System and method for quantitative competition and recording medium having recorded thereon program for implementing them

Abstract: A bid-opening device 20 receives from each bidding device 10 m a bidding price index γ m = g(h k (IV m )) (where IV m is an initial value, h k indicates k-times processing with a one-way function h and g is a one-way function) and its identifier ID m . The received information is stored on a common bulletin board 25. D m = h k (IV m ) (where D m = g(D m ) is generated with k set as the upper limit value K of the bidding price. The bulletin board 25 is checked for γ m which matches this D m . If no match is found, m is incremented by one, and the check for γ m -D m matching is made for each of m bidding devices. Upon completion of the matching for all the bidding devices, k is decremented by one, and a check is made for for γ m which matches D m = g(h k (IV m )), and the index k for which they match is determined as the highest price bid. The bid-opening device 20 outputs that k and the identifier ID m of γ m .

Description: BACKGROUND OF THE INVENTION The present invention relates to a quantitative competition system and method which compare aimed or intended values of participants and publish the highest or lowest value but hide the other values, and a recording medium on which there is recorded a program for implementing the method. More particularly, the invention pertains to a system and method for quantitative comparison of secret values, for example, in electronic lottery, electronic voting, or electronic sealed-bid auction under circumstances where many participants are allowed to access the server online across the Internet. Fig. 1 is depicts in block form the configuration of a conventional electronic competitive bidding or sealed-bid auction system. A notice of auction is input to bidding devices 10 1 to 10 M , which send their identifiers ID 1 to ID M and bidding prices PR 1 to PR M to a bid-opening device 20. In this transfer of each data, if it is desirable to keep the data secret, the bidding price is sent in compressed form, and if it is necessary to identify the data, it is sent together with a digital signature of the bidder. The bid-opening device 20 detects the highest (or lowest) one of the bid values received from the bidding devices 10 1 to 10 M , and outputs the winning bid and the identifier corresponding thereto. This scheme is disclosed, for example, in Japanese Patent Application Laid-Open Gazette No. 118876/90 entitled "Electronic Bidding System." On the other hand, schemes which publish only the winning bid but hide other information involved are disclosed, for example, in Kazuo SAKO, "Auction Protocol Which Hides Bids of Losers," The Symposium on Cryptography and Information Security, SCIS'99, and in Shingo MIYAZAKI and Kouichi SAKURAI, "A Bulletin-Board Base Digital Auction Scheme with Bidding Down Strategy," The Symposium on Cryptography and Information Security SCIS'99. The sealed-bid auction scheme to which the present invention is applied satisfies the requirements mentioned below. Only bidding prices are used to determine the winning bid; that is, the successful bidding device is not determined using other information. The bid-opening device and a provisional identifier registration device neither cheat nor malfunction. The document XP000677143, PayWord And Micromint: Two Simple Micropayment Schemes by R.L. Rivest and A. Shamir, pages 69 - 87, 10-04-1996 discloses generating a chain of paywords by applying a one-way function to a randomly selected initial value a number of times. The initial value represents an n-th payword and the last (0-th) element of this chain obtained after having applied the one-way function n times is sent to a vendor at the begin of a purchasing action but does itself not form a payword . The first to n-th paywords are each worth the same unit value and are sent to the vendor piece by piece as payment for respective services. The vendor can verify each payword by applying the same one-way function to the respectively received payword and comparing the result with the previously received payword . At the end of the purchasing action the vendor can calculate the total amount of the received paywords by multiplying the number of received paywords , i.e., the number of applications of the one-way-function, with the unit value. SUMMARY OF THE INVENTION It is therefore an object of the present invention to provide a quantitative competition system and method which ensure the fairness of a sealed-bid auction, the validity of a successful bid value, the undeniability of bidders, the anonymity of the bidders and the freedom from double bidding. The fairness of an auction means that it is possible to prevent bidders from obtaining information on other bidding devices prior to bidding in an attempt to make bids with unfair advantage. The validity of a successful bid value means that it is possible to prevent an unfair determination of the winning bid. The undeniability of bidders means that it is possible to prevent them from denying their bid values at the bid-opening time. The anonymity of bidders means that it is possible to prevent leakage of identification information of every bidding device. This object is achieved by the method and system as claimed in claims 1 and 24, respectively, and their preferred embodiments as claimed in the dependent claims. According to the present invention, each bidding device transforms its bidding price to bidding price information through utilization of a one-way function, and the bid-opening device uses the bidding price information to determine the successful bid without revealing the prices bid by unsuccessful bidders. BRIEF DESCRIPTION OF THE DRAWINGS Fig. 1 is a block diagram showing the general configuration of an electronic competitive bidding system; Fig. 2 is a flowchart showing the general procedure of an electronic sealed-bid auction; Fig. 3 is a block diagram depicting a bidding device which uses one-way functions to generate a bid index; Fig. 4 is a block diagram depicting a concrete example of a bidding price transformer using one-way function processors; Fig. 5 is a flowchart showing the procedure for generating the bid index through the use of one-way functions; Fig. 6 is a block diagram illustrating the functional configuration of a bid-opening device 20 in a first embodiment of the present invention which is designed to determine the highest price bid as the winning bid; Fig. 7 is a flowchart showing the operation of the bid-opening device 20 depicted in Fig. 6; Fig. 8 is a block diagram illustrating the functional configuration of a bid-opening device in a second embodiment of the present invention; Fig. 9 is a flowchart showing the operation of the bid-opening device depicted in Fig. 8; Fig. 10 is a block diagram illustrating the functional configuration of a bid-opening device in a third embodiment of the present invention which is designed to determine the highest price bid as the winning bid; Fig. 11 is a flowchart showing the bid-opening procedure of the bid-opening device depicted in Fig. 10; Fig. 12 is a block diagram illustrating the functional configuration of a bidding device in a fourth embodiment of the present invention; Fig. 13 is a block diagram illustrating the functional configuration of a bid-opening device in the fourth embodiment; Fig. 14 is a flowchart showing the bid-opening procedure of the bid-opening device depicted in Fig. 13; Fig. 15 is a block diagram illustrating the functional configuration of a bidding device in a fifth embodiment of the present invention; Fig. 16 is a flowchart showing the bidding procedure of the bidding device depicted in Fig. 15; Fig. 17 is a block diagram illustrating the functional configuration of a bid-opening device in a fifth embodiment of the present invention; Fig. 18 is a flowchart showing the bid-opening procedure of the bid-opening device depicted in Fig. 17; Fig. 19 is a flowchart showing the bid-opening procedure in a modified form of the fifth embodiment; Fig. 20 is a block diagram illustrating the functional configuration of a bidding device in a sixth embodiment of the present invention; Fig. 21 is a flowchart showing the bidding procedure of the bidding device depicted in Fig. 20; Fig. 22 is a block diagram illustrating the functional configuration of a bid-opening device in the sixth embodiment; Fig. 23 is a flowchart showing the bid-opening procedure of the bid-opening device depicted in Fig. 22; Fig. 24 is a flowchart showing a modified form of the bid-opening procedure depicted in Fig. 23; Fig. 25 is a block diagram illustrating the functional configuration of a bidding device in a seventh embodiment of the present invention; Fig. 26 is a flowchart showing the bidding procedure of the bidding device depicted in Fig. 25; Fig. 27 is a block diagram illustrating the functional configuration of a bid-opening device in the seventh embodiment; Fig. 28 is a flowchart showing the bid-opening procedure of the bid-opening device depicted in Fig. 27; Fig. 29 is a flowchart showing a modified form of the bid-opening procedure depicted in Fig. 28; Fig. 30 is a block diagram illustrating the functional configuration of a bidding device in an eighth embodiment of the present invention; Fig. 31 is a block diagram illustrating the functional configuration of a bid0opening device in the eighth embodiment; Fig. 32 is a flowchart showing the bid-opening procedure of the bid-opening device depicted in Fig. 31; Fig. 33 is a block diagram illustrating an electronic sealed-bit auction system which utilizes a provisional identifier; and Fig. 34 is a flowchart showing the bidding procedure in the system depicted in Fig. 33. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS First Embodiment The electronic competitive sealed-bid auction system according to this embodiment comprises, as is the case with Fig. 1, M bidding devices 10 1 to 10 M and a bid-opening device 21, which are accessible to each other. This system operates as depicted in the flowchart of Fig. 2. Step S1: Input a notice of auction to each of the bidding devices 10 1 to 10 M . The notice of auction describes bidding conditions such as the name of an object for auction, the highest and/or lowest price bid, the term for bidding and qualifications for bidding, and a method of access to the bid-opening device. The bidding conditions differ with objects for auction. Step S2: In response to the notice of auction which describes the details of the auction, each bidding device 10 m (where m = 1, 2, ..., M) generates a bidding price, and sends to the bid-opening device 20 the bidding price PR m and an identifier ID m by which the bidding device 10 m can be identified: Step S3: Having received from the respective bidding device 10 1 to 10 M their bidding prices PR 1 to PR M and identifiers ID 1 to ID M , the bid-opening device 20 computes the highest (or lowest) one of the bidding prices and determines it to be the winning bid. Step S4: The bid-opening device 20 outputs the winning bid and the identifier of the bidding device having submitted it. The above procedure is the same as in the prior art. Since the communication channel between each bidding device 10 m and the bid-opening device 20 is not an anonymous channel, in particular, such a procedure as shown in Fig. 2 allows each bidding device 10 m to learn the bidding prices of the other bidding devices by wiretapping. To avoid this, according to the present invention, each bidding device 10m uses one-way functions to transform the bidding price to bidding price information, which is sent to the bid-opening device 20. Referring to Fig. 3, the bidding device 10m comprises an input part 11, an information storage part 12, a control part 13 and a bidding price transformer 14. The control part 13 follows a control procedure stored in the information storage part 12 to control operations of the respective parts. The notice of auction, which describes the bidding conditions, is provided in advance via a channel to the bidding device 10 m from the bid-opening device 20 or some other device. In the information storage part 12 there are also stored the identifier ID m of this bidding device 10m and an initial value IV m which is used for bidding. At the time of bidding, the bidding price PR m provided to the input part 11 is fed via the control part 13 into the bidding price transformer 14. The bidding price transformer 14 uses one-way functions to transform the bidding price PR m to bidding price information γ m in integral form. The bidding price information γ m and the identifier ID m are sent from a transceiver (not shown) to the bid-opening device 20 via a communication channel. Based on the bidding price information from every bidding device 10 m , the bid-opening device 20 decides the winning bid. In this embodiment, the bidding price transformer 14 is supplied with the initial value IV m read out of the information storage part 12 by the control part 13 and the bidding price PR m , and processes the initial value IV m by a one-way function by the number of times corresponding to the bidding price PR m as described below. A one-way function is such a function that, given a variable x, y(=h(x)) can easily be computed from x, but when given y, very high computational complexity is needed to compute x such that y=h(x). An output value, obtained by processing the initial value IV m with the one-way function h by k times like h(...h(h(h(IV m )))...), will hereinafter be represented by h k (IV m ). A concrete example of the bidding price transformer 14 is depicted in Fig. 4 and its procedure in Fig. 5. The bidding price transformer 14 comprises a data storage part 14S, conversion table memory 15, a one-way function h processor 16 and a one-way function g processor 17. Incidentally, the examples of the bidding device 10 m depicted in Figs. 3 and 4 will also be referred to in describing second and third embodiments of the invention later on, in which case the broken-lined data storage part 14S is provided to store the result D m of each computation by the one-way function h and the final result C m of computation. A conversion table, which is stored in the conversion table memory 15, is contained as one bidding condition in the notice of auction, for instance. Each bidding device 10m reads out the conversion table from a bulletin board 25 of the bid-opening device 20 which will be described later on with reference to Fig. 6. The conversion table is a table on which there are shown appropriately-separated K bidding prices from the lowest to the highest bidding price and a series of corresponding indexes k = 1, 2, ..., K. Each bidding device 10 m selects a desired bidding price from the conversion table and uses the corresponding index k as a bidding price index k m . In the Fig. 4 example the bidding prices corresponding to the indexes k = 1, 2, ..., K are arranged to monotonously increase, but by arranging them to monotonously decrease, the lowest bidding price can be made a winning bid. The one-function h processor is supplied with the initial value IV m and the price index k m corresponding to the bidding price PR m read out of the conversion table memory 15, and repeatedly processes the initial value IV m with the one-way function h by the number of times corresponding to the price index k m . As a result, h km (IV m ) is obtained. The one-way function g processor 17 further processes the output value h km (IV m ) with a one-way function g to compute γ m = g(h km (IV m )), and