PP 3 ′, . . . to each of the bidders by using the secret key K Sc  of the successful bidding decider.  
         [0133]     In operation  1027 , the bidder decrypts the transmitted encrypted bidding function value E 2  by using the public key of the successful bidding decider to establish the bidding function values PP 1 ′, PP 2 ′, PP 3 ′, . . .  
         [0134]     In operation  1030 , the bidder verifies whether or not the bidding function values PP 1 ′, PP 2 ′, PP 3 ′, . . . include the bidding function values PP 1 , PP 2 , PP 3 , . . . , which the bidder has previously transmitted to the successful bidding decider in operation  1015 .  
         [0135]     In operation  1035 , all of the bidders encrypt their bidding prices v″(v 1 ″, v 2 ″, v 3 ″ . . . ) and the bidding coefficients C″ (C 1 ″, C 2 ″, C 3 ″. . . ) by using the public key K Ps  of the successful bidding decider, and then transmit the encrypted bidding prices v″ (v 1 ″, v 2 ″, v 3 ″ . . . ) and the bidding coefficients C″ (C 1 ″, C 2 ″, C 3 ″ . . . ) to the successful bidding decider.  
         [0136]     In operation  1037 , the successful bidding decider decrypts the encrypted and transmitted bidding price and bidding coefficient E 3  by using the secret key K Ss  of the successful bidding decider to establish the bidding price (v″) and the bidding coefficient (c″).  
         [0137]     In operation  1040 , the successful bidding decider verifies whether or not the bidding function value f(C″), which is established by inputting the bidding coefficient C″ to the second lowest bidding function f(C), is identical with the bidding function value PP received from the bidder in operation  1015 .  
         [0138]     In operation  1050 , the successful bidding decider compares the bidding prices of the bidders with each other to determine the successful bidding price V ref .  
         [0139]     In operation  1055 , the successful bidding decider uses the secret key K Ss  of the successful bidding decider to encrypt, and then transmits the successful bidding price V ref , all bidding prices v 1 ′, v 2 ′, v 3 ′ . . . , and the bidding coefficients c 1 ′, c 2 ′, c 3 ′ . . . to each of the bidders.  
         [0140]     In operation  1057 , the bidder uses the public key K Ps  of the successful bidding decider to decrypt the encrypted and transmitted successful bidding price V ref , all bidding prices v 1 ′, v 2 ′, v 3 ′, . . . , and the bidding coefficients c 1 ′, c 2 ′, C 3 ′ . . .  
         [0141]     In operation  1060 , the bidder compares the received bidding function values PP 1 , PP 2 , PP 3 , . . . of the operation  1025  with the input function values f(c 1 ′), f(c 2 ′), f(c 3 ′) . . . which are generated by inputting the decrypted bidding coefficients c 1 ′, c 2 ′, c 3 ′ . . . of operation  1057  to the second lowest bidding function f(C), to verity whether or not the successful bidding decider fallaciously reported the bidding price to the bidder. That is, in operation  1060 , it is verified whether or not the successful bidding decider manipulated the bidding price.  
         [0142]     The embodiment of  FIG. 10  is different from the embodiment of  FIG. 9  in that the data transmission of the operation  1015 , the operation  1025 , the operation  1035  and the operation  1055  are performed at the encrypted state and in that the successful bidding decider receiving data through the operation  1015 , the operation  1025 , the operation  1035  and the operation  1055  decrypts the encrypted data.  
         [0143]     The encryption of the operations  1015  and  1035  is performed using the public key K Ps  of the successful bidding decider, and the successful bidding decider being at a receiving side uses the secret key K Ss  of the successful bidding decider to decrypt the received data. The above process has an effect (message certification effect) of securing that the third party does not copy data. This is because anyone can obtain the public key of the successful bidding decider, but only the successful bidding decider has the secret key of the successful bidding decider.  
         [0144]     The encryption of the operations  1025  and  1055  is performed using the secret key K Ss  of the successful bidding decider, and the bidder being at the receiving side decrypts the received data by using the public key K Ps  of the successful bidding decider.  
         [0145]     The embodiments of the invention can be written as computer programs and can be implemented in general-use computers that execute the programs using a computer-readable recording medium. Examples of the computer readable recording medium include magnetic storage media (e.g. , ROM, floppy disks, hard disks, etc.), optical recording media (e.g., CD-ROM or DVDs), and storage media such as carrier waves (e.g. , transmission through the Internet).  
         [0146]     As described above, the invention can preliminarily transmit the bidding function value, which can distinguish the bidding prices of all other bidders, to each bidder through the successful bidding decider, thereby preventing the successful bidding decider from manipulating the bidding price.  
         [0147]     Further, the invention uses the bidding function value to verify whether or not the bidding price of the bidder is true, thereby preventing the bidder from denying his/her transmitting bidding price.  
         [0148]     While this invention has been particularly shown and described with reference to exemplary embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims. The exemplary embodiments should be considered in descriptive sense only and not for purposes of limitation. Therefore, the scope of the invention is defined not by the detailed description of the invention but by the appended claims, and all differences within the scope will be construed as being included in the invention.