Abstract:
A shuffle with proof having a method for proof generating with small computational resources proportionate to the number of input encrypted messages and a corresponding method for verification. Shuffle is represented by a generalized transformation. Combining a proof that the transformation information is retained and a proof of a condition under which the transformation is met constitute the proof for shuffle. The two proofs are short proportional to the number of input encrypted messages. Transformation information retention is proved in such a manner that, since the response is generated from challenge value in dependency upon transformation, the condition under which the transformation is met is reflected in the response-challenge value relation. If the condition under which the transformation corresponding to the shuffle is selected as the condition for proof, the two proofs may constitute the proof for shuffle.

Description:
FIELD OF THE INVENTION 
   This invention relates to a technique for shuffle for guaranteeing the presence of one-to-one correspondence between input and output encrypted messages, such as is used in constructing an anonymous communication path, as the one-to-one correspondence is kept confidential, and to a technique of verifying the shuffle. 
   BACKGROUND OF THE INVENTION 
   Background Art (1) 
   As for the background art for shuffle with proof, reference is had to e.g., the JP Patent Kokai JP-A-08-263575 (publication 1).  FIG. 1  shows the structure described in this publication 1. Meanwhile, in the drawings of the present application, confluent arrows indicate that the information corresponding to the originating point of the arrows are all collected and sent to a location corresponding to the points of the respective arrows, whilst diverging arrows indicate that all or part of the information at the originating points of the arrows are sent to a location corresponding to the points of the arrows. On the other hand, broken lines indicate that these depend on the input message generating method used. 
   In  FIG. 1 , 160 pseudo output encrypted messages  103  represent commitment for zero-knowledge proving. Challenge values are generated from the input/output encrypted messages and the commitment, whilst the response (reply) represents designation of the mapping, responsive to bit values of the challenge values, from the input encrypted message or the output encrypted message, indicated by solid or arrows, to the pseudo output encrypted message. 
   Referring to  FIG. 1 , there is introduced a technique of permuting (re-arranging) plural ElGamal input cipher-texts  100  followed by re-encryption and for outputting the re-encrypted cipher-texts. This technique is termed “shuffle”. For guaranteeing that this processing is authentic, the above publication introduces the following technique: That is, secret random numbers for permuting and re-encryption are made to be different each time and an operation similar to the shuffle is repeated a number of times equal to the number of safe variables (about 160) to output pseudo output encrypted messages so as to be used as commitment for proving the authenticity. As challenge values  105 , Hash values of the commitments and the input/output encrypted messages are output. 
   The bit sequences of these challenge values are read sequentially from the upper side and designation of permutation (mapping representing the permutation) from the encrypted input message for the bit “0” and that from the encrypted output message for the bit “1” and the re-encryption (the random number used in re-encryption) is made into the response  106 . 
   The aforementioned commitment, challenge values and response are output as a proof text of the shuffling. The method for designating the relation of correspondence responsive to the bit values of the Hash values is termed a Cut and Choose method. 
   Background Art (2) 
   As another prior-art technique, reference is had to “A mix-network on permutation networks”, termed[Publication 2], publicized by Abe in Paper of Asiacrypt&#39; 99 (LNCS 1716 258–273 Springer 1999), herein termed the Publication 2. In this Publication 2, permutation of a pair of encrypted input message is repeated to realize the permutation of plural encrypted input messages, in their entirety, as shown for example in  FIG. 2 . In this Publication 2, permutation of a pair of encrypted input message is repeated to realize the permutation of plural encrypted input messages, in their entirety, as shown for example in  FIG. 2 . By constructing the proving of the permutations of the respective encrypted input messages by a method other than the cut-and-choose method, the shuffling with proof may be improved in efficiency when the number of the encrypted input messages is smaller than a preset number. That is, the sequence of the encrypted input messages is re-arranged (permuted) in its entirety by permutation of individual encrypted input messages. Although the proving of the individual permutations is efficient, it is necessary to provide a large number of permutations. 
   SUMMARY OF THE DISCLOSURE 
   The above-described background arts suffer from the following deficiencies: 
   In the background art (1), shuffling needs to be performed a number of times corresponding to the safety variable (about 160) for commitment generation. Each shuffling is in need of computation which consume large amount of computational resource involving modular exponentiation twice as many as the number of re-encrypted input messages. 
   On the other hand, verification is in need of computation which consume large amount of computational resource involving modular exponentiation twice as many as the number of re-encrypted input messages. 
   Moreover, in the background art (2), the commitment of permutation of a pair of encrypted input messages and its proof is in need of a sum total of 16 modular-exponentiation computations. 
   The computational resources per permutation is small as compared to the computational resources per two encrypted input messages of the background art (1) (=320), permutation of paired encrypted input messages is retained to be performed a number of times which enables permutation of any sort of the entire encrypted input messages, this number being n logn-n+1, where n is the number of encrypted input messages. 
   So, the computational resources is increased with the increasing number of the encrypted input messages. 
   It is therefore an object of the present invention to provide a method and a system in which the required computational resources for proving can be diminished without dependency on the number of encrypted input messages, and a program product. 
   It is another object of the present invention to provide a method and a system for reducing the required computational resources for verification as in the case of proving. Other objects, advantages and features of the present invention will be apparent from the entire disclosure including the following description. 
   According to a first aspect of the invention, there is provided a method for shuffle with proof in which an input message sequence which is comprised of encrypted messages and one or more public-keys, and shuffle information are input, and in which an encrypted output message sequence obtained by processing permutation of the encrypted messages and re-encryption by the public key or keys, and a shuffle proof text as a proof text for the processing, are output. 
   The method comprises: 
   (a) a transformation information retention commitment generating step of generating an output encrypted message sequence from an input message sequence and generating a commitment pertinent to retention of the transformation information from the input message sequence to the output encrypted message sequence, termed as “transformation information retention commitment”; 
   (b) a transformation condition commitment generating step of generating a commitment pertinent to a condition to be met by the transformation, termed as “transformation condition commitment”; and 
   (c) a response generating step of generating a response from the shuffle information and challenge value; 
   wherein 
   (d) the transformation information retention commitment, the transformation condition commitment and the response are output as the shuffle proof text; and 
   wherein 
   (e) the shuffle information includes the manner of permuting the input encrypted message, variables used for permuting and random numbers. 
   According to a second aspect of the invention, there is provided a shuffle verifying method in which an input message sequence, an output encrypted message sequence and a shuffle proof text are input, and a result of verification indicating acceptance or non-acceptance is output. 
   The method comprises: 
   (a) a transformation information retention verifying step of verifying the retention of the transformation information on transformation from an input message sequence to an output encrypted message sequence from the input message sequence, output encrypted message sequence, transformation information retention commitment pertinent to retention of the transformation information from the input message sequence to the output encrypted message sequence, a response and challenge value; and 
   (b) a transformation condition verifying step of verifying the condition to be met by transformation from the input message sequence to the output encrypted message sequence, by the transformation condition commitment pertinent to the condition to be met by the transformation, the response and the challenge value; wherein 
   (c) acceptance is output as the result of the shuffle verification if both the verification of the transformation information retention verifying step and the verification of the transformation condition verifying step are accepted, and non-acceptance is output otherwise. 
   According to a third aspect of the invention, there is provided an apparatus for shuffle with proof in which input message sequence, which is including a plurality of input encrypted messages and one or more public keys, and the shuffle information including the manner of permuting the input encrypted messages, variables used for re-encryption and random numbers is input, and an output encrypted message sequence obtained on permutation of the encrypted message and re-encryption by the public key and a shuffle proof text are output. 
   The apparatus comprises: 
   (a) a transformation information retention commitment generating unit for generating the output encrypted message sequences from the input message sequence and for generating a commitment pertinent to retention of the transformation information from the input message sequence to the output encrypted message sequences, termed as “transformation information retention commitment”; 
   (b) a transformation condition commitment generating unit for generating a commitment pertinent to a condition to be met by the transformation, termed as “transformation condition commitment”; and 
   (c) a response generating unit for generating a response from the shuffle information and challenge value; 
   wherein 
   (d) the transformation information retention commitment, the transformation condition commitment and the response are output as the shuffle proof text. 