output it as bidding price information. Step S1: Input the bidding price PR m and the initial value IV m . Step S2: Read out of the conversion table memory 15 the price index k m corresponding to the bidding price PR m . Step S3: In the one-way function h processor 16, set the initial value of a variable t at 0 and the initial value IV m as γ m , and repeat steps S4 and S5. Step S4: Determine if t is smaller than k m . Step S5: If t < k m , update γ m with the value of h(γ m ) at that time, then increment t by one, and return to step S4. Step S6: When t becomes no longer smaller than k m , γ m = h km (IV m ), which is processed by the one-way function g processor 17 such that γ m ← g(γ m ). Step S7: Output, as the bidding price information, γ m obtained in step S6. BID-Opening Device (Successful Bid: Highest Bidding Price Bid) Fig. 6 is a block diagram of the bid-opening device 20, which comprises a control unit 21, and a one-way function h processor 22, a one-way function g processor 23, an adder/subtractor 24, a bulletin board 25, a program storage part 27S, an information storage part 35 and an update data storage part 36 which are all connected to the control unit 21. In the program storage part 27S there is stored a bid-opening procedure as a program, and the control unit 21 follows the program to control operations of the respective parts. The one-way function h processor 22 and the one-way function g processor 23 are series-connected. The parts except the bulletin board 25 constitute a winning bid selector 27. The number of bidding devices are M, and the initial value IV m unique to each bidding device 10 m (where m = 1, 2, ..., M) is prestored by some means in the information storage part 35. In the bulletin board 25 there are also recorded other bidding conditions including a conversion table 25A; furthermore, verification information and the results of auction to be published are written in the bulletin board 25. The pieces of information are accessible from any external devices as well as from the bidding devices. In this case, however, assume that only the bid-opening device 20 is allowed to write in the bulletin board 25 (This applies to other embodiments). In Fig. 7 there is depicted the procedure of operation of the bid-opening device 20. In this example the bidding price corresponding to the largest price index k m is made a successful bid as follows. The index value is decremented one by one from the largest index K corresponding to the upper limit of the bidding price specified in the conversion table; a check is made for the presence of the bidding price information γ m (where m = 1, 2, ..., M) corresponding to each index k is present; and when the bidding price information γ m corresponding to the index k is detected for the first time, the detected bidding price information γ m is determined to be the winning bid, and m (the bidding device number) at that time is computed, with which the bid-opening procedure ends. Step S0: The bidding device 10 m inputs its bidding price information γ m and identifier ID m (where m = 1, 2, ..., M) to the bid-opening device 20. Step S1: The control unit 21 records the bidding price information γ m and the identifier ID m in the bulletin board 25 accessible from arbitrary external devices (the bidding devices 10 1 to 10 M and other devices which are not bidders). Step S2: To precalculate D m (1) , D m (2) , ..., D m (K) such that D m (k) =h(D m (k-1) ) in the subsequent steps S3 to S6, the control unit 21 sets the bidding price index k←1, then reads out ID m from the information storage part 35, and substitutes it for D m (0) . Step S3: The control unit 21 determines whether k > K. Step S4: If not k > K, process D m (k-1) by the one-way function h processor 22 to obtain D m (k) . Step S5: Store D m (k) in the update data storage part 36. Step S6: Increment k by the adder/subtractor 24 such that k←k+1. Step S7: If k > K in step S3, it means that D m (1) =h(D m (0) )=h(IV m ), D m (2) =h(D m (1) )=h 2 (IV m ) ..., D m (K) =h K (IV m ) have all been obtained, then set k ←K. Step S8: Read out D m (K) =h K (IV m ) from the update data storage part 36 and process it by the one-way function g processor 23 to obtain D m (k) ←g(D m (k) ). Step S9: For m = 1, 2, ..., M, the control unit 21 makes a check to see if there exists on the bulletin board 25 the bidding price information γ m of the same value as D m (k) . Step S10: If there does not exist on the bulletin board 25 the information γ m of the same value as D m (k) , the control unit 21 inputs k to the adder/subtractor 24 to decrement k by one (i.e., k← k-1), then returns to step S8, and in steps S8 and S9 it makes a check to see if there exists on the bulletin board 25 the bidding price information γ m of the same value as D m (k) =g(h k (IV m )) for m = 1, 2, ..., M. Step S11: If the information γ m of the same value as D m (k) is found on the bulletin board 25 in step S9, then the control unit 21 regards it as the successful price bid and outputs the index k at this time and the bidding device identifier ID m corresponding to the information γ m , with which the procedure ends. As described above, a check is made first for a bidding device 10 m having input the bidding price information γ m equal to D m (k) (=g(h k (IV m ))) when the price index k is at the upper limit value K; if no such a bidding device is found, then a check is made for a bidding device 10 m having input the bidding price information γ m equal to D m (k) when the index k is decremented by one, i.e. k←k-1; and if no such a bidding device is found, then a check is made for a bidding device 10 m having input the bidding price information γ m equal to D m (k) when the index is further decremented by one, i.e. k←k-1. By making a check, for each decremented value of k, for the bidding device 10 m having input the bidding price information γ m equal to D m (k) (=g(h k (IV m ))), it is possible to find out the price information γ m corresponding to the largest index k, that is, the highest bidding price PR m . Furthermore, each bidding device 10m sends h km+1 (IV m ) to the bid-opening device 20 in advance, and the bid-opening device 20 publishes it on the bulletin board 25 and, after determining the successful bidding price, publishes on the bulletin board 25 the highest bidding price index k and the identifier ID m of the successful bidding device and the verification information h k (IV m ); in this instance, since k ought to equal k m , it is possible for an arbitrary device, except the bid-opening device 20, to verify the validity or fairness of the bid-opening process by processing he published verification information h k (IV m ) with the one-way function h to obtain h k+1 (IV m ) and making sure that it matches h km+1 (IV m ) published prior to the determination of the winning bid. Alternatively; the verification of the bid-opening procedure can be made by publishing the verification information h k (IV m ) of each bidding device on the bulletin board 25 in advance and making a check to see if the value, obtained by processing h km (IV m ) published on the bulletin board 25 with the one-way function h by (K-k m ) times, matches h k (IV m ). In the case where the lowest price bid is the winning bid, the bidding prices corresponding to the price indexes k on the conversion table are arranged so that they monotonously increase from the lowest value, as referred to previously. In the embodiment described above, all the bidding device 10 1 to 10 M send h km+1 (IV m ) to the bid-opening device 20 in advance for verification of the results of bid-opening, and the bid-opening device 20 writes it on the bulletin board. It is easy to compute h k+1 (z) from h k (z) in the computation h k+1 (z)=h(h k (z)) using the one-way function, but the computation of h k (z) from h k+1 (z) involves too enormous computational complexity to deal with practically. Accordingly, even if the bidding device publishes in advance h kv (IV m ) computed using an index k v larger than the bidding price index k m , there is no possibility of other devices learning the index k m unless the bid-opening is performed in terms of the index k m . The value k v of the verification information h kv (IV m ) may be a value k v =k m +A which is obtained by adding the bidding price index k m with a predetermined integer A equal to or greater than I, or a value k v =K+A which is obtained by adding the maximum index K with the integer A. In the former case, h km (IV m ) is processed A times with the one-way function h and the resulting value is checked for matching the published information h kv (IV m ). In the latter case, h km (IV m ) is processed K+A-k m times with the one-way function h and the resulting value is checked for matching the published information h kv (IV m ). In the embodiment described above, the bidding devices 10 1 to 10 M each convert the bidding price PR to the index k by the conversion table and the initial value IV m is processed with the one-way function by the number of times corresponding to the index k, but the use of the conversion table is not essential to the present invention. For example, the index k itself may be the bidding price, in which case neither of the bidding devices the bid-opening device requires the conversion table and any one of a series of value k which monotonously increase one by one within a predetermined range defined by the upper and lower limits given as one of the conditions for auction is chosen as the bidding price PR. SECOND EMBODIMENT In the first embodiment the bid-opening device 20 must be informed of the initial value IV m of each bidding device 10 m . In this embodiment, however, the bid-opening device 20 requires the bidding device 10 m to send D m =h k (IV m ) corresponding to the price index k, and hence it does not need the initial value IV m . Since the price index k varies one by one such that k = 1, 2, ..., K, this embodiment utilizes the fact that h k+1 (IV m ) = h(h k (IV m )) holds; the bid-opening device 20 verifies the response h k (IV m ) from the bidding device in reply to an inquiry k, then computes g(h k (IV m )), and checks it for matching the bidding price information γ m (where m = 1, 2, ..., M). A description will be given of the case where the highest bid is the successful bid. The bidding device 10 m has a configuration in which the data storage part 14S indicated by the broken line in Fig. 4 is added to the bidding price transformer 14. The bidding device 10 m precomputes D m = h k (IV m ) by the one-way function h processor 16 for each value k = 1, 2, ..., K+1, and stores it in the data storage part 14S. When inquired about k from the bid-opening device 20, the bidding device 10 m reads out of the data storage part 14S the data h k (IV m ) corresponding to k and sends it as the response D m = h k (IV m ) to the bid-opening device 20. Furthermore, the bidding device 10 m sends to the bid-opening device 20 the data h K+1 (IV m ) as the verification information C m , together with the bidding price information γ m . As depicted in Fig. 8, the bid-opening device 20 comprises a control unit 21, a one-way function h processor 22, a one-way function g processor 23, a subtractor 24, a program storage part 27S and a bulletin board 25 accessible from other devices. On the bulletin board 25 there are published a conversion table 25A along with conditions for auction. The bid-opening device 20 operates as depicted in Fig. 9. The bidding devices 10 m each send h K+1 (IV m ) as the verification information C m in advance to the bid-opening device 20 together with the bidding price information γ m and the identifier ID m . The bid-opening device 20 publishes these pieces of data on the bulletin board 25. Let it be assumed here that M bids are submitted, and hence m =1, 2, ..., M. Step S1: Write and hence publish on the bulletin board 25 the plural pieces of data sent to the bid-opening device 20, that is, the bidding price information γ m (=g(h km (IV m ))), the verification information C m and the identifier ID m . Step S2: Set the initial value of the index k at the upper limit value K by control unit 21. Step S3: Request each bidding device 10 m to present D m =h k (IV m ). The bidding device 10 m reads out of the data storage part 14S the data h k (IV m ), and sends it to the bid-opening device 20 as D m . Step S4: Record and hence publish on the bulletin board 25 the data D m sent from every bidding device 10 m . Step S5: Generates h(D m ) for each D m by the one-way function h processor 22. Step S6: Make a check to see if all the pieces of data h(D m ) match all the pieces of verification information C m on the bulletin board 25; if no mismatch is found, then determine that they are all correct or valid, and continue the subsequent steps. If a mismatch is found, then sends an error message to every bidding device 10 m and finish the bid-opening operation. Step S7: When no mismatch is found in step S6, generate data g(D m ) by the one-way function g processor 23. Step S8: Make a check to see if each data g(D m ) matches the bidding price information γ m ; if no match is found, update C m on the bulletin board 25 with D m (C m ←D m ). Step S10: Set k←k-1 by the subtractor 24 and return to step S3. Step S11: If even one of the pieces of information γ m is found to match with g(D m ) in step S8, decide k at that time as the price index corresponding to the winning bid, and output the index k and the identifier ID m of the bidding device 10 m having offered the successful bid. In this instance, the values of losing bids other than the winning one are kept secret from even the bid-opening device 20. In this embodiment, too, the lowest price bid can be chosen as the winning bid simply by arranging the bidding prices in such a manner as to monotonously decrease relative to a monotonously increasing index k on the conversion table, or by arranging the bidding prices in such a manner as to monotonously increase relative to a monotonously decreasing index k. THIRD EMBODIMENT In the second embodiment described above, as depicted in Fig. 9, the bid-opening device 20 requires all the bidding devices to present h k (IV m ) and receive it for each value of the price index k until the first detection of a match between g(D m ) and γ m in step S3. Accordingly, the sealed-bit auction scheme according to the second embodiment has the defect of a large communication cost. This embodiment requires the bid-opening device 20 to be informed of nothing but the initial value IV m , publishes only the winning bid, that is, hides the losing bids, and reduces the communication cost. The bidding device 10 m is identical in construction as in the second embodiment, and in the constructions of figs. 3 and 4 the data storage part 14S is provided in which there are prestored D m (1) , ..., D m (K) and C m = D m (k+1) . The bid-opening device 20, shown in Fig. 10, is basically identical in construction with the bid-opening device of Fig. 8, and no description will be repeated, but as will be described below with reference to Fig. 11, the operation of the bid-opening device in this embodiment differs from the operation depicted in Fig. 9. Step S0: The bidding devices 10 1 to 10 M each send in advance the verification information C m (= h k+1 (IV m )) to the bid-opening device 20 together with the bidding price information γ m (=g(h km (IV m )) and the identifiers ID m . Step S1: The bid-opening device 20 first publishes the plural pieces of bidding price information γ m and verification information C m