   According to a fourth aspect, there is provided a shuffle verification apparatus which (a) receives inputs, and in which (b) the result of verification, i.e., acceptance or non-acceptance is output; 
   the inputs (a) comprising: 
   (a1) an input message sequence, made up of a plurality of encrypted messages and one or more public keys, input to a device for shuffle with proof, which is fed with the input message sequence and a shuffle information as input, and which outputs an encrypted output message sequence obtained on permutation of the encrypted messages and re-encryption by the public key or keys, and a shuffle proof text, 
   (a2) the output encrypted message sequence output from the device for shuffle with proof, and 
   (a3) a shuffle proof text output from the device for shuffle with proof, the shuffle proof text including the transformation information retention commitment pertinent to retention of the transformation information from the input message sequence to the output encrypted message, a transformation condition commitment pertinent to a condition to be met by the transformation, and the response. 
   The apparatus further comprises: 
   (c) a transformation information retention verifying unit for testifying retention of the transformation information on transformation from the input message sequence to the output encrypted message sequence based on the input message sequence, output encrypted message sequence, transformation information retention commitment, response and challenge value; and 
   (d) a transformation condition verifying unit for verifying the condition to be met by transformation from the input message sequence to the output encrypted message sequence based on the transformation condition commitment, the response and the challenge value; 
   wherein 
   (e) acceptance is output as the result of the shuffle verification if the verification by the transformation information retention verifying unit and the transformation condition verifying unit are both accepted and non-acceptance is output otherwise. 
   According to a fifth aspect of the present invention, there is provided an input message sequence generating method. The method generates an input message sequence, input to a device for shuffle with proof, in such a manner that a portion of the generated input message sequence is in the form of numerical values corresponding to the public key and the input encrypted message sequence transformed by the pseudo random numbers. According to the present invention, the input encrypted message sequence; public key and the pseudo random numbers may be combined into one input message sequence. 
   According to a sixth aspect, there is provided a machine readable program so formulated that a computer, as a shuffle apparatus, in which an input message sequence, which is including a plurality of input encrypted messages and one or more public keys, and the shuffle information, including the manner of permuting the input encrypted message, variables used for re-encryption and random numbers, are input, and in which an encrypted output message sequence obtained on permutation of said encrypted messages and re-encryption by said public key or keys, and a shuffle proof text, are output, is caused to perform the processing comprising: 
   (a) transformation information retention commitment generating processing of generating said output encrypted message sequences from said input message sequence and generating a commitment pertinent to retention of the transformation information from said input message sequence to said output encrypted message sequences, termed as “transformation information retention commitment”; 
   (b) transformation condition commitment generating processing of generating a commitment pertinent to a condition to be met by said transformation, termed as “transformation condition commitment”; and 
   (c) response generating processing of generating a response from said shuffle information and challenge value; and 
   (d) processing of outputting said transformation information retention commitment, transformation condition commitment and said response as said shuffle proof text. 
   According to a seventh aspect, there is provided a machine readable program so formulated that a computer, as a shuffle verifying apparatus, in which an input message sequence, an output encrypted message sequence output by a device for shuffle verifying with proof, the transformation information retention commitment, output from a device for shuffle with proof, pertinent to retention of the transformation information from said input message sequence to said output encrypted message sequence, a transformation condition commitment, pertinent to the condition to be met by said transformation, and a shuffle proof text including a response, are input, and a result of verification indicating acceptance or non-acceptance is output, to perform the processing comprising: 
   (a) transformation information retention verifying processing of verifying the retention of the transformation information from said input message sequence to said output encrypted message sequence from the input message sequence, output encrypted message sequence, transformation information retention commitment pertinent to retention of the transformation information from said input message sequence to said output encrypted message sequence, a response and challenge value; 
   (b) transformation condition verifying processing of verifying the condition to be met by transformation from said input message sequence to said output encrypted message sequence from the transformation condition commitment pertinent to the condition to be met by said transformation, said response and the challenge value; and (c) processing of outputting acceptance as the result of the shuffle verification if both the verification of the transformation information retention verifying processing and the verification of the transformation condition verifying processing are accepted, and of outputting non-acceptance if otherwise. 
   According to a eighth aspect, there is provided a method for generating a public key sequence with proof comprising: 
   generating a public key sequence having a pseudo random number sequence uniquely determined from a given input as generators, having a public key, corresponding to the same secret key, as generators, and 
   generating a proof text proving the correspondence to the same secret key; 
   wherein the generations of said public key sequence and the proof text are performed in cooperation by provers owning the secret key in a scattered fashion. 
   According to a ninth aspect, there is provided an apparatus for generating a public key sequence with proof wherein a public key sequence having a pseudo random number sequence uniquely determined from a given input as generators, corresponding to the same secret key and having the public key as the element, and a proof text proving the correspondence to the same secret key are generated in cooperation by provers owning the secret key in a scattered fashion. 
   According to a tenth aspect, there is provided a machine readable program for allowing a computer to perform the processings of: 
   generating a public key sequence having a pseudo random number sequence uniquely determined from a given input as generators, said public key sequence corresponding to the same secret key and having a public key as element, and generating a proof text proving the correspondence to the same secret key by cooperation of provers owning the secret key in a scattered fashion. 
   In the following, the basic concept of the invention will be explained. 
   According to the present invention, the proof that the shuffle is represented by a sort of more general transformation and the information on this transformation is retained, and the proof for the condition to be met by the transformation are combined together to constitute the proof for shuffle. 
   These two proofs are each simpler than the proof of the conventional shuffle such that the computational resources is diminished without dependency on the number of the input encrypted messages. This asset is not lost in the proof of the shuffle consisting in the combination of the two proofs. 
   The proof that the information on transformation is retained is acquired by generating a response from the challenge value, after generation of the output encrypted message sequence and the transformation information retention commitment, depending upon the aforementioned transformation and upon the random numbers used in generating the transformation information retention commitment. 
   Since the transformation is reflected on the relation between the response and the challenge value, the relation, in terms of equation(s), to be met, based on the condition met by the transformation, by the response and the challenge value exists without dependency on the challenge value. This relation (equation) is committed to prove the condition to be met by the transformation. 
   If the condition to be met by the transformation representing the shuffle is selected as the condition to be met by the transformation to be proved, the proof of the shuffle can be constituted by the two proofs. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows the structure of the prior-art technique 1. 
       FIG. 2  shows the structure of the prior-art technique 2. 
       FIG. 3  shows information input/output between the structure of a device for shuffle with proof and a shuffle verifying device in an Embodiment of the present invention. 
       FIG. 4  shows details of the device for shuffle with proof of Embodiment 1 of the present invention. 
       FIG. 5  shows details of the shuffle verifying device of Embodiment 1 of the present invention. 
       FIG. 6  shows details of the device for shuffle with proof of Embodiment 2 of the present invention. 
       FIG. 7  shows details of the shuffle verifying device of Embodiment 2 of the present invention. 
       FIG. 8  shows details of the device for shuffle with proof of Embodiment 3 of the present invention. 
       FIG. 9  shows details of the shuffle verifying device of the Embodiment 3 of the present invention. 
       FIG. 10  shows details of the device for shuffle with proof of Embodiment 4 of the present invention. 
       FIG. 11  shows details of the shuffle verifying device of the Embodiment 4 of the present invention. 
       FIG. 12  shows details of an input message sequence-generating device of Embodiment 5 of the present invention. 
       FIG. 13  shows details of an input message sequence-generating device of Embodiment 6 of the present invention. 
       FIG. 14  shows details of a pre-processing device in the Embodiment 6 of the present invention. 
       FIG. 15  shows details of an input message sequence-generating device of Embodiment 7 of the present invention. 
       FIG. 16  shows details of an input message sequence-generating device of the Embodiment 7 of the present invention. 
       FIG. 17  shows details of the device for shuffle with proof of the Embodiment 7 of the present invention. 
       FIG. 18  shows details of the shuffle verifying device of the Embodiment 7 of the present invention. 
       FIG. 19  shows details of a device for shuffle with proof in the Embodiments 6 and 7 of the present invention. 
       FIG. 20  shows details of a device for shuffle in the Embodiments 6 and 7 of the present invention. 
   