and the identifiers ID m on the bulletin board 25. Step S2: Set the count value k of a counter 24 at the upper limit value K. Step S3: Inquire of all the bidding devices 10 m (where m = 1, 2, ..., M) about whether they have bid the price k. Step S4: Decide whether there is bidding device having bid with the price index k. Step S5: If no bidding device have bid the price, decrement k by the subtractor 24 to k←k-1 and return to step S3. Step S6: When it is found that any one of the bidding devices 10 m has bid with the price index k, request all the bidding devices 10m to present h k (IV m ). Each bidding device 10 m reads out of the data storage part 14S the data D m = h k (IV m ) precalculated corresponding to k as depicted in Fig. 4, and sends the read-out data D m to the bid-opening device 20. Step S7: Publishes all the responses D m from the bidding devices 10 m on the bulletin board 25. Step S8: For all the responses D m , generate h K-k+1 (D m ) by the one-way function h processor 22. Step S9: Check whether each data h K-k+1 (D m ) matches the corresponding verification information C m on the bulletin board 25. If no mismatch is found, decide that the responses Dm are all correct, and proceed to the subsequent steps. If a mismatch is found, send an error message to each bidding device 10 m and finish this bid-opening session for the price index k. Step S10: When no mismatch is found in step S9, generate g(D j ) by the one-way function g processor 23 for the response D j (where 1≤j≤M) from the bidding device 10 j which has identified itself as having bid with the price k. Step S11: Check whether g(D j ) matches the bidding price information γ j of the bidding device 10j on the bulletin board 25. If they match, decide that the bidding device 10 j has bid the price k. If they do not match, send an error message to all the bidding devices and discontinue the session. As described above, in this embodiment, when no bidding device identifies itself as having bid the price k in reply to the inquiry in step S3, the bidding devices 10 m are not required to send D m (= h k (IV m )) to the bid-opening device 20, and hence the communication cost is small as compared with that in the second embodiment accordingly.As described below, a check is made, if necessary, to see if another bidding device 10m (different from the device 10j) has bid a value higher than k. Step S12: For t such that k≤t≤K and all m's except j (1≤m≤M m≠j), it is checked whether g(h t-k (h k (IV m ))) matches the bidding price information γ m on the bulletin board 25 for t = k, k+1, ..., K, and if a match is found, then send an error message to each bidding device concerned and discontinue the bid-opening session. Step S13: If no match is found, then output the winning bidding price k and the identifier ID j of the bidding device having offered the winning bid. With this check or verification, if a bidding device makes a false response to the inquiry about whether it has bid, for example, with k, this dishonest bidding device is detected; that is, this embodiment satisfies the "undeniability" requirement. When this checking does not takes place, the bid-opening device 20 outputs k and ID j when deciding in step S11 that the bidding device 10 j has bid with k. In this way, this embodiment keeps losing bids, except the winning one, secret from even the bid-opening device 20 and reduces the communication cost. While in the above the highest price bid is the winning bid, the lowest price bid can be chosen as the successful bid simply by arranging monotonously decreasing (or increasing) values of the bidding price PR on the conversion table in a one-to-one correspondence with monotonously increasing (or decreasing) values of the price index k. FOURTH EMBODIMENT This embodiment is intended to permit simplification of the processing in the bid-opening device 20 and reduction of the communication cost. As depicted in Fig. 12, each bidding device 10 m has a construction in which a random generator 18 and an operation part 19 are added to the bidding device shown in Figs. 3 and 4. The random generator 18 generates a random number R m known only to the bidding device 10 m . The operation part 19 uses the random number R m and the bidding price PR m to perform a predetermined operation such as a bit concatenation, exclusive ORing, addition, or multiplication (which operation will hereinafter be denoted by (+)), then processes the operation result with the one-way function h, and sends the processed output h(PR m (+) R m ) to the bid-opening device 20, together with the bidding price information γ m = g(h km (IV m )) and the identifier ID m . Fig. 13 illustrates in block form the functional configuration of the bid-opening device 20 in this embodiment, and Fig. 14 is a flowchart for explaining its operation. The bid-opening device 20 has a construction in which one-way function h processors 22' and 22" and an operation part 26 are added to the device of Fig. 10 and a subtractor 37 is replaced for the counter 24. For simplicity sake, the program storage part 27S is not shown. At the time of bidding every bidding device 10 m (where m = 1, 2, ..., M) sends h K+1 (IV m ) to the bid-opening device 20, along with the bidding price information γ m (= g(h km (IV m ))), the identifier ID m and the random number information h(PR m (+) R m ). Step S1: Receive γ m , C m (=h K+1 (IV m )), h(PR m (+) R m ) and ID m from each bidding device 20 and write and hence publish them on the bulletin board 25. Step S2: Request all the bidding devices 10 m to present their bidding prices PR m and random numbers R m . Step S3: Upon receiving PR m and R m from each bidding device 10m, compute (PR m (+)R m ) in the operation part 26, then compute h(PR m (+)R m ) by the one-way function h processor 22", and make a check to determine whether the value matches h(PR m (+)R m ) published on the bulletin board 25. Step S4: At this point the bid-opening device 20 learns all the bidding prices. The bid-opening device 20 obtains from the conversion table 25A a price index k mx corresponding to the highest value among the bidding prices PR m . The bidding device (the successful bidding device) which has bid k mx will hereinafter be identified by 10 j . At this point each bidding device cannot learn the bidding prices of the other bidding devices. Step S5: Send k mx to all the bidding devices 10 m and request them to present D m = h kmx (IV m ). Step S6: Each bidding device 10m reads out D m = h kmx (IV m ) corresponding to k mx from the data storage part 14S described previously with respect to Fig. 4, and sends the read-out Dm to the bid-opening device 20, which writes on the bulletin board 25 all the pieces of data D m sent from the bidding devices 10 m . Step S7: For each Dm, calculates K+1-k mx by a subtractor 37 and generate h K+1-kmx (D m ) by the one-way function h processor 22 from K+1-k mx and D m . Step S8: Verify whether h K+1-kmx (D m ) matches the corresponding verification information C m on the bulletin board 25. If no mismatch is found, decides that all the responses Dm are correct, and proceed to the subsequent steps. If a mismatch is found, sends an error message to each bidding device and discontinue further operation. Step S9: If no mismatch is found in step S8, generate g(D j ) by the one-way function g processor 23 for D j presented by the successful bidding device 10 j . Step S10: Verify whether g(D j ) matches the bidding price information γ j in the bidding data of the bidding device 20 j on the bulletin board 25. If they match, decide that the bidding device 10 j has bid with the price index k mx . If they do not match, send an error message to each bidding device and discontinue the current bid-opening session. As described below, a check is made, if necessary, to see if another bidding device 10m (different from the device 10j) has bid a value higher than k mx . Step S11: For t = k mx + 1, k mx +2, ..., K, compute h t-kmx (D m ) by the one-way function processor 22' and process the computed result by the one-way function g processor 23 to obtain g(h t-kmx (D m )), and verify whether this g(h t-kmx (D m )) matches the bidding price information γ m on the bulletin board 25 for all m's except j (1≤m≤M, m≠j). Step S12: If a match is found, then send an error message to each bidding device concerned and discontinue the current bid-opening session. If no match found, then output the winning bidding price index k mx and the identifier ID j of the bidding device 10 j having offered the winning bid. For example, in the case where the bidding device 10m bids with the random number information h(PR m (+)R m ) and the bidding price information γ m = g(h km (IV m ) (where PR m '>PR m ) and presents a price PR m ' higher than the bidding price PR m and the random number R m in reply to the request for the presentation of PR m and R m in step S2 and the price PR m ' becomes the highest bid value, if the bidding device presents D m = h kmx (IV m ) in reply to the request for the presentation of h kmx (IV m ) in step S5, it will pass the verifications in steps S7 and S8, incurring the possibility that the price PR m lower than PR m ' becomes the winning bid. However, this can be prevented because a mismatch is found between g(D m ) = g(h kmx (IV m )) and γ m = g(h km (IV m ) in steps S10 and S11. With the bid-opening device 20, it is possible to specify the winning bid and the corresponding successful bidding device from PR m , R m and ID m with ease, reduce the communication cost and verify that the winning bidding price and the winning device determined from PR m , R m and ID m are true. FIFTH EMBODIMENT In this embodiment each bidding device sends to the bid-opening device bidding price information containing select information which indicates whether to bid for each of K kinds of biddable prices. As depicted in Fig. 15, the bidding price transformer 14 of the bidding device 10m comprises a control part 41, a random generator 42, a one-way function h processor 43, a select information generator 45, a (+) operation part 46 and an adder 47. Let PR m represent that one of the K kinds of price indexes 1, 2, ..., K which the bidding device 10 m is to actually to bid. The select information denoted by b m (k) is defined as follows: b m (k) = 0 when the bidding device 10 m does not bid for the price index k such that 1≤k≤K, and b m (k) = 1 when the bidding device 10 m bids for the price index k. Accordingly, only one piece of information b m (k) exists for k such that 1≤k≤K. As mentioned above, the two values of the information b m (k) may be any values as long as they differ from each other. The information b m (k) is known only to the bidding device 10 m , which uses it as a random number for each value of the price index k. The bidding device 10 m outputs {ID m , h(1(+)b m (1) (+)R m (1) ), h(2(+)b m (2) (+)R m (2) ), ..., h(K(+)b m (K) (+)R m (K) } as bidding price information. A description will be given, with reference to Fig. 16, of the procedure by which K pieces of bidding price information are prepared. Step S1: The bidding price PR m is determined and input. Step S2: Set the initial value of the price index k to 1. Step S3: Make a check to see if k≤K. Step S4: If k≤K, generate the random number R m (k) by the random generator 42. Step S5: Generate the rice select information b m (k) by the select information generator 45 in accordance with the bidding price PR m . Step S6: The (+) operation part 46 inputs thereinto these pieces of information R m (k) and b m (k) , and compute k(+)b m (k) (+)R m (k) . (+) indicates a predetermined appropriate operation, which may be a concatenation, addition, subtraction, multiplication, or the like. Step S7: Process the result of this operation by the one-way function h processor 43 to generate h(k(+)b m (k) (+)R m (k) ). Step S8: Increment k by one by the adder 47, followed by a return to step S3. Step S9: When k exceeds K and the processing for all values of k ends, the bidding device 10 m sends the identifier ID m and the bidding price information {h(1(+)b m (1) (+)R m (1) ), ..., h(K(+)b m (K) (+)R m (K) )} to the bid-opening device 20. By generating the bidding price information for each price through utilization of the one-way function and sending all the pieces of bidding price information to the bid-opening device 20 as described above, the aimed bidding price can be kept secret. Fig. 17 illustrates in block form the bid-opening device 20 which performs the bid-opening process based on such bidding price information sent thereto. The bid-opening device 20 comprises a control part 21, a one-way function h processor 22, a subtractor 24, an information storage part 35, a bulletin board 25 and a (+) operation part 51. As is the case with Fig. 8, the program storage part 27S is also provided but is not shown for the sake of brevity. Fig. 18 is a flowchart for explaining the operation of this bid-opening device 20. Step S0: Receive the bidding price information {ID m , h(1(+)b m (1) (+)R m (1) ), h(2(+)b m (2) (+)R m (2) ), ..., h(K(+)b m (K) (+)R m (K) )} from the bidding device 10 m (where m = 1, 2, ..., M). Step S1: Write and hence publish the bidding price information on the bulletin board 25. Step S2: Set the price index k to its highest value K. Step S3: Request each bidding device 10 m to present the random number R m (k) . Step S4: The bidding devices 10 m each send the random number R m (k) to the bid-opening device 20, which writes all the random numbers R m (k) on the bulletin board 25. Step S5: Set b m (k) = 1, then input b m (k) and R m (k) to the (+) operation part 51 to perform an operation k(+)1(+)R m (k) , and process the result of operation by the one-way function h processor 22 to obtain h(k(+)1(+)R m (k) ). Step S6: For each m, make a check to see if a match is found between the calculated h(k(+)1(+)R m (k) ) and h(k(+)1(+)R m (k) ) on the bulletin board 25. Step S7: If no match is found, decrement k such that k←k-1, and return to step S3. Step S8: If a match is found, determine the k as the successful price index k, then output the price index k and the identifier ID m of the bidding device 10 m concerned, and discontinue the current bid-opening session. As described above, according to this embodiment, the bidding price information corresponding to each price index is generated using the one-way function and all the pieces of bidding price information are sequentially opened in decreasing order from the highest value. This enables non-winning bids to be kept secret. In the case of choosing the lowest price bid as the successful bid in this embodiment, the bidding device 10 m similarly sends the bidding price information {ID m , h(1(+)b m (1) (+)R m (1) ), h(2(+)b m (2) (+)R m (2) ), ..., h(K(+)b m (K) (+)R m (K) )} to the bid-opening device 20. The bid-opening device 20 carries out about the same processing as depicted in Fig. 18, but in step S2 the lowest value 1 of the K kinds of biddable price indexes is set to k and in sep S6 the index k is incremented by one such that k←k+1. In the case of choosing the highest price bid as the winning bid in this embodiment, since random numbers R m (t) received from all the bidding devices for t (where k≤t≤K) are recorded on the bulletin board 25 in step S4, it is possible to detect that b m (k) = 0 for the index larger than the winning price index k, by calculating h(t(+)b m (t) (+)R m (t) ) and comparing it with h(k(+)b m (k) (+)R m (k) ) published on the bulletin board 25. Accordingly, any one can determine if the winning price index k is