   PREFERRED EMBODIMENTS OF THE INVENTION 
   For clarifying the above and other objects, features and advantages of the present invention, preferred embodiments of the present invention are now explained in detail with reference to the drawings. 
   First, the matter, which forms the premises underlying the present invention, is explained. The encryption method used in the present invention is a method belonging to the public key crypt-system which also belongs to a probabilistic crypto-system, such as ElGamal crypto-system. 
   In the method for shuffle with proof, according to the present invention, a prover performing the shuffle with proof cannot falsify or disguise the proof message for shuffling unless all of the formulators of the encrypted input messages divulge the secret variables used for creating the encrypted input messages to the prover. By using the input message sequence generating method according to the present invention, in combination, it is similarly possible to prevent falsification or disguise of the proof message even if the formulators of the encrypted input messages would act in collusion with the prover. 
   The method for shuffle with proof, according to the present invention, is comprised of a transformation information retention (holding) commitment generating processing, for generating the transformation information retention commitment, a transformation condition commitment processing for generating the transformation condition commitment, and a response generating processing for generating the response and sub-response. The proof text (verifying text) is made up of the commitment generated by the above three sort of processings and the response (response and sub-response). 
   The method for shuffle and verification according to the present invention is comprised of a transformation information retention verification processing for verifying the retention of the transformation information from the input message sequence, encrypted output message sequence, transformation information retention commitment and the response, and a transformation condition verification processing for verifying the condition satisfied by the transformation from the transformation condition commitment, response and the sub-response. 
   [Transformation Information Retention Commitment Generating Processing] 
   The transformation information retention commitment generating processing, forming the method for shuffle with proving is now explained. 
   The transformation information retention commitment generating processing performs transformation corresponding to shuffle from the input message sequence to generate an encrypted output message sequence, while performing general transformation using random numbers to generate a transformation information retention commitment. 
   If any other component than the encrypted input message sequence and the public key is contained in the input message sequence, this component transformed in association with the shuffle is also regarded as the transformation information retention commitment. 
   If plural responses are to be generated, general transformation by different random numbers is executed a number of times to generate a number of sets of the transformation information retention commitments. 
   In this transformation, the output encrypted message sequence and the transformation information retention commitment can be generated as a representation of the variables and random numbers used for re-encryption and values associated with the permutation with respect to a basis comprised of the input message sequence. 
   This representation associates the basis with a represented value, and the method needs to be such as to render the computation of the representation from the basis and the value of representation difficult with respect to the computational resources. For this representation method, modular exponentiation may be used. 
   For example, let the encrypted input message sequence g[i, ┌]; i=1, . . . , n; ┌=0, . . . , l, 
   the public key being g[i, ┌]; i=n+1, . . . , n+m; ┌=0, . . . , l, 
   other components of the input message sequence being g[i, ┌]; i=1, . . . , n+m; ┌=l+1, . . . , l′, 
   random numbers associated with general transformation, referred to below as the information hiding factor, being A[μ, j]; μ=1, . . . , n+m, j=n+1, . . . , n+m′, 
   the variable for re-encryption being A[i, j]; i=n+1, . . . , n+m, j=1, . . . , n, 
   the variable for transformation corresponding to permutation being A[i, j]; i, j=1, . . . , n, and 
   output encrypted message sequence being g″[i, ┌]; i=1, . . . , n; ┌=1, . . . , l, 
   it is possible to generate an output encrypted message sequence g″[i, ┌]; i=1, . . . , n; ┌=1, . . . , l as 
   g″[i, ┌]=             j=1   n g[j, ┌] A[j, i]             j=n+1   n+m g[j, ┌] A[j, i] /F* p  i=1, . . . , n ┌=1, . . . , l,
   the transformation information retention commitment as g″[i, ┌]=             j=1   n g[j, ┌] A[j, i]             j=n+1   n+m g[j, ┌] A[j, i] /F* p  i=n+1, . . . , n+m′ ┌=1, . . . , l, and
   the transformation information retention commitment, in case g[i, ┌]; i=1, . . . , n+m; ┌=l+1, . . . , l′ is included in the input message sequence, as g″[i, ┌]=             j=1   n g[j, ┌] A[j, i]             j=n+1   n+m g[j, ┌] A[j, i] /F* p  i=1, . . . , n+m′ ┌=l+1, . . . , l′.
   The above can collectively be represented by g″[i, ┌]=             j=1   n+m g[j, ┌] A[j, i] /F* p  i=1, . . . , n+m′ ┌=1, . . . , l′.
   Here, g″[i, ┌]; i=1, . . . , n+m; ┌=1, . . . , l is termed as “output message sequence”, where g″[μ, ┌]; μ=1, . . . , n+m′; ┌=1, . . . , l′ is the represented value, A[μ, ν]; μ=1, . . . , n+m; ν=1, . . . , n+m′ is the representation and g[μ, ┌]; μ=1, . . . , n+m; ┌=1, . . . , l′ is the basis. 
   If plural sets of the transformation information retention commitments are to be generated depending on the number of the responses, plural different A[μ, j]; μ=1, . . . , n+m, j=n+1, . . . , n+m′ are provided and generated. 
   The fact that a prover is able to generate the transformation information retention commitment, input message sequence, and response corresponding to the output encrypted message sequence and challenge value, in such a manner as to satisfy the verification formulas, presents a proof that the knowledge of transformation from the input message sequence to the output encrypted message sequences is possessed. 
   [Transformation Condition Commitment Generating Processing] 
   The processing for generating the transformation condition commitments forming the shuffle method with proving is now explained. 
   The condition met by the transformation from the input message sequence to the output message sequence and the transformation information retention commitment is reflected on the relation between the response and the challenge value. So, there exists the relation (correlative equation) between the response and the challenge value, which holds without dependency on the challenge value. The transformation condition commitment is the commitment of this relation, which serves for representing the condition met by the transformation. 
   If plural responses are to be generated, the difference in the knowledge-hiding factor is reflected in the relation. For example, it is possible that this relation is determined as an identity as a polynomial of the responses and challenge values and the coefficients are committed. Alternatively, certain terms of the polynomial may be regarded as sub-responses, and coefficients of the sub-response may be committed to serve as transformation condition commitments. It is sufficient if a response and a sub-response are generated after determination of the challenge values. 
   The respective components of the response are polynomials of challenge values. The embodiments employ identities intending the relation that the square sums of certain terms of certain polynomials and square sums of certain components of the challenge values become equal to each other without dependency on (i.e., irrespective of) the challenge values, or identities intending the relation that the cubic sums of certain terms of certain polynomials and cubic sums of certain components of the challenge values become equal to each other without dependency on the challenge values. 
   The corresponding identities used in the embodiments are those which intend the relation
 