correct. In this embodiment it is also possible for each bidding device 10 m to send the set of random numbers {R m (k) }(k=1,...,K) to the bid-opening device 20 in advance together with ID m and h(k(+)b m (k) (+)R m (k) ). In this case, however, the random number R m (k) is not written on the bulletin board 25 but is stored in the broken-lined information storage part 35. In Fig. 19 there is shown a flowchart for explaining the processing which the bid-opening device 20 performs in such an instance. Sep S0: Receive the bidding price information {ID m , h(1(+)b m (1) (+)R m (1) ), h(2(+)b m (2) (+)R m (2) ), ..., h(K(+)b m (K) (+)R m (K) )} and random number information {R m (1) , R m (2) , ..., R m (K) } from the bidding devices 10 m (where m = 1, 2, ..., M). Step S1: Write and hence publish only the bidding price information on the bulletin board 25 and store the random number information in the storage part 35 indicated by the broken line. Step S2: Set the highest possible value K for the price index k. Step S3: For all m's, generate h(k(+)1(+)R m (k) ) read out of the storage part 35 with b b (k) =1. Step S4: Verify whether a match is found between h(k(+)1(+)R m (k) ) and the pieces of bidding rice information written on the bulletin board 25. Step S5: If no match is found, decrement the price index k by one to k←k-1 and return to step S3. Step S6: For the index k for which the mach was found; write and publish all the random numbers R m (t) (k≤t≤K) on the bulletin board 25. Step S7: Determine k at this time as the winning price index, then output this index k and the identifier ID m of the bidding device 10 m having submitted the winning bid, and discontinue the current bid-opening session. When the lowest bidding price bid is chosen as the successful bid in this case, the index k is set to the smallest possible value, 1 in this example, in step S2 in Fig. 19, then in step S4 the index k is incremented by one, and in step S5 all random numbers R m (t) which are 1≤t≤k are published on the bulletin board 25. The other operations are the same as those described above with reference to Fig. 19. While in the above the select information b m (k) has been described to be "0" or "1," they may be any values as long as they are two predetermined different integers. SIXTH EMBODIMENT In the second embodiment described previously, at the time of bidding each bidding device 10 m sends the bidding price information γ m = g(h km (IV m )) and the verification information C m = h k+1 (IV m ) to the bid-opening device 20, and at the time of bid-opening the bid-opening device 20 requests the bidding device 10 m to send the information D m = h k (IV m ), and verifies whether h(D m ) and C m match for the information D m received from the bidding device 10 m ; if they match, then the bid-opening device 20 further verifies whether g(D m ) matches the bidding price information γ m . These verifications need to be made for all of m = 1, 2, ..., M. Accordingly, the amount of data to be processed increases with an increase in the number M of bidding devices used. In addition, the larger the highest value K of the index k, the more the amount of multiple processing by the one-way function h; hence it is necessary to process an enormous amount of data. To avoid this, the sixth embodiment utilizes one-way functions to transform the bidding price index k m to gm=H m (K) = h K-km )g xm (IV m )), which is sent to the bid-opening device 20. x m is an arbitrary integer equal to or greater than 1 which is determined by the bidding device 10 m . When inquired about whether it has bid with the index k, the bidding device 10 m presents H m (k-1) = h k-km-1 (g xm (IV m )) if k is not its aimed bidding price index k m , and if it is the aimed value, the bidding device 10 m presents H m (k-1) = g xm-1 (IV m ). The bid-opening device 20 verifies the former with h(H m (k-1) ) = H m (K) and the latter with g(H m (k-1) ) = H m (K) . The M bidding devices 10 m indirectly bit the bidding price index values k m to the bid-opening device 20, which detects the largest one of the index values k m bit by the bidding devices 10 m . Each bidding device 10 m and the bid-opening device 20 have a conversion table in which K bidding prices PR 1 , PR 2 , ..., PR K are held in a one-to-one correspondence with K kinds of index values 1, 2, ..., K in increasing order. In this embodiment, as the index value k (where k = 1, 2, ..., K) increases, the value PR k becomes larger; for example, PR 1 = 10, PR 2 = 15, PR 3 = 30, ..., PR K = 3000. Fig. 20 illustrates in block form the functional configuration of the bidding device 10 m and Fig. 21 is a flowchart showing its bidding procedure. The bidding device 10 m comprises an input part 11, a control part 13, a bidding price transformer 14, a transceiver 18 and a storage part 12 all of which are interconnected via a common bus10B. The bidding price transformer 14 is made up of a conversion table 15, a one-way function h processor 16 and a one-way function g processor 17. The control part 13 reads out a bidding procedure program stored in the storage part 12, and follows it to control the respective parts to perform bidding as described below. Step S1: The bidding device 10 m receives via the input part 11 a value PR m selected from values PR 1 , PR 2 , ..., PR K , an initial value (a positive integer) IV m and an arbitrary integer x m equal to or greater than 1. Step S2: The control part 13 stores IV m and x m in the storage part 12. Step S3: The control part 13 refers to the conversion table 15 with PR m to convert it to an index value k m and stores it in the storage part 12. Step S4: The initial value IV m and the bidding price x m are read out from the storage part 12, then input to the one-way function g processor 17, wherein the initial value is processed x m times with the one-say function g to obtain g xm (IV m ). Step S5: If necessary, g xm (IV m ) is stored in the storage part 12. Step S6: g xm (IV m ), K and k m are input to the one-way function h processor 16, wherein g xm (IV m ) is processed K-k m times with the one-way function h to obtain H m (K) = h K-km (g xm (IV m )). Step S7: H m (K) is sent as the bidding price information to the bid-opening device 20 via the transceiver 18. In this way, the bidding device 10 m sends the aimed bidding price index value k m to the bid-opening device 20. Step S8: Upon beginning the bid-opening procedure by the bid-opening device 20 after the bidding, each bidding device 10 m receives an inquiry with the price index k from the bit-opening device 20 by the transceiver 18 as depicted in Fig. 21. Step S9: A check is made to see if the inquiry index value k is equal to the bidding price index k m stored in the storage part 12. Step S10: If not equal, g xm (IV m ) is read out of the information storage part 12 and is input to the one-way function h processor 16 together with k m and k, wherein g xm (IV m ) is processed k-k m -1 times with the one-way function h to obtain response information H m (k-1) by the following equation: H m (k-1) = h k-km-1 (g xm (IV m )) Step S11: The bidding device 10 m sends the response information H m (k-1) to the bid-opening device 20 via the transceiver 18 and waits for the next inquiry about k. Step S12: If the inquiry about k corresponds to its aimed value, that is, if the index k is equal to k m in he storage part 12, the bidding device 10 m inputs the initial value IV m and x m-1 to the one-way function g processor 17, wherein the initial value IV m is processed xm-1 times to obtain response information H m (k-1) = g xm-1 (IV m ). Step S13: The bidding device 10 m sends the response information H m (k-1) to the bid-opening device 20. The functional configuration of the bid-opening device 20 is depicted in Fig. 22 and its bid-opening procedure in Fig. 23. The bid-opening device 20 comprises a control part 21, a program storage part 27S, a one-way function h processor 22, a one-way function g processor 23, a counter 24, a bulletin board 25, a conversion table 25A, a transceiver 28, and an information storage part 25, all of which are interconnected via a common bus 20B. Step S1: Upon receiving the bidding information γ m =H m (K) from each bidding device 10 m by the transceiver 28, the control part 21 writes the bidding information H m (K) on the bulletin board 25 to publish it. Step S2: Set K as the initial value of k in the counter 24. Step S3: Send the index value k on the counter 24 to each bidding device 10 m , and send thereto an inquiry about bidding via the transceiver 28. Step S4: Receive the response information H m (k-1) by the transceiver 28 from all the bidding devices 10 m . Step S5: Input H m (k-1) to the one-way function h processor 22 to process it with the one-way function h to obtain h(H m (k-1) ). Step S6: Check whether the result of operation h(H m (k-1) ) is equal to the bidding information A m for m in the storage part 35. Step S7: If h(H m (k-1) ) = H m (K) , store the received information H m (k-1) as updated bidding information H m (K) in the storage part 35 to update H m (K) . Step S8: Decrement the count value k of the counter 24 by one, then return to step S3, and inquire of each bidding device 10 m about whether it has made a registration for the value k. Step S9: When it is found in step S6 that the response information H m (k-1) from the bidding device 10 m is g xm-1 (IV m ) which is equal to the aimed index value, h(H m (k-1) ) does not become equal to H m (K) = h ek+1-km-1 (g xm (IV m )) = g xm (IV m ) in step S9. Then, input H m (k-1) to the one-way function g processor 23 to process H m (k-1) with the one-way function g to obtain g(H m (k-1) ). Step S10: Make a check to see if g(H m (k-1) ) is equal to H m (K) corresponding to m stored in the storage part 35. Step S11: When the bidding device 10 m has performed correct processing, h km+1-km-1 (g xm (IV m )) = g xm (IV m ) = H m (K) by the process of step S7 at the time of previous inquiry for k = k m+1 ; hence, g(H m (k-1) ) = H m (K) holds. In the inquiry processing for the index k which is decremented one by one, the index k for which g(H m (k-1) ) = H m (K) holds for the first time is the maximum value among the bidding price indexes of all the bidding devices, and this index k is converted to the value PR k by referring to the conversion table 25A. Step S12: Output the value PR k as the highest one of the bid values, and if necessary, output the number m of the winning bidding device 10 m as well. In the bidding procedure of Fig. 21, if k = k m in step S9 when inquired about whether it has bid k, the bidding device 10 m may send to the bid-opening device 29 in step S13 not only the response information H m (k-1) but also information indicating k = k m (which information will hereinafter be referred to as identity information). In this instance, upon receiving the response information H m (k-1) in step S4 in Fig. 23, the bid-opening device 20 makes a check for the identity information; if not having received it, the device 20 performs the operation h(H m (k-1) ) and if having received the information, g(H m (k-1) ). Incidentally, in the event that g(H m (k-1) ) ≠ H m (K) in step S10 in Fig. 23, the bid-opening device 20 may discontinue the current bid-opening session for the reasons of dishonesty by the bidding device concerned and request all the bidding devices 10 m to rebid; alternatively, the bid-opening device 20 may abandon inquiring that bidding device but may continue inquiring the other bidding device. In the case where the bidding prices PR k are arranged on each of the conversion tables 15 and 25A in such a decreasing order as 3000, 2950, 2900, ..., 100 in correspondence with the K index values 1, 2, ..., K, the price PR k corresponding to the index value k for which g(H m (k-1) ) = H m (K) holds for the first time in the inquiry about bidding k represents the lowest one of the aimed bidding prices of all the bidding devices 10 m . The maximum value of the index may also be obtained without using the conversion table 15. In this instance, the bidding device 10 m sends the index value k m , and after step S10 in Fig. 23 the bid-opening device 20 outputs the index value k as the highest value and, if necessary, together with the number m (or ID m ) of the bidding device 10 m having sent the index value k m . In the bid-opening procedure depicted in Fig. 23, the bid-opening device 20 inquires every bidding device 10 m for each value of the index k = K, K-1, K-2, ..., and receives H m (k-1) , and then processes it using the one-say function h in step S5; hence this procedure requires high communication cost and computational complexity. In Fig. 24 there is depicted a modified form of the above-mentioned bid-opening procedure whose communication cost and computational complexity are reduced using the scheme described previously with reference to Fig. 11. No description will be given of the bidding procedure since it is exactly the same as in the Fig. 11 embodiment. Step S0: Receive bidding price information H m (K) = h K-km (g xm (IV m )) and the identifier ID m from each bidding device. Step S1: Publish the received H m (K) and ID m on the bulletin board 25. Step S2: Set the initial value of the index k at K. Step S3: Inquire of all the bidding devices 10 m about whether they bid for the index k. Step S4: Check the responses for YES (which means that the bidding device concerned has bid for the index k). Step S5: If no YES is found, decrement k by one and returns to step S3 to repeat the same processing. Step S6: If YES is found, request all the bidding devices 10 m to send their bidding price information H m (k-1) . The response of the bidding device to this request is the same as described previously in connection with Fig. 21. Step S7: Publish the received information H m (k-1) on the bulletin board 25. Step S8: For the response H m (k-1) of an m (representing the bidding device 10 m ) having bid for the index k, compute H m = h K-k (g(H m (k-1) ) and for the responses H m (k-1) from the other m's (the other bidding devices), compute H m = h K+1-k (H m (k-1) ). Step S9: Check whether H m = H m (k) holds for all the m's, and if a mismatch is detected for even one of them, send an error message to all the bidding devices and discontinue the current bid-opening session. Step S 10: If a match is detected for every one of them, determine that the bidding device 10 m of the number m having responded YES is the winning bidder, and output the index k at that time and its identifier ID m for publication The bidding device 10m and the bid-opening device 20 can each be implemented by a computer. SEVENTH EMBODIMENT In the fifth embodiment described previously, each bidding device 10 m generates the select information b m (k) indicating whether to bid for every price index k = 1, 2, ..., K, and these pieces of select information are contained in K pieces of information h(k(+)b b (k) (+)R m (k) ), where k = 1, 2, ..., K. In this embodiment, however, the plural pieces of select information are contained in a single piece of bidding price information through multiple processing with the one-way function h. Fig. 25 illustrates in block form the functional configuration of each bidding device 10 m . Assume that K bidding prices PR and the k price indexes k = 1, 2, ..., K, which have a one-to-one correspondence with each other, are both arranged in monotone increasing order in the same direction. The bidding select information for each index k will be identified as b (k) . In this embodiment a select information generating