Σ i=1   n (Σ j=1   n   A[i, j]c[j] ) 2 =Σ i=1   n   c[i]   2   /F   q 
 
or the relation
 
Σ i=1   n (Σ j=1   n   A[i, j]c[j] ) 3 =Σ i=1   n   c[i]   3   /F   q 
 
using the challenge values c[i] and the response r[i].
 
   Meanwhile,
 
Σ j=1   n   A[i, j]c[j]/F   q   i= 1 , . . . , n 
 
is a portion of a polynomial
 
Σ j=1   n+m′   A[i, j]c[j]/F   q   i= 1 , . . . , n 
 
of the challenge value forming r[i].
 
   For example, these relations reflect the properties that A[i, j ]; i, j=0, . . . , n in the variables A[μ, ν]; μ=0, . . . , n+m; ν=0, . . . , n+m′ defining the transformation from the input message sequence to the output encrypted message sequences and to the transformation information retention commitment is an orthonormal matrix or a quasi-permutation matrix. 
   The “permutation matrix” is such a square matrix in each column and in each row of which only one nonzero element exists which is of a value of 1. A matrix, which is simultaneously an orthonormal matrix and a sub-permuted matrix, is a permutation matrix. 
   The “quasi-permutation matrix” is the above-mentioned permutation matrix, whose element equal to “1” is replaced by one of cubic roots of 1. It is also possible to replace the respective components by different cubic roots of 1. In such case, the transformation corresponding to the permutation matrix corresponds to the shuffle. That is, the transformation can be proved to be the shuffle by proving the condition met by the transformation by the transformation condition commitment generating processing. 
   Examples of the identities intending the above relation include
 
Σ i=1   n   r[i]r[i]+Σ   μ=1   n+m   ρ′[μ]r[μ]/F   q =Σ i=1   n   c[i]c[i]+Σ   μ=1   n+m′   φ[μ]c[μ]/F   q 
 
and
 
Σ i=1   n   r[i]r[i]r[i]+ρ″r′+Σ   μ=1   n+m   ρ′[μ]r[μ]/F   q =Σ i=1   n   r[i]r[i]r[i]+ρ″ (λ[0]+Σ i=1   n   λ[i]r[i]r[i] )+Σ μ=1   n+m   ρ′[μ]r[μ]/F   q =Σ i=1   n   c[i]c[i]c[i]+Σ   i=1   n   ψ[i]c[i]c[i]+Σ   μ=1   n+m′   φ[μ]c[μ]/F   q .
 
   Here, the coefficients of the identity ρ″, ρ′[i], φ[μ], ψ[i] need to be determined so that the relation corresponding to the conditions to be met by the transformation. 
   There are also occasions wherein sub-equation coefficients λ[μ]; μ=0, . . . , n are committed, with a portion of the identity
 
 r′=λ[ 0]+Σ i=1   n   λ[i]r[i]r[i]/F   q 
 
as a sub-response.
 
   As transformation condition commitments, coefficients of identities or those coefficients partly or entirely committed, and sub-equation coefficients or these coefficients partly or entirely committed, are generated, are generated. In an embodiment, a portion of an identity is committed to
 
v, v φ[0] /F* p 
 
for example, and the sub-equation coefficients are committed to
 
 u, u   λ[μ]   /F*   p  μ=0 , . . . , n 
 
   Committing the coefficients of the identity and using the sub-response are effective for diminishing the information for a verifier to identify the shuffle from the response and the commitment. 
   [Response Generating Processing] 
   The response generating processing of constructing the shuffle method with proving is hereinafter explained. 
   In the response generating processing, the transformation information retention commitment, transformation condition commitment, an input message sequence and an output encrypted message sequences are input to a challenge value generating function (unit) to acquire a challenge value. 
   It is noted that the “challenge value generating function” is such a function in which it is computationally difficult to find input from an output or to determine input with the relation among different output components in mind. This assures that a challenge value has been generated after determination of the input, commitment and the output, without taking the intention of the prover into account. 
   If the challenge value generating function is not used, the challenge value is acquired by arbitrary selection by a verifier after the input, output and the commitment have been shown. 
   From the challenge value, the response or the sub-response, reflecting the shuffle method and the information-hiding factor is generated. 
   If plural responses and sub-responses are generated, the respective responses need to reflect different information hiding factors. 
   For example, it suffices to generate an response such that the value having represented by the challenge value with respect to the basis comprising of the output encrypted message sequences and the transformation information retention commitment will be equal to the value having represented by the response value with respect to the basis comprising of the input message sequence. 
   For example, the response r[μ]; μ=1, n+m is generated such as
 
 r[μ]=Σ   ν=1   n+m′   A[μ, ν]c[ν]/F   q μ=1 , . . . , n+m 
 
with sub-response,
 
 r′=λ[ 0]+Σ i=1   n   λ[i]r[i]r[i]/F   q ,
 
using the challenge value c[μ]; μ=1, . . . , n+m′.
 
[Transformation Information Retention Verification Processing]
 
   The transformation information retention verification processing, forming the shuffle verification method, is hereinafter explained. 
   It is verified that the relation among the input message sequence, output encrypted message sequences and the transformation information retention commitment is reflected by the relation between the response and the challenge value. For example, it is confirmed that there exists the relationship between the response and the challenge value such that a represented value represented by the challenge value with respect to the basis of the output encrypted message sequences and the transformation information retention commitment is equal to a represented value represented by the response with respect to the basis of an input message sequence. 
   For example, it is confirmed that the challenge value c[i]; i=1, . . . , n+m′ and the response r[i]; i=1, . . . , n+m satisfy the relation:
 
             i=1   n+m′   g″[i, ┌]   c[i] =           i=1   n+m   g[i, ┌]   r[i]   /F*   p ┌=1 , . . . , l′. 

   The same value of the challenge value as that used in formulating a proof message is used. This is possible because, in using a challenge value generating function, an input to the challenge value generating function exists in the proof message, input message sequence and the output encrypted message sequence. 
   [Transformation Condition Verification Processing] 
   The transformation condition verification processing, forming the shuffle verification method, is now explained. 
   From the transformation condition commitment, it is verified that the challenge value and the response meet the relation reflecting the condition met by the transformation. 
   For example, the response and the challenge value or the response, challenge value and the sub-response is substituted into an identity connoting the condition to be met by the transformation to confirm that the identity holds. In case where there is a sub-response, the authenticity of the sub-response is also confirmed based on the committed response, sub-response and sub-equation coefficients. 
   For coefficients, e.g., ρ″, ρ′[μ], φ[μ], ψ[i], as the transformation condition commitment, the challenge value c[i]; i=1, . . . , n+m′ and the response r[i]; i=1, . . . , n+m are confirmed from the fact that the identity
 
Σ i=1   n   r[i]r[i]+Σ   μ=1   n+m   ρ′[μ]r[μ]=Σ   i=1   n   c[i]c[i]+Σ   μ=1   n+m′   φ[μ]c[μ]/F   q 
 
or the identity
 
Σ i=1   n   r[i]r[i]r[i]+ρ″r′+Σ   μ=1   n+m   ρ′[μ]r[μ]/F   q =Σ i=1   n   r[i]r[i]r[i]+ρ″ (λ[0]+Σ i=1   n   λ[i]r[i]r[i] )+Σ μ=1   n+m   ρ′[μ]r[μ]/F   q =Σ i=1   n   c[i]c[i]c[i]+Σ   i=1   n   ψ[i]c[i]c[i]+Σ   μ=1   n+m′   φ[μ]c[μ]/F   q 
 
hold, while the authenticity of the sub-response is confirmed from the fact that the equation of verification
 
 u   r′   =u [0]             i=1   n   u[i]   r[l]r[l]   /F*   p 
 
holds.

   If the coefficients of the identity are partially committed, it is confirmed that, instead,
 
 v^{Σ   i=1   n   r[i]r[i]+Σ   μ=1   n+m   ρ′[μ]r[μ]}/F*   p =v^{Σ i=1   n   c[i]c[i]+Σ   μ=1   n+m′   φ[μ]c[μ]}/F*   p 
 
or
 
 v^{Σ   i=1   n   r[i]r[i]r[i]+ρ″r′+Σ   μ=1   n+m   ρ′[μ]r[μ]}/F*   p   =v^{Σ   i=1   n   r[i]r[i]r[i]+ρ″ (λ[0]+Σ i=1   n   λ[i]r[i]r[i] )+Σ μ=1   n+m   ρ′[μ]r[μ]}/F*   p   =v^{Σ   i=1   n   c[i]c[i]c[i]+Σ   i=1   n   ψ[i]c[i]c[i]+Σ   μ=1   n+m′   φ[μ]c[μ]}/F*   p 
 
holds. In the above equations, [^] denotes exponential processing.
 
[Input Message Sequence Generating Method]
 
   In the shuffle method with proving, according to the present invention, the transformation from the input message sequence to the output encrypted message sequences and the transformation information retention commitment needs to be reflected in the relation between the response and the challenge value. To this end, the response that can be generated given a challenge value needs to be limited. However, if the prover knows the generating information of the input encrypted message, there is a risk that this limitation be violated. The method to obstruct this risk is the input message sequence (string) generating method. 
   The input message sequence generating method according to the present invention generates pseudo random numbers to transform the input message sequence, or the pseudo random number is added to the input message sequence to generate an input message sequence which cannot be determined even by the formulator of the input encrypted message. 
   [Input Message Sequence Generating Method (1)] 
   Pseudo random numbers are generated and added to the encrypted input message sequence and to the public key to serve as an encrypted input message sequence. The pseudo random numbers are determined from a preset input to assure reproducibility. 
   For example, if the encrypted input message sequence is g[i, ┌]; i=1, . . . , n; ┌=0, . . . , l and the public key is g[i, ┌]; i=n+1, . . . , n+m; ┌=0, . . . , l, pseudo random numbers of (n+m) ×(l′−l), where l′−l≧1, are generated from the preset input such that
 
 g[i, ┌]; i= 1 , . . . , n+m; ┌=l+ 1 , . . . , l′ 
 
whilst the input message sequence is set to
 
 g[i, ┌]i= 1 , . . . , n+m; ┌= 1 , . . . , l′. 
 