part 54 does not generate the select information b m (k) for all the indexes k = 1, 2, ..., K but instead generates the select information b m (k) only for indexes higher than the bidding price index k m . H m (k) = h(H m (k-1) ∥b m (k) ) is calculated, with H m (km-1) = IV m as its initial value, for each value of the price index k (=k m , k m +1, ..., K) by a one-way function h processor 16, a concatenator 55 and a buffer 56, and the calculation result H m (k) for each value of k is stored in a data storage part 14S. The final calculation result H m (K) will hereinafter be called bidding price information. In this embodiment, a signature Sig m (H m (K) ) is generated by a signature generator 14S for the bidding price information 57. Fig. 26 is a flowchart showing the bidding procedure of the bidding device 10 m depicted in Fig. 25. In this embodiment, letting k m be the index that the bidding device 10 m bids, a select information generating part 54 generates K-k m +1 pieces of select information b m (km) , b m (km+1) , ..., b b (K) corresponding to the indexes larger than the bidding price index k m (step S1). Then, H m (km-1) , H m (km) , H m (km+1) , ..., H m (K) are successively generated by the following equations H m (km-1) = IV m H m (km) = h(H m (km-1) ∥b m (km) ) H m (km+1) = h(H m (km) ∥b m (km+1) ) . . H m (t) = h(H m (t-1) ∥b m (t) ) . . H m (K) = h(H m (K-1) ∥b m (K) ) and they are stored in the data storage part 14S (step S2). Further, a signature σ m = Sig m (H m (K) ) of the bidding device 10 m corresponding to the bidding price information H m (K) is generated by the signature generator 14S (step S3). The bidding device 10 m sends H m (K) , σ m and ID m as bidding information to the bid-opening device 20 (step S4). After this, the bid-opening procedure is started, and upon receiving an inquiry for each value of the index k, the bidding device reads out {H m (k-1) , b m (k) } from the data storage part 14S, and sends it to the bid-opening device (step S5). The bid-opening device 2- comprises, as depicted in Fig. 27, a control part 21, a one-way function h processor 22, a counter 24, a bulletin board 25, an information storage part 35, comparators 58 and 59 and a signature verifier 33. The pieces of bidding information H m (K) , σ m and ID m are written on the bulletin board 25. Then, the received information H m (k-1) is processed by the one-way function h processor 22 to obtain H m (k) = h(H m (k-1) ), which is compared by the comparator 58 with H m (k) read out of the bulletin board 25 to thereby verify whether they match. Further, it is checked by the comparator 59 whether the received select information b m (k) is 1, and if it is 1, then the bidding device 10 m corresponding to m at that time is determined as the winning bidder. Fig. 28 is flowchart showing the bid-opening procedure of the bid-opening device 20. The bid-opening device 20 verifies respective signatures σ m (where m = 1, 2, ..., M) and writes and publish the received pieces of information H m (K) , σ m and ID m on the bulletin board 25. The bid-opening device 20 sends k to the bidding device in the order [k = K, K-1, K-2, ...] and asks for information {H m (k-1) , b m (k) } corresponding to each index k. The information {H m (k-1) , b m (k) } received from each bidding device 10 m is published on the bulletin board 25, then it is checked whether H m (k) =h(H m (k-1) ,b m (k) ) holds for the information H m (k) received previously, and if it holds, it is checked whether the select information b m (k) is 1. When b m (k) is 0 for all of m = 1, 2, ..., M, the bid-opening device 20 decrements k to k←k-1, then sends again k to the bidding device and asks for the information {H m (k-1) , b m (k) }, thereafter performing the same operation as described above. If the select information b m (k) is 1 for any one of m = 1, 2, ..., M, then the bidding device 10m corresponding to that m is determined as the winning bidder. The bid-opening device 20 publishes the successful bidding price index k m and the identifier ID m of the wining bidding device 10 m on the bulletin board 25. According to this embodiment, since each bidding device does not perform the multiple processing with the one-way function for indexes smaller than the bidding price index k m , the computational complexity is low accordingly. The bid-opening device performs the bid-opening procedure as described below. Step S0: Receive the pieces of bidding information H m (K) , σ m and ID m from each bidding device, then verifies respective signature σ m (where m = 1, 2, ..., M), and writes and publishes the pieces of information H m (K) , σ m and ID m on the bulletin board 25. Step S1: Set the initial value of the index k to K. Step S2: Send k to each bidding device 10 m and ask for {H m (k-1) , b m (k) }. Step S3: Write and publish on the bulletin board 25 the information {H m (k-1) , b m (k) } received from each bidding device 10 m . Step S4: Process the received information {H m (k-1) , b m (k) } by the one-way function h processor 22, thereby generating H = h(H m (k-1) ∥b m (k) ). Step S5: Verify by the comparator 58 whether H generated in step S4 matches the previously received H m (k) . If a mismatch is detected, then send an error message to the bidding device and discontinue the current bid-opening session. Step S6: If no mismatch is found in step S5, then decides that the response from every bidding device is valid, and make a check by the comparator 59 to see if the received select information b m (k) is 1. Step S7: When the select information b m (k) from all bidding devices 10 m is 0, update k with k-1 and return to step S2. Step S8: When any one of M pieces of select information b m (k) is 1 in step S6, determine that the bidding device 10 m corresponding to m, which provides that information b m (k) , is the winning bidder, and write and publish its identifier ID m and the index k k of the winning bid on the bulletin board 25, and end the bid-opening procedure. In this embodiment the prices bid by the unsuccessful bidders are each kept secret from the other participants. The lowest price bid can be made a successful bid by using a conversion table wherein the price PR monotonously increases in the direction opposite to that in which the index k monotonously increases. Incidentally, it is also possible to set H m (0) = IV m , H m (1) = h(H m (0) ∥b m (1) ), ..., H m (K) =h(H m (K-1) ∥b m (K) ), where b m (1) =b m (2) = ... = b m (km-1) = 0. The communication cost and computational complexity involved in the bid-opening procedure of Fig. 27 can also be reduced, as depicted in Fig. 27', by the application of the scheme described previously with reference to Fig. 11. The bidding procedure is exactly the same as in Fig. 25, and hence it will not be described. Step S0: Receive the bidding price information H m (K) = h(...h(h(H m (km-1) ∥ b m (km) )∥ b m (km+1) )...∥ b m (K) ) and the identifier ID m from each bidding device. Step S1: Publish the received H m (K) and ID m . Sep S2: Set the initial value of the index k at K. Step S3: Inquire of all the bidding devices 10 m about whether they bid for the index k. Step S4: Check the responses for YES (which means that the bidding device concerned has bid for the index k). Step S5: If no YES is found, decrement k by one and returns to step S3 to repeat the same processing. Step S6: If YES is found, request all the bidding devices 10 m to send their bidding price information H m (k-1) = h(...h(h(H m (km-1) ∥ b m (km) )∥ b m (km+1) )...∥ b m (K) ). The response of the bidding device to this request is the same as described previously in connection with Fig. 21. For the bidding device having bid for the current index k, k = k m ; therefore, H m (k-1) = H m (km-1) = IV m . Step S7: Publish the received information H m (k-1) on the bulletin board 25. Step S8: For the response H m (k-1) of an m (representing the bidding device 10 m ) having bid for the index k, compute H m = h(...h(h(H m (k-1) ∥ 1)∥ 0)∥... 0) and for the responses H m (k-1) from the other m's (the other bidding devices), compute H m = h(...h(h(H m (k-1) )∥ 0)∥ 0)...∥ 0). Step S9: Check whether H m = H m (k) holds for all the m's, and if a mismatch is detected for even one of them, send an error message to all the bidding devices and discontinue the current bid-opening session. Step S10: If a match is detected for every one of them, determine that the bidding device 10 m of the number m having responded YES is the winning bidder, and output the index k at that time and its identifier ID m for publication. EIGHTH EMBODIMENT In each of the embodiments described above, the bidding prices for the bidding price indexes k = 1, 2, ..., K in the conversion table are discrete values such as 1000, 2000, 3000, ..., 10000, and each bidding device bids any one of the K prices but cannot bid a value intermediate between such discrete prices unless K prices are set in "1" increments. This embodiment is intended to permit bidding an arbitrary price intermediate between the discrete values. In this embodiment, an arbitrary bidding price PR within the range defined by the highest and lowest bidding prices is expressed by the following equation. PR =F(k) + Q where: 0 ≤ Q < F(k+1) - F(k) F(k) represents the price corresponding to the index k specified on the conversion table, and Q will hereinafter be referred to as a fraction. This embodiment is one that the bidding price PR defined above is applied to the first embodiment described previously. Bidding price information g(h k (IV m )) and Q m are sent to the bid-opening device 20. Further, h K+1 (IV m ) is sent as verification information to the bid-opening device 20 together with the bidding price information. These pieces of information are published on the bulletin board as is the case with the first embodiment. Fig. 30 is a block diagram illustrating the configuration of each bidding device 10 m , which is identical with the bidding device of the first embodiment except that the bidding price transformer 14 is provided with a fraction calculating part 14a. The fraction calculating part 14A calculates, for a desired bidding price PR m , Q m = PR m -F(k m ) from the index k m read out of the conversion table 15. As is the case with the first embodiment, the one-way function h processor 16 and the one-way function g processor 17 are used to compute γ m = g(h km (IV m )) from the index k m which satisfies F(k m +1)>PR m ≥F(k m ) and the initial value IV m . In this embodiment, γ m , Q m and ID m are sent as bidding information to the bid-opening device 20. The bid-opening device 20 has, as depicted in Fig. 31, a sequencer 61 added to the bid-opening device 20 in Fig. 6 embodiment. The pieces of bidding information γ m , Q m and ID m are written and published on the bulletin board 25. The sequencer 61 reads out Q 1 , Q 2 , ..., Q M from the bulletin board 25, then sequences them in descending order, and sequences m = 1, 2, ..., M corresponding to the sequenced Q m , thereafter outputting them in the sequential order. The control part 21 checks, for each index k, whether ID m and γ m match each other in the sequence of m defined by the sequencer 61. The bidding device 10 m , which corresponds to m for which the match was detected for the first time, is determined as the winning bidder. Fig. 32 is a flowchart depicting the bid-opening procedure of the bid-opening device 20 shown in Fig. 31, which steps S0 to S7 are identical with steps S0 to S7 in Fig. 7 but differ in step 8 and the subsequent steps. Roughly speaking, steps S8 and S9 are new steps added between steps S7 and S8 in Fig. 7. Step S8 and the subsequent steps will be described below. In step S8 the bid-opening device 20 makes a check to determine whether the matching in steps S6 to S11 has been completed for all fractions Q m for the current value of k. If not completed, the procedure goes to step S9. In step S9, the bid-opening device 20 selects the largest one of the remaining fractions Q m to determine the corresponding m. In step S10, the device 20 reads out of the data storage part 36 updated data D m (k) =h k (IV m ) corresponding to that m, and processes the read-out data by the one-way function g processor 23, thereby updating it such that D m (k) ←g(D m (k) ). In step S11, the control device 21 makes a check to see if the updated data D m (k) corresponding to m matches the bidding price information γ m corresponding to m on the bulletin board 25, and if not, then the procedure returns to step S8. When it is decided in step S8 that the matching for all the fractions Q m for the current k has been completed, the bid-opening device 20 decrements k by one in step S12, and returns to step S8, repeating the same procedure as mentioned above. When it is decided in step S11 that the updated data D m (k) and the bidding price information γ m match, the bid-opening device 20 determines in step S13 that the index k at that time is the successful index, and outputs the identifier ID m corresponding to that information γ m and the successful bidding price F(k)+Q m (or merely the index k). Since D m (k) and gm are matched for the current value of the index k in the descending order of the fractions Q m as described above, the highest bidding price can be determined at the first detection of matching between D m (k) and γ m . The value m selected in step S9 is determined by the sequencer 61 which sequences in descending order the fractions Q 1 , Q 2 , ..., Q M read out of the bulletin board 25, and sequences m correspondingly, as explained before. In the case where the lowest bidding price bid is a successful bid, the bidding prices are arranged in monotonous increasing order in the direction opposite to that in which the values of the index k are arranged in monotonous decreasing order, and the sequencer 61 sequences the fractions Q 1 , Q 2 , ..., Q M in ascending order, and sequences m correspondingly In step S9 the smallest one of the unprocessed fractions Q m is chosen, and the corresponding m is determined. The sequencer 61 determines the order of m. The technique that uses may fractions Q m as described above can be applied not only to the first embodiment but also to the other embodiments explained heretofore. Electronic Competitive Auction System With Provisional Identifier Device In the embodiments described above, since the bidding device identifier ID m is contained in the bidding information and sent to the bid-opening device 20, it can be said that these embodiments employ what is called a sealed-bit auction system. Fig. 33 illustrates in block form an electronic competitive auction system which uses a provisional identifier registration device to implement a secret-bit auction. Each bidding device 10 m is adapted to communicate with a provisional identifier registration device 70. Referring to Fig. 34, the operation of this electronic competitive auction system will be described below. Step S0: Notice of auction is input to each bidding device 10 m . Step S1: Upon receiving the notice of auction, the bidding device 10 i outputs the identifier ID m to the provisional identifier registration device 70. Step S2: For the identifier ID m provided thereto, the provisional identifier registration device 70 issues a provisional identifier AID m , then stores ID m and AID m in pair form on a recording medium 71, and sends the provisional identifier AID m to the bidding device 10 i . Step S3: The bidding device 10 m : generates the bidding price PR m ; Step S4: Transforms the bidding price PR m to the bidding price information γ m ; and Step S5: Sends the bidding price information γ m and the provisional identifier AID m to the bid-opening device 20. Step S6: The bid-opening device 20: calculates a successful bid from the bidding price indexes received from all the bidding deices; and Step S7: Outputs the successful bid and the provisional identifier of the bidding device having offered the successful bid. Thus, it is possible to keep the auction participants secret. Incidentally, when inquiring the provisional identifier registration device 70, the bid-opening device 20 can learn the identifier ID m corresponding to the successful bid from the provisional identifier AID m . This method can be applied to all of the embodiments explained hereinbefore. The bidding device 10 m and the bid-opening device 20 can be implemented by interpreting and executing programs with a computer. EFFECT OF THE INVENTION As described above, according to the present invention, since the bidding price is transformed to a bidding price information through utilization of a one-way function and is then sent to the bid-opening device, it is possible to keep the bidding price of each bidding device secret from the other bidding devices prior to bidding, ensuring fairness of auction. By publishing h K+1 (IV m ) and h kmx (IV m ) corresponding to a winning bidding price on the bulletin board, it is possible for all bidding devices to verify whether the winning bidding price was correctly bid without being altered. Furthermore, the present invention permits determination of the winning bid while hiding the other bidding prices. According to the second embodiment, the bids of losing bidders, except the winning bid, are kept secret even from the bid-opening device. According to the third embodiment, the bids of losing bidders, except the winning bid, are kept secret even from the bid-opening device, and in addition, the amount of data for communication with the bidding device can be reduced. According to the fourth embodiment, there is no fear of the bidding price of each bidding device being revealed to the other devices, and at the bid-opening time the successful bid can easily be specified directly from the bidding price; furthermore, the validity of the winning bid can be verified. According to the fifth embodiment, bidding price information including each bidding price index is generated using a one-way function, and the bidding price information for all bidding price indexes is sent to the bid-opening device; hence, the aimed bidding price can be kept secret. According to the sixth embodiment, either h(H m (k-1) ) = H m (K) or g(H m (k-1) ) = H m (K) needs only to be verified for each index value and for each bidding device. According to the seventh embodiment, each bidding device generates the bidding price information including select information for only indexes larger than the bidding price index k m in the fifth embodiment, the computational complexity can be reduced. According to the eighth embodiment, since a desired bidding price is expressed by the sum of fractions Q of a price F(k) corresponding to the index k, an arbitrary price can be bid with a small number of indexes.

Claims: A quantitative competition method for a system in which a plurality of participant devices (10 1 -10 M ) send their secret aimed values to a server device (20) and said server device (20) determines which of said plurality of participant devices (10 1 -10 M ) has offered the maximum or minimum one of said aimed values received, said method comprising the steps wherein: (a) each of said participant devices (10 1 -10 M ) repeatedly applies a predetermined one-way function a number of times corresponding to said secret aimed value to an initial value peculiar to the respective participant device (10 m ) to generate aimed value information; (b) either each participant device (10 m ) or said server device (20) repeatedly applies said one-way function a given number of times to the initial value peculiar to the respective participant device (10 m ) to generate for each participant device (10 m ) a sequence of updated initial values; (c) said server device (20) successively checks whether, for any of the participant devices (10 1 -10 M ), there is a match between one of the respective updated initial values and the respective aimed value information; and (d) upon first detection of matching in said step (c), said server device (20) decides that said aimed value of that one of said participant devices (10 1 -10 M ) which corresponds to said updated value having matched said aimed value information is maximum or minimum. The method of claim 1, wherein step (c) is performed each time a new updated initial value has been generated for each participant device (10 m ) in step (b) and step (b) is repeated to generate the next updated initial value for each participant device (10 m ) when no matching is detected in said step (c). The method of claim 2, wherein said step (b) is performed by said server device (20). The method of claim 3, wherein: there is provided a conversion table showing the relationships between a sequence of values selectable as said aimed values and a sequence of indexes k of integral values corresponding thereto; said step (a) includes a step wherein, letting M be the number of participant devices (10 1 -10 M ), an m-th participant device m computes said aimed value information by γ m = g(h km (IV m )) where h is said predetermined one-way function, g is also a one-way function, IV m is said initial value, k m is an index corresponding to said aimed value of said participant device m, and h km (IV m ) indicates that said one-way function h has been applied k m times to said initial value IV m ; said step (b) includes a step wherein said server device (20) generates said updated initial value by D m = g(h k (IV m )); and said step (c) includes a step of making a check for said aimed value information γ m = D m . The method of claim 2, wherein said step (b) includes a step wherein said each participant device (10 m ) generates said updated value by the number of times requested by said server device (20) and sends said updated initial value to said server device (20). The method of claim 5, wherein: there is provided a conversion table showing the relationships between a sequence of values selectable as said aimed values and a sequence of indexes k of integral values corresponding thereto; and said step (a) includes the steps wherein: (a-1) letting M be the number of participant devices (10 1 -10 M ), an m-th participant device m computes said aimed value information by γ m = g(h km (IV m )) where h is said predetermined one-way function, g is also a one-way function, IV m is said initial value, k m is an index corresponding to said aimed value of said participant device m, and h km (IV m ) indicates that said one-way function h has been applied k m times to said initial value IV m ; (a-2) said each participant device (10 m ) generates verification information C m = h K+1 (IV m ) and sends it to said server device (20) together with said aimed value information γ m ; and (a-3) said server device (20) publishes said aimed value information γ m and said verification information C m = h K+1 (IV m ) received from said each participant device (10 m ) on a bulleting board accessible from all of said participant devices (10 1 -10 M ). The method of claim 6, wherein: said step (b) includes the steps wherein: (b-1) said server device (20) requests said each participant device (10 m ) to send D m = h k (IV m ) corresponding to said index k; and (b-2) said each participant device (10 m ) responds to the request of said server device (20) to generate D m = h k (IV m ) corresponding to said index k as said updated initial value and sends it to said server; and said step (c) includes a step of generating g(D m ) and making a check for said aimed value information γ m = g(D m ). The method of claim 7, wherein: said step (b) includes a step (b-0) wherein said server device (20) sets said index k to an upper limit value K; and said step (c) includes the steps wherein said server device (20): (c-1) publishes said D m = h k (IV m ) received from said each participant device (10 m ) on said bulletin board (25); (c-2) computes said g(D m ) for said each participant device m; (c-3) makes a check to see if said g(D m ) matches said aimed value information; (c-4) if no match is detect for any of said m in said step (c-3), replaces said D m with said C m , decrementing said index k by one and returns to said step (b-1); and (c-5) if a match is detected in said step (c-3), determines that the value corresponding to said index k at that time is said maximum or minimum aimed value. The method of claim 8, wherein said step (c-1) includes a step wherein said server device (20) generates h(D m ) and makes a check to see if h(D m ) = C m holds. The method of claim 8, wherein said step (c-5) includes a step wherein said server device (20) publishes said determined maximum or minimum aimed value and said h km (IV m ) on said bulletin board (25). The method of claim 5, wherein: there is provided a conversion table showing the relationships between a sequence of values selectable as said aimed values and a sequence of indexes k of integral values corresponding thereto; and said step (a) includes the steps wherein: (a-1) letting M be the number of participant devices (10 1 -10 M ), an m-th participant device m computes said aimed value information by γ m = g(h km (IV m )) where h is said predetermined one-way function, g is also a one-way function, IV m is said initial value, k m is an index corresponding to said aimed value of said participant device m, and h km (IV m ) indicates that said one-way function h has been applied k m times to said initial value IV m ; (a-2) said each participant device (10 m ) generates verification information C m = h K+1 (IV m ) and sends it to said server device (20) together with said aimed value information γ m ; and (a-3) said server device (20) publishes said aimed value information γ m and said verification information C m = h K+1 (IV m ) received from said each participant device on a bulleting board accessible from all of said participant devices (10 1 -10 M ); said step (b) includes the steps wherein: (b-0) said server device (20) sets said index k to an upper limit value K; (b-1) said server device (20) inquires said each participant device m about whether said aimed value information corresponds to said index; (b-2) said server device (20) checks whether an answer from said each participant device (10 m ) acknowledges said correspondence; (b-3) if the answer from all participant devices (10 1 -10 M ) denies said correspondence, said server device (20) decrements said k by one and returns to said step (b-1); (b-4) if said answer from any participant device j acknowledges said correspondence, said server device (20) requests each participant device (10 m ) to present an updated initial value h k (IV m ); (b-5) each participant device (10 m ) generates D m = h k (IV m ) using its initial value IV m and sends said D m to said server device (20); and (b-6) said server device (20) publishes said D m from said each participant device (10 m ) on said bulletin board (25); and said step (c) includes the steps wherein said server device (20): (c-1) generates h(D m ) for each D m received in step (b-5); (c-2) checks whether each h(D m ) matches the corresponding verification information C m on said bulletin board (25); (c-3) if no mismatch is detected in said step (c-1), generates g(D j ) for D j presented by said participant device j; (c-4) checks whether said g(D j ) matches said aimed value information γ j of said participant device j on said bulletin board (25); and (c-5) if said g(D j ) matches said γ j in said step (c-4), decides that said participant device j sent to said server device (20) said aimed value information corresponding to said aimed value k. The method of claim 1, wherein: there is provided a conversion table showing the relationships between a sequence of values selectable as said aimed values and a sequence of indexes k of integral values corresponding thereto; and said step (a) includes the steps wherein: (a-1) letting M be the number of participant devices (10 1 -10 M ), an m-th participant device m computes said aimed value information by γ m = g(h km (IV m )) where h is said predetermined one-way function, g is a one-way function, IV m is said initial value, k m is an index corresponding to said aimed value of said participant device m, and h km (IV m ) indicates that said one-way function h has been applied k m times to said initial value IV m ; (a-2) said each participant device (10 m ) generates verification information C m = h K+1 (IV m ); (a-3) said each participant device m generates h(PR m (+)R m ) from a random number R m and said aimed value PR m and sends said h(PR m (+)R m ) to said server device (20) together with said γ m and said C m , said (+) represents a predetermined arbitrary operation; said step (b) includes the steps wherein: (b-1) said server device (20) publishes said γ m , h(PR m (+)R m ) and an identifier ID m as received from each participant device (10 m ) on said bulletin board (25); (b-2) said server device (20) requests said each participant device (10 m ) to present said aimed value PR m and said random number R m ; (b-3) said each participant device (10 m ) sends said aimed value PR m and said random number R m to said server device (20); (b-4) said server device (20) determines from said PR m and R m as received from each participant device (10 m ) an index k corresponding to the maximum one of said aimed values and a participant device j having sent said aimed value information corresponding to said index k; (b-5) said server device (20) request all of said participant devices (10 1 -10 M ) to present h k (IV m ); and (b-6) said each participant device (10 m ) generates D m =h k (IV m ) as said updated initial value using its initial value IV m , and sends said D m =h k (IV m ) to said server device (20); said step (c) includes the steps wherein said server device (20): (c-1) publishes all of the D m 's received in step (b-6) on said bulletin board (25); (c-2) generates h(D m )'s for all of said D m 's; (c-3) checks whether said h(D m )'s match said verification information C m 's on said bulletin board (25), respectively; (c-4) if no mismatch is detected in said step (c-3), generates g(D j ) for D j presented by said participant device j having sent said maximum aimed value determined in said step (b-4); and (c-5) checks whether said g(D j ) matches said aimed value information γ j of said participant device j on said bulletin board (25); and said step (d) includes a step wherein if a match is detected in said step (c-5), said server device (20) decides that said participant device j sent said aimed value information for said aimed value k. The method of claim 12, wherein said step (d) includes the steps of: (d-1) generating E m = g(h t-k (h k (IV m )) for t such that k≤t≤K and for all of said m's except said j; (d-2) checking whether these E m 's match said aimed value information γ m on said bulletin board (25); and (d-3) if no match is detected in said step (d-2), deciding that said k is an index corresponding to the maximum or minimum aimed value, and outputting said k and said identifier ID m of said participant device j having presented said aimed value. The method of claim 12 or 13, wherein: said each participant device m adds to the computation using said aimed value