[Input Message Sequence Generating Method (2)]
 
   The respective encrypted messages, forming an encrypted input message sequence, and the public key, are re-encrypted by respective public keys forming a public key sequence generated from the input message sequence and the public key as inputs, and are combined together to form an input message sequence. 
   The “public key sequence” are prepared by uniquely generating a number of pseudo random numbers from an input which is the same as the number of the public keys forming the public key sequence so that the any of the random numbers represents certain element of the respective public keys. 
   For example, if the public key sequence is g′[i, ┌]; i=1, . . . , n+m; ┌=1, . . . , l, the input encrypted message sequence is η[i, ┌]; i=1, . . . , n; ┌=0, . . . , l, and the public key is η[i, ┌]; i=n+1, . . . , n+m; ┌=0, . . . , l, an input message sequence g[i, ┌]; i=1, . . . , n+m; ┌=l+1, . . . , l is represented by g[i, ┌]=η[i, ┌]g′[i, ┌] s[l] /F* p  using an optional positive integer s[i]; i=1, . . . , n+m which is apparent for a verifier. As s[i], e.g., s[n+m]=0, s[j]=1; j=1, . . . , n+m−1 is selected. 
   [Input Message Sequence Generating Method (3)] 
   Each input plain message (text) is encrypted using each associated public key forming a public key sequence, and proof is made of the fact that this public key has been used for encryption. 
   The encrypted message, which has received this proof, and the public key, are combined to an input message sequence. 
   If, for example, the public key sequence is g′[i, ┌]; i=1, . . . , n+m; ┌=0, 1, and the plain text is m[i]; i=1, . . . , n; ┌=0, 1, the input encrypted message
 
 η[i,  0 ]=g′[i,  0] s[i]   /F*   p   i= 1 , . . . , n 
 
 η[i,  1 ]=m[i]g′[i,  1] s[l]   /F*   p   i= 1 , . . . n 
 
is generated, at the same time as the knowledge of s[i] such that
 
 η[i,  0 ]=g′[i,  0] s[l]   /F*   p 
 
is proved to give a proof message encrypted using g′[i, 0].
 
   From the encrypted message for which the proof error message is verified, the input message sequence is made into
 
 g[i, ┌]=η[i, ┌]i= 1 , . . . , n; ┌= 0, 1
 
 g[i, ┌]=g′[i, ┌]i=n+ 1 , . . . , n+m; ┌= 0, 1.
 
[Method for Generating Public Key Sequence with Proof]
 
   From a given input, a pseudo random number sequence is uniquely generated, and plural public keys which includes values created by a given procedure from respective random numbers as components and which have the same secret key are generated in a plurality of numbers in association with the respective random numbers. Simultaneously, a proof message that all the public keys have the same secret key is produced. 
   If the secret key is owned discretely by plural persons, each person prepares the public key sequence with each secret key and combines them together to generate a public key sequence. 
   For example, a pseudo random number generator Hash (*) is accorded and an output is prepared from an input *. An output is fed to input. This process is repeated to generate a pseudo random number recursively. A number of public keys g′[i, ┌]; i=1, . . . , n+m; ┌=0, . . , l, having, as generators, each value of a number sequence g′[i, 0]; i=1, . . . , n+m made up of n+m generators obtained on removing 0 and 1 from a number sequence resulting from raising respective generators of the number sequence to the k&#39;th power, and having the same secret key, are generated in association with the respective random numbers. 
   If the secret key is x[┌]; ┌=1, . . . , l, the public key sequence may be represented by
 
 g′[i,  0 ]=g′[i,  0]
 
 g′[i, ┌]=g′[i,  0] x[┌]   /F*   p   i= 1 , . . . , n+m; ┌= 1 , . . . , l. 
 
   A proof message that the above public key sequence has correctly been generated is generated. 
   If the secret key is owned in scattered state, each person creates a public key sequence corresponding to the discrete secret key, and the respective public key sequences are finally combined together to create a public key sequence associated with the secret key. 
   DETAILED DESCRIPTION OF THE EMBODIMENTS 
   Referring to the drawings, the present invention is explained with reference to Examples employing the ElGamal cipher-texts. In the drawings, abbreviations are used. For example, “retention commit” is the transformation information retention commitment, “condition commit” is the transformation condition commitment, “identity commit” is the commitment of the identity coefficients, “sub-response commit” is the commitment of the coefficients of the sub-response, “retention processing” is the processing for generating the transformation information retention commitment, “condition processing” is the processing for generating the transformation condition commitment, “response processing” is the processing for generating the response, “retention verification processing” is the transformation information retention verification processing, and “condition verification processing” is the transformation condition verification processing. 
     FIG. 3  shows input/output in the embodiment of the present invention for the shuffle device and a shuffle verification device. 
   In a preferred embodiment of the present invention, shown in  FIG. 3 , an input message sequence  300 , made up of plural input message sequences  322  and the public key  323 , a shuffle matrix  304 , made up of a shuffle matrix  307 , determining the permuting method, a re-encryption secret random number  305 , as a variable for re-encryption, and an information hiding factor  306  as random numbers for generating the transformation information retention commitment, and a shuffle information  303 , comprehending element (generator) coefficients  308  as seeds of coefficients of the identity, quasi-equation coefficients  309  as coefficients of the equation determining sub-response  319  as part of the identity and various constants for generating the transformation condition commitment made up of coefficients basis  310  for committing the coefficients, are input to a shuffle device with proof  312 , and an output encrypted message sequence  313  and a shuffle proof message  314  are issued as output. 
   The shuffle proof message  314  comprehends a transformation condition commitment  316 , including coefficients of the identity, committed coefficients of the identity and committed coefficients of the sub-response, a transformation information retention commitment  315 , a response  317  and a sub-response  318 . 
   The input message sequence  300 , output encrypted message sequence  313  and the shuffle proof message  314  are input to a shuffle verification device  319 , which then outputs the result of verification  322  in the form of acceptance or non-acceptance. 
   The shuffle device with proof is unable to falsify the shuffle proof message if and only if the prover is unaware of the input encrypted message generating information. The method added for inhibiting this falsification under any condition is the input message sequence generating method. Three sorts of the input message sequence generating methods are hereinafter explained along with the method for generating the public key sequence with proof used in two of these three input message sequence-generating methods. 
   In the following, the matter to be premised as a presupposition in common for the shuffle method with proof, an input message sequence generating method and an individual public key sequence generating method with proof, is explained in order. 
   First, the ElGamal domain parameters are explained. 
   These variables are two prime numbers (generators) p, q satisfying the relation
 
 p=kq+ 1
 
where k is an integer.
 
[Challenge Value Generating Function and Basis Generating Function]
 
   The challenge value generating function and basis generating function are explained. These are
 
Hash[μ; μ=0 , . . . , n] (*), Hash′[μ; μ=0 , . . . , n] (*).
 