PR m and said random number R m in said step (a-3) additional information I m about its aimed value information, and outputs h(PR m (+)I m (+)R m ); and in said step (b-2) said server device (20) requests all of said participant devices (10 1 -10 M ) to present said additional information I m as well as said aimed value PR m and said random number R m . The method of claim 1, wherein: there is provided a conversion table showing the relationships between a sequence of values selectable as said aimed values and a sequence of indexes k of integral values respectively corresponding thereto; said step (a) includes the steps wherein: (a-1) each said participant device (10 m ) generates h(H m (K) ) as its aimed value information by repeating, for each of a sequence of index values k from at least k m corresponding to an aimed value of said each participant device (10 m ) to an upper limit index value K, processing of: combining an initial value of said each participant device (10 m ) with select information to provide combined information and applying said one-way function h to said combined information to generate an updated value, said select information indicating whether said each index value k is an aimed value or not; and sends said aimed value information H m (K) to said server device (20); (a-2) said server device (20) publishes said received aimed value information H m (K) on a bulletin board (25) accessible from all of said participant devices (10 1 -10 M ); said step (b) includes the steps wherein: (b-1) said server device (20) requests said each participant device (10 m ) to send an updated initial value {H m (k-1) , b m (k) } corresponding to each value of said index k in the descending order from K; and (b-2) said each participant device (10 m ) generates and sends {H m (k-1) , b m (k) } as said updated initial value to said server device (20); said step (c) includes the steps wherein said server device (20): (c-1) publishes on said bulletin board (25) said updated initial value {H m (k-1) , b m (k) } received from said each participant device (10 m ); (c-2) processes said updated initial value H m (k) with a one-way function h to generate H m (k) = h(H m (k-1) ∥b m (k) ); (c-3) checks whether said updated initial value H m (k) matches H m (k) received previously; and (c-4) if a match is detected in said step (c-3), decides whether said select information b m (k) represents that the corresponding index k is the index k m of said aimed value; and said step (d) is a step wherein if the result of decision in said step (c-4) is true, said server device (20) outputs said index concerned and the corresponding participant device (10 m ) number m, and if the result of decision is false, said server device (20) returns to said step (b) and repeats processing for the next index value k. The method of claim 1, wherein: there is provided a conversion table showing the relationships between a sequence of values selectable as said aimed values and a sequence of indexes k of integral values respectively corresponding thereto; said step (a) includes the steps wherein: (a-1) upon each processing of said initial value with said one-way function h, said each participant device (10 m ) generates an updated initial value by adding to said processed initial value select information b m (k) indicating whether said processed initial value is an aimed value for one value of said index k, and generates H m (k) by repeatedly performing this processing from at least that index k m of a sequence of indexes which corresponds to said aimed value to the upper limit value K, and sends said H m (k) as said aimed value information to said server device (20); (a-2) said server device (20) publishes the received aimed value information H m (K) on a bulletin board (25) accessible from all of said participant devices (10 1 -10 M ); said step (b) includes the steps wherein: (b-1) for each value of said index k in descending order from K, said server device (20) inquires said each participant device (10 m ) about whether it has bid for said index k, and said each participant device (10 m ) responds YES or NO to said inquiry; (b-2) upon first detection of the response YES, said server device (20) requests said each participant device (10 m ) to send its updated initial value H m (k-1) ; and (b-3) said each participant device (10 m ) generates and sends (H m (k-1) = h(H m (k-2) ∥ b m (k-1) ) as said updated initial value to said server device (20); said step (c) includes the steps wherein said server device (20): (c-1) publishes on said bulletin board (25) said updated initial value (H m (k-1) ) received from said each participant device (10 m ); (c-2) letting a and a represent predetermined values of said select information b m (k) indicating bidding and not bidding, respectively, generates, for said participant device m having bid for the current index k, H m = h(...h(h(H m (k-1) ∥ a)∥ a )...∥ a ) through the use of said updated initial value H m (k-1) , and for every one of the other participant devices m, generates H m = h(...h(h(H m (k-1) ∥ a )∥ a )...∥ a ) through the use of said updated initial values H m (k-1) ; (c-3) checks whether said H m for said each participant device (10 m ) matches said H m (k) published on said bulletin board (25); and (c-4) if a match is detected in said step (c-3), determines that said participant device (10 m ) having responded YES is the winning participant device (10 m ), and publishes the current value of said index k as the index K m of the aimed value of said winning participant device (10 m ). The method of claim 1, wherein: there is provided a conversion table showing the relationships between a sequence of values selectable as said aimed values and a sequence of indexes k of integral values respectively corresponding thereto, and, letting the initial value of said each participant device (10 m ) be represented by IV m , where m = 1, 2, ..., M, M being an integer equal to or greater than 2: said step (a) is a step wherein said each participant device (10 m ) generates H m (K) = h K-km (g xm (IV m )) using a predetermined positive integer x m , said initial value IV m and one-way functions h and g, and sends said H m (K) as said aimed value information to said server device (20); said step (b) includes the steps wherein: (b-1) said server device (20) publishes said aimed value information A m on a bulletin board (25) accessible from all of said participant devices (10 1 -10 M ); (b-2) said server device (20) sends said index k to said each participant device (10 m ) to ask for its updated initial value; (b-3) said each participant device (10 m ) decides whether the received index k is the index k m corresponding to said aimed value; (b-4) if the result of decision in step (b-3) is k = k m , said each participant device (10 m ) generates and sends H m (k-1) = g xm-1 (IV m ) as said updated initial value to said server device (20); and (b-5) if the result of decision in said step (b-3) is not k = k m , said each participant device (10 m ) generates and sends H m (k-1) = h k-km-1 (g xm (IV m )) as said updated initial value to said server device (20); said step (c) includes the steps wherein said server device (20): (c-1) processes said updated initial value H m (k-1) with said one-way function h to generate h(H m (k-1) ); (c-2) decides whether said h(H m (k-1) ) is equal to said aimed value information H m (K) ; (c-3) if it is decided in said step (c-2) that they are equal, updates said aimed value information H m (K) with said updated initial value H m (k-1) , then decrements said index k by one and returns to said step (b-2); (c-4) if it is decided in said step (c-2) that they are not equal, processes said updated initial value H m (k-1) with said one-way function g to generate g(H m (k-1) ); and (c-5) decides whether said g(H m (k-1) ) matches said aimed value information H m (K) ; and said step (d) is a step wherein if a match is detected in said step (c-5), decides that the aimed value of said participant device (10 m ) corresponding to m and k having provided said match is the maximum or minimum. The method of claim 1, wherein: there is provided a conversion table showing the relationships between a sequence of values selectable as said aimed values and a sequence of indexes k of integral values respectively corresponding thereto; said step (a) includes the steps wherein: (a-1), letting the initial value of said each participant device (10 m ) be represented by IV m , where m = 1, 2, ..., M, M being an integer equal to or greater than 2, said each participant device (10 m ) generates H m (K) = h K-km (g xm (IV m )) using a predetermined positive integer x m , said initial value IV m and one-way functions h and g, and sends said H m as said aimed value information to said server device (20); and (a-2) said server device (20) publishes the received aimed value information H m (K) on a bulletin board (25) accessible from all of said participant devices (10 1 -10 M ); said step (b) includes the steps wherein: (b-1) for each value of said index k in descending order from K, said server device (20) inquires said each participant device (10 m ) about whether it has bid for said index k, and said each participant device (10 m ) responds YES or NO to said inquiry; (b-2) upon first detection of the response YES, said server device (20) requests said each participant device (10 m ) to send its updated initial value H m (k-1) ; (b-3) said each participant device (10 m ) decides whether its received k is the index k corresponding to said aimed value; (b-4) if the result of decision in said step (b-3) is k = k m , said each participant generates and sends H m (k-1) = g xm-1 (IV m ) as said updated initial value to said server device (20); and (b-5) if the result of decision in said step (b-3) is not k = k m , said each participant device (10 m ) generates and sends H m (k-1) = g xm-1 (IV m ) as said updated initial value to said server device (20); and said step (c) includes the steps wherein said server device (20): (c-1) for said updated initial value H m (k-1) received from said participant device (10 m ) having responded YES, generates H m = h K-km g(H m (k-1) ) and for said updated initial value received from said each participant device (10 m ) having responded NO, generates H m = h K+1-k g(H m (k-1) ) (c-2) checks whether all of said H m are equal to said aimed value information H m (K) published on said bulletin board (25); and (c-3) if it is decided in said step (c-2) that they are equal, determines that said participant device (10 m ) having responded YES is the winning participant device (10 m ), and publishes the current value of said index k as the index k of the aimed value of said winning participant device (10 m ). The method of claim 1, wherein: there is provided a conversion table showing the relationships between a sequence of values selectable as said aimed values and a sequence of indexes k of integral values respectively corresponding thereto, and letting the initial value of said each participant device (10 m ) be represented by IV m , where m = 1, 2, ..., M, M being an integer equal to or greater than 2; said step (a) is a step wherein said each participant device (10 m ) generates H m (K) = h K-km (g xm (IV m )) using a predetermined positive integer x m , said initial value IV m and one-way functions h and g, and sends said H m (K) as said aimed value information to said server device (20); said step (b) includes the steps wherein: (b-1) said server device (20) publishes said aimed value information H m (K) on a bulletin board (25) accessible from all of said participant devices (10 1 -10 M ); (b-2) said server device (20) sends said index k to said each participant device (10 m ) to ask for its updated initial value; (b-3) said each participant device (10 m ) decides whether the received index k is the index k m corresponding to said aimed value; (b-4) if the result of decision in step (b-3) is k = k m , said each participant device (10 m ) generates H m (k-1) = g xm-1 (IV m ) as said updated initial value and sends it to said server device (20) together with a flag indicating that said k and k m are equal; and (b-5) if the result of decision in said step (b-3) is not k = k m , said each participant device (10 m ) generates and sends H m (k-1) = h k-km-1 (g xm (IV m )) as said updated initial value to said server device (20); said step (c) includes the steps wherein said server device (20): (c-1) checks whether the received updated initial value H m (k-1) has said flag added; (c-2) if it is decided in said step (c-1) that said flag is added, processes said updated initial value H m (k-1) with said one-way function g to generate g(H m (k-1) ); (c-3) decides whether said g(H m (k-1) ) matches said aimed value information H m (K) ; (c-4) if it is decided in said step (c-1) that no flag is added, processes said updated initial value H m (k-1) with said one-way function h to generate h(H m (k-1) ); (c-5) decides whether said h(H m (k-1) ) matches said aimed value information H m (K) ; (c-6) if it is decided in said step (c-5) that they are equal, updates said aimed value information H m (K) with said initial value H m (k-1) , then decrements said index k by one and returns to said step (b-2); and (c-7) if it is decided in said step (c-5) that they are not equal, processes said initial value H m (k-1) with said one-way function g to generate g(H m (k-1) ) and returns to said step (c-3); and said step (d) is a step wherein if a match is detected in said step (c-3), decides that the aimed value of said participant device (10 m ) corresponding to m and k having provided said match is the maximum or minimum. The method of any one of claims 4, 12, 15, 16, 17, 18, and 19 wherein said sequences of indexes k and values selectable as said aimed values are both monotonous increasing values in the same direction, and in said step (c) said server device (20) determines the maximum aimed value. The method of any one of claims 4, 12, 15, 16, 17, 18, and 19, wherein said sequences of indexes k and values selectable as said aimed values are both monotonous increasing values in opposite directions, and in said step (c) said server device (20) determines the minimum aimed value. The method of any one of claims 4, 12, 15, 16, 17, 18, and 19, wherein letting an arbitrary aimed value be represented by PR = F(k)+Q, where F(k) is a value in said conversion table corresponding to said index k and Q is a fraction which is a positive integer which satisfies F(k+1)-F(k)>Q≥0; said step (a) includes a step wherein, letting said aimed value PR m of said each participant device m be represented by PR m = F(k m )+Q m , said each participant device m generates said aimed value information by applying said one-way function h a number of times corresponding to k m to said initial value, sends said aimed value information to said server device (20) together with said fraction Q m , where m = 1, 2, ..., M, M being an integer equal to or greater than 2; said step (b) includes a step wherein said server device (20) publishes said aimed value information and said fraction Q m on a bulletin board (25) accessible from all of said participant devices (10 1 -10 M ); said step (c) includes a step where said server device (20) makes a check for matching between said updated initial value and said aimed value information for each index value in an ascending or descending order of said fraction Q m where m = 1, 2, ..., M; and said step (d) includes a step wherein upon first detection of a match in said step (c), said server device (20) finishes said check and determines, from k m and m at the time of detecting the match, that PR m = F(k m )+Q m is said maximum or minimum aimed value. The method of any one of claims 1, 4, 12, 15, 16, 17, 18, and 19, further comprising the steps wherein: (0-1) said each participant device m sends its identifier ID m to a provisional identifier registration device; (0-2) said provisional identifier registration device issues a provisional identifier AID m for said identifier ID m , and stores said identifier ID m and said provisional identifier AID m in pair form in storage means; (0-3) said provisional identifier registration device sends said provisional identifier AID m to said each participant device (10 m ); and (0-4) said each participant device m sends said provisional identifier AID m as said identifier to said server device (20), together with said aimed value information. A quantitative competition system which comprises a server device (20) and M participant devices (10 1 -10 M ) each connected via a communication channel to said server device (20) and in which each participant device (10 m ) sends its aimed value to said server device (20) and said server device (20) determines which of said M participant devices (10 1 -10 M ) has offered the maximum or minimum one of said aimed values received, each of said participant devices (10 1 -10 M ) comprising: aimed value generating means for generating a secret aimed value PR m ; aimed value transforming means (14) provided with a conversion table memory (15) having stored therein a first conversion table which defines the relationships between a sequence of values selectable as said secret aimed value and a sequence of indexes corresponding thereto, for converting said secret aimed value PR m to the corresponding index k m , and for applying a predetermined one-way function repeatedly by a number of times corresponding to said secret aimed value to an initial value peculiar to the respective participant device (10 m ) to obtain aimed value information; and sending means for sending to said server device (20) said aimed value information and an identifier identifying the respective participant device (10 m ); and said server device (20) comprising: a conversion table memory (25A) which has stored therein a second conversion table which is the same as said first conversion table; a bulletin board (25) on which said server device (20) is adapted to write said aimed value information an said identifier received from each participant device (10 m ); updated initial value acquiring means adapted to acquire sequences of updated initial values, each sequence obtained by applying said one-way function repeatedly in correspondence with values of an index k, which are predetermined consecutive positive integers, to the initial value peculiar to a respective one of said participant devices (10 1 -10 M ); a counter (24) for updating said index k one by one; and control means (21) adapted to compare for each participant device (10 m ), upon each updating of said index k, the respective updated initial value with the respective aimed value information on said bulletin board (25) to check whether they match, and to determine m and k at the time of first detection of a match, 1≤m≤M. The system of claim 24, wherein said each participant device (10 m ) has initial value updating means for applying a one-way function h to said initial value to generate an updated initial value and for sending said updated initial value to said server device (20), and said updated initial value acquiring means of said server device (20) is adapted to receive said updated initial value from said each participant device (10 m ). The system of claim 24, wherein said updated initial value acquiring means of said server device (20) is adapted to apply a one-way function h to said initial value to generate said updated initial value. The system of claim 25 or 26, wherein said aimed value transforming means (14) of said each participant device (10 m ) comprises: a one-way function h processor (16) which processes the initial value IV m of the respective participant device (10 m ) with the one-way function h by the number of times corresponding to said index k m to obtain an output h km (IV m ); and a one-way function g processor (17) which processes said output from said one-way function h processor (16) with a one-way function g to obtain g(h km (IV m )) as said aimed value information. The system of claim 27, wherein said aimed value transforming means (14) of said each participant device (10 m ) includes verification information generating means for processing said initial value IV m with said one-way function h by K+1 times to generate C m = h K+1 (IV m ) as verification information and for sending said verification information to said server device (20); wherein said server device (20) is adapted to publish on said bulletin board (25) C m = h K+1 (IV m ) received from said each participant device (10 m ) in advance, said server device (20) further comprising a one-way function h processor (22) for processing a response D m with a one-way function h to generate h(D m ), and wherein said control means (21) is adapted to check whether C m = h(D m ) holds, and if not, to rewrite said C m with said D m and update said index k in said counter (24). The system of claim 24, wherein said each participant device (10 m ) comprises: a random generator (18) for generating a random number R m ; an operating device (19) for operating said random number R m and said aimed value PR m to obtain PR m (+)R m , where (+) represents a predetermined arbitrary operation; one-way function h processing means (19) for processing said PR m (+)R m with a one-way function h to obtain h(PR m (+)R m ); verification information generating means for processing said initial value IV m with said one-way function h K+1 times to generate C m = h K+1 (IV m ); and means for sending said h(PR m (+)R m ) and said verification information to said server device (20) together with said aimed value information; and said server device (20) comprises: means for publishing on said bulletin board (25) C m = h K+1 (IV m ), said aimed value information γ m = h k (IV m ) and h(PR m (+)R m ) received from said each participant device (10 m ), said PR m and said R m being an aimed value and a random number of said each participant device m; said control means (21) adapted to decide the maximum or minimum aimed value from said aimed values PR m and said random numbers R m received from said participant devices (10 1 -10 M ), and to determine the index k mx corresponding to said maximum or minimum aimed value and to request said each participant device (10 m ) to send (D m ) = h kmx (IV m ) corresponding to said index k mx ; said updated initial value acquiring means comprising a one-way function h processor (16) for processing D m with a one-way function h to generate h K+1-kmx (D m ), and a one-way function g processor (17) for processing D i with a one-way function g to generate g(D j ); and wherein said control means (21) is further adapted to check if said h K+1-kmx (D m ) matches C m on said bulletin board (25) and if said g(D j ) matches said γ m on said bulleting board. The system of claim 24, wherein said aimed value transforming means (14) of said each participant device (10 m ) comprises: a conversion table memory (15) which has stored therein a conversion table which defines indexes k = 1, 2, ..., K corresponding to K kinds of values selectable as aimed values, for converting said aimed value PR m to the corresponding index k m ; initial value updating means adapted to process, for each index k, an initial value with a one-way function h and to add to the processed initial value select information b m (k) for said index k to obtain an updated initial value and to repeat this processing until k = K is reached, thereby generating H m (K) ; a one-way function h processor (16) for outputting said updated initial value H m (k) from said initial value updating means as said aimed value information; select information generator (54) for generating said select information b m (k) in accordance with whether said aimed value corresponds to the respective index k from at least k m to K; a concatenator (55) for concatenating said H m (k-1) from said one-way function h processor (16) and said select information b m (k) to generate H m k = h(H m (k-1) ∥b m (k) ); a buffer (56) for temporarily holding the output from said concatenator and for outputting said output for the next value of said index k; a storage part (14S) for storing, for each value of said index k, H m (k) corresponding thereto; and a second one-way function h processor for processing H m (k) , obtained by repeating processing until k = K, with a one-way function h to generate h(H m (k) ), said h(H m (k) ) being output as said aimed value information; wherein said sending means is a means adapted to responding to a request of said server device (20) for said index k to read out H m (k) from said storage part and sending said H m (k) to said server device (20); and in said server device (20): there are published on said bulletin board (25), as said aimed value information, h(H m (k) ) obtained by processing, with a one-way function h, H m (k) generated by said each participant device (10 m ) which, upon each processing of said initial value with the one-way function h, added the processed value with select information b m (k) indicating whether said value was an aimed value for each value of said index k and repeated this processing from at least the index k m corresponding to said aimed value to the upper limit value K of said index k; said initial value updating means includes a one-way function h processor by which {H m (k-1) , b m } received from said each participant device (10 m ) in answer to an inquiry for said index k is processed with the one-y function h to generate H m (k) = h(H m (k-1) ∥ b m (k) ); and said server device (20) includes an updated initial value comparator for making a check to see if said H m (k) matches H m (k) in {H m (k) , b m (k+1) } previously received. The system of claim 24, wherein said aimed value transforming means (14) of said each participant device (10 m ) comprises: a conversion table memory which has stored therein a conversion table which defines indexes k = 1, 2, ..., K corresponding to K kinds of values selectable as aimed values, for converting said aimed value PR m to the corresponding index k m ; a one-way function g processor for processing said initial value IV m with a one-way function g by a predetermined number of times x m to generate g xm (IV m ); a one-way function h processor for processing said g xm (IV m ) with a one-way function h K-k m times to generate H m (K) = h K-km (g xm (IV m )) as said aimed value information; and response generating means adapted to respond to a request from said server device (20) for k to decide whether k=k m , and if true, to generate H m (k-1) = h k-km-1 (g xm (IV m )), and if false, to generate H m (k-1) = g xm-1 (IV m ); and wherein said sending means is adapted to send said H m (k-1) in response to said request from said server device (20) for said k; and said server device (20) comprises: means for publishing on said bulletin board (25) H m (K) = h K-km (g xm (IV m )) as said aimed value information received from said each participant device (10 m ), a one-way function h processor for processing H m (k-1) received from said each participant device (10 m ) as an answer to an inquiry for said k with the one-way function h to generate h(H m (k-1) ), and a one-way function g processor (17) for processing said answer H m (k-1) with a one-way function g to generate g(H m (k-1) ); wherein said control means (21) is adapted to check if said h(H m (k-1) ) matches said aimed value information H m (K) published on said bulletin board (25); if a match is detected, to update said aimed value information H m (K) with said H m (k-1) and to decrement said index k of said counter (24) by one; and if a mismatch is detected, to check if said g(H m (k-1) ) matches said aimed value information H m (K) ; and if a match is detected, to determine, based on k and m at that time, the maximum or minimum aimed value PR m and the participant device m having offered said value PR m . The system of any one of claims 24, 25, 26, 29, 30 and 31, wherein said aimed value transforming means (14) of said each participant device (10 m ) comprises: a conversion table memory (15) which has stored therein a conversion table which defines the relationships between a sequence of values selectable as said aimed values and a sequence of indexes k of integral values respectively corresponding thereto, for converting said aimed value PR m to the corresponding index k m ; a fraction calculating part (14A) for calculating said fraction Q m = PR m -F(k m ) based on F(k m ) obtained from said conversion table and said aimed value PR m , wherein an arbitrary aimed value PR is represented by PR = F(k)+Q, F(k) is a value in said conversion table corresponding to said k and Q is a fraction which is a positive integer which satisfies F(k+1)-F(k)>Q≥0; and a one-way function h processor (16) for processing said initial value IV m with said one-way function h by the number of times corresponding to said k m to generate said aimed value information; and wherein said sending means is adapted to send said fraction Q m to said server device (20) together with said aimed value information; and said server device (20) comprises: means for publishing on said bulletin board (25) said fraction Q m received from said each participant device (10 m ) together with said aimed value information; a sequencer (61) for deciding the sequence of said fractions Q m on said bulletin board (25); and a select information comparator for checking whether said select information b m (k) is equal to a value b indicating the selection of the aimed value corresponding to said index k in said decided sequence of fractions Q m . The system of any one of claims 24, 25, 26, 29, 30 and 31, wherein said sequence of index values k and said sequence of values selectable as said aimed values on said conversion table are monotone increasing values in the same direction, and said server device (20) is adapted to determine the maximum aimed value. The system of any one of claims 24, 25, 26, 29, 30 and 31, wherein said sequence of index values k and said sequence of values selectable as said aimed values on said conversion table are monotone increasing values in opposite directions, and said server device (20) is adapted to determine the minimum aimed value. A recording medium readable by a computer, the medium carrying a program of instructions executable by the computer to perform the function of the server device in the method according to any one of claims 1 to 23. A recording medium readable by a computer, the medium carrying a program of instructions executable by the computer to perform the function of a participant device in the method according to any one of claims 1 to 23.

Inventors: KOBAYASHI KUNIO, MORITA HIKARU, SUZUKI KOUTAROU, KOBAYASHI, KUNIO, MORITA, HIKARU, SUZUKI, KOUTAROU, Kobayashi, Kunio c/o Nippon Telegr.&amp; Teleph. Corp., Morita, Hikaru c/o Nippon Telegr.&amp; Teleph. Corp., Suzuki, Koutarou c/o Nippon Telegr.&amp; Teleph. Corp.
Applicants: NIPPON TELEGRAPH & TELEPHONE, NIPPON TELEGRAPH AND TELEPHONE CORPORATION
Application year: 2000
Main classification: G06F17
IPC codes: G06Q30