   The Greek letter μ, as suffix of each function, is a value from 0 to μ. If an argument[*] is input, (n+1) element vector is output. 
   An output of the challenge value generating function is (n+1) integers other than 1 and 0 not larger than q, whilst an output of the basis generating function is (n+1) integers other than 1 and 0 not larger than p, and is an integer which is the generator (element) of F* p  of orders q (generator of the sub-group whose order being q of the multiplication group of orders p−1). 
   These functions are those for which the argument cannot be determined by number-theoretically intending the relation between input and output and between different components of the output. 
   As an illustrative method for constructing the basis generating function, one Hash function Hash (*) outputting |p| bits is provided to compute
 
Hash (*)
 
with the computed result being input to the argument of the Hash function to derive the computed results. This operation is repeated to recursively generate the number sequence h[0], h[1], h[2], . . . to find number sequence h[0] k , h[1] k , h[2] k , . . . by raising each numerical value to the k&#39;th power. Among these, (n+1) generators other than 1, 0 are sequentially selected.
 
   As for the challenge value generating function, a number sequence is found using the Hash function outputting |q| bits and, among the generators of this sequence, those which are other than 1, 0 are selected. In this case, the operation of raising the values to the k&#39;th power is unnecessary. 
   [Public Key] 
   The public key is explained. The public key is two values η[0, 0], η[0, 1], with η[0, 0] being an generator of F* p  having the number of order of q. As for the η[0, 1], it is computed using a secret key x by
 
η[0, 1]=η[0, 0] x   /F*   p .
 
[Input Encrypted Message]
 
   The input encrypted message is explained. The plain message is selected from generators of the F* p  not more P, whose order equal to q, and is termed M. Using a secret random number r, generated by a pseudo random number generator, the input encrypted message is computed as being
 
(η[0, 0] r , M η[0, 1] r )/F* p .
 
[Re-encryption]
 
   The re-encryption is explained. Given the ElGamal cipher-texts text (η[0, 0] r , M η[0, 1] r )/F* p , an optional random number s is selected and transform is carried out such that
 
(η[0, 0] r   , Mη[ 0, 1] r )→(η[0, 0] r η[0, 0] s   , Mη[ 0, 1] r η[0, 1] s ) /F* p =(η[0, 0] r+s   , Mη[ 0, 1] r+s )/ F*   p .
 
   This processing is called “re-encryption”. The above transform can be executed without knowing the value of r. The decoded result of the cipher-texts text, re-encoded by this transformation, remains unchanged. The random number s at this time is called “re-encryption secret random number”. 
   [Permutation Matrix] 
   The permutation matrix is explained. In the permutation matrix, there exists only one non-zero component in any row or column and assumes the value of 1, provided that it is on Fq in the preferred embodiment. The following is given as an example.
 
0 0 0 1 0
 
1 0 0 0 0
 
0 0 0 0 1
 
0 0 1 0 0 /F q .
 
[Quasi-permutation Matrix]
 
   The quasi-permutation matrix is hereinafter explained. The [quasi-permutation matrix] is defined as being ones resulting from permutation of one of the permutation matrix by one of three cubic roots of 1 on F* p . These being w, w 2 , 1, an example of the quasi-permutation matrix is given as follows:
 
0 0 0 w 2  0
 
w 0 0 0 0
 
0 w 2  0 0 0
 
0 0 0 0 1
 
0 0 w 0 0 /F q .
 
[Shuffle]
 
   The shuffle is explained. The input encrypted message sequence η[i, 0], η[i, 1]; i=1, . . . , n is shuffled in sequence to generate an encrypted message sequence η ′[i, 0], η′[i, 1]; i=1, . . . , n. Then, using n secret random number s[i]: i=1, . . . , n and public keys η[0, 0] and η[0, 1], an output encrypted message sequence g″[i, ┌]; i=1, . . . , n, ┌=0, 1 are computed by
 
 g″[i, ┌]=η′[i, ┌]η[ 0, ┌] s[i]   /F*   p   i= 1 , . . . , n, ┌= 0, 1.
 
   This is an output result of the shuffle, and is termed [output encrypted message sequence]. 
   [Shuffle matrix] 
   The shuffle matrix is explained. The [shuffle matrix] is a n+1 row by n+1 column matrix with the generators A[μ, ν]; μ, ν=0, . . . , n being such that
 
A[μ, ν]=
 
 A[i, j]i, j= 1 , . . . , n  “permutation matrix”  307 
 
 A[ 0 , j]∈   R   j= 1 , . . . , n  re-encryption secret random number  305 
 
 A[i,  0]∈ R   i= 1 , . . . , n  information hiding factor  306 
 
A[0, 0]∈R information hiding factor  306 
 
[Shuffle Matrix Transformation]
 
   The shuffle matrix transformation is explained. This acts on the input message sequence g[μ, ┌] in the following manner to output an output message sequence g″[μ, ┌].
 
 g″[μ, ┌]=               ν=0   n   g[ν, ┌]   A[ν, μ]   /F*   p μ=0 , . . . , n, ┌= 0, 1.

   If the “shuffle matrix” is a permutation matrix, the output encrypted message sequence is g″[i, 0], g″[i, 1]; i=1, . . . , n, and expanded,
 
 g″[j,  0 ]=g[i,  0]η[0, 0] A[0, j]   /F*   p 
 
 g″[j,  1 ]=g[i,  1]η[0, 1] A[0, j]   /F*   p 
 
is obtained for a permutation (i, j|π(i)=j). This represents an output of the shuffle.
 
   If the “permutation matrix” is a quasi-permutation matrix,
 
 g″[j,  0 ]=g[i,  0] W[l] η[0, 0] A[0, j]   /F*   p 
 
 g″[j,  1 ]=g[i,  1] W[i] η[0, 1] A[0, j]   /F*   p 
 
is output as a result of quasi-shuffle (the quasi-shuffle is defined as giving shuffle on raising each output encrypted message to the first, wth or to the w 2 th power). Here, w[i]; i=1, . . . , n assumes any one of cubic roots of 1 on Fq.
 
   EMBODIMENT 1 
   Referring to  FIGS. 4 and 5 , the shuffle method with proof, and a verification method, according to Embodiment 1 embodying the Present invention, are explained. Meanwhile, ┌ is assumed to be 0, 1. 
   As the shuffle information  401 , the permutation matrix  402 , coefficient basis  404  and the generator coefficient  403  are prepared as follows: 
   As for the permutation matrix  402 , numbers of 1 to n are arrayed in order. A Pseudo random number generator, not shown, is used n times to generate n sequences of numbers and an ith number of each number sequence is divided by n−i+1 to find a remainder which is set to π′(i). 
   It is noted that i is in the order from 1 to n, with the π′(i) th number counted from the lower side of the number sequence being set to π(i). The operation of removing this number from the number sequence is executed to determine π (i); i=1, . . . , n. The ith row of the shuffle matrix is 1 only for the component of the π(i) column with the remaining values being 0. In this manner, the permutation matrix is generated. 
   The components of the shuffle matrix other than the permutation matrix are generated as follows: First, 2n+1 numbers on F q  are generated by the pseudo random number generator and allocated to A[i, 0], A[0, j], A[0, 0]; i, j=1, . . . , n. The above numbers are combined to a shuffle matrix. 
   As for the generation of the coefficient basis  404  v, generator coefficient  403  r′[0], numbers on F q  other than 1, 0 are generated by a pseudo random number generator and set as r′[0]. By the pseudo random number generator, generators of /F* p  are generated and are raised to the power of k on F* p  to select numbers other than 1, 0, to generate generators of F* p  having the number of orders q. These generators are set to v. 
   From
 
 r′[ 0]∈ R   F   q , ≠0, 1
 
 v∈   R   F*   p , ≠1, s.t.  v   q =1 /F*   p 
 
and from the input encrypted message sequence η[i, 0], η [i, 1]; i=1, . . . , n, and the public key η[0,0], η[0, 1], an input message sequence  400 
 
 g[μ┌]; μ= 0 , . . . , n; ┌= 0, 1 is set to:
 
 g[ 0, ┌]=η[0, ┌]┌=0, 1
 
 g[i, ┌]=η[i, ┌]/F*   p   i= 1 , . . . , n, ┌= 0, 1.
 
   In the following, the shuffle method with proof is used. 
   By a shuffle matrix operation  405  in the transformation information retention commitment generating processing  419 , the shuffle matrix  402  is caused to act on the input message sequence  400  in the following manner to generate an output message sequence  406  g″[μ, ┌]; μ=0, . . . , n; ┌=0, 1, by
 
 g″[μ, ┌]=               ν=0   n   g[ν, ┌]   A[ν, μ]   /F*   p μ=0 , . . . , n, ┌= 0, 1.

   It is noted that g″[i, ┌]; i=1, . . . , n; ┌=0, 1 is an output encrypted message sequence, and g″[0, ┌]; ┌=0, 1 is the transformation information retention commitment  408 . 
   By the identity coefficient calculation  409  in the transformation information retention commitment generating processing  420 , an identity coefficients  410  φ[μ], r′[0] is generated, using the generator (element) coefficient  403  r′[0] and a shuffle matrix  402 , to
 
 r′[ 0 ]=r′[ 0]
 
φ[0]=Σ j=1   n   A[j,  0 ]A[j,  0 ]+r′[ 0 ]A [0, 0 ]/F   q 
 
 φ[i]= 2Σ j=1   n   A[j,  0 ]A[j, i]+r′[ 0 ]A[ 0 , i]/F   q   i= 1 , . . . , n. 
 
   Using the coefficient basis  404  v, the identity coefficient  410  r′[0], φ[0] are committed to
 
 v   1   =v   r′[0]   /F*   p 
 
 ω=v   φ[0]   /F*   p 
 
   by the hiding processing  411 . φ[i], . . . might be hidden as v^φ[i], . . . also. 
   By the above, φ[i], ω, v 1 , v constitute the transformation condition commitment  412 . 
   Here, the commitment  40  A is the transformation information retention commitment  408  and the transformation condition commitment  412 . 
   By the response generating processing  421 , the aforementioned input message sequence  400 , output encrypted message sequence  417  and the commitment  409  are arguments of the challenge value generating function  413  to generate a challenge value  414  as
 
 c[ 0]=1,
 
 c[i]= Hash[i]( g[ν,  0 ], g[ν,  1 ], g″[ν,  0 ], g″[ν,  1 ], v, φ[ν], ω, v′; ν= 0 , . . . , n ) i= 1 , . . . , n 
 
   from which a response  416  is generated at  415  as
 
 r[μ]=Σ   ν=0   n   A[μ, ν]c[ν]/F   q μ=0 , . . . , n 
 
   using shuffle matrix  02 . 
   The above commitment  40  A and the response  416  are output as a shuffle proof  418  to output an output encrypted message sequence  417  as a result of the shuffle. 
   The verifying method is explained with reference to  FIG. 5 . 
   By the shuffle verifying method, 
   an input message sequence  400  g[μ, ┌]; μ=0, . . . , n; ┌=0, 1, 
   an output encrypted message sequence  417  g″[i, ┌]; i=1, . . . , n; ┌=0, 1, 
   a transformation information retention commitment  408  g″[0, ┌]; ┌=0, 1 which is a commitment  409  in the shuffle proof message  418 , and 
   a transformation condition commitment  412  φ[ν], ω, v′, v; ν=0, . . . , n; ┌=0, 1 are substituted into a challenge value generating function  500  to generate a challenge value  501  as
 
 c[ 0]=1
 
 c[i ]=Hash[ i] ( g[ν, ┌], g″[ν, ┌], φ[ν], ω, v′, v; ν= 0 , . . . , n; ┌= 0, 1) i= 1 , . . . , n. 
 
   By the transformation information retention verifying processing  505 , it is verified  502  that the verifying equation
 
             μ=0   n   g[μ, ┌]   r[μ] =           μ=0   n   g″[μ, ┌]   c[μ]   /F*   p ┌=0, 1
 
holds using this challenge value  501 , input message sequence  400 , transformation information retention commitment  408 , an output message sequence  406  which is the output encrypted message sequence  417 , and the response  416 .

   By the transformation condition verifying processing  506 , it is verified  503  that, using the challenge value  501 , response  416  and the transformation condition commitment  412 , the verifying equation
 
 v′   r[0]   v^{Σ   i=1   n   r[i]r[i]}=ωv^{Σ   i=1   n ( c[i]c[i]+φ[i]c[i] )} /F*   p 
 
holds.
 
   If the above verifying equations hold in their entirety, the proof message is accepted  504 . 
   The above-described shuffle method with proof has the effect of assuring that the shuffle matrix transformation for the input message sequence has been carried out by a shuffle matrix at least having the “permutation matrix” belonging to the orthonormal matrix. 
   On the input encrypted message and on the output encrypted message, there are imposed limitations, so that, if this effect is able to assure the authenticity of the shuffle, the preferred embodiment is able to construct the shuffle with proof. 
   It is assumed, for example, that the input encrypted message has been proved to have been selected from a limited number of candidates, and that these candidates cannot be expressed using others as basis. If, after shuffle and decoding of the input encrypted message, any (or each) decoded message has been selected from correct candidates, it may be said from the proof message of the present embodiment that the shuffle is authenticated. 
   Meanwhile, the processing and the function of the transformation information retention commitment  419 , transformation condition commitment processing  420  and the response generating processing  421  are realized by a program executed on a computer. The transformation information retention verifying processing  505 , transformation condition verifying processing  506  in  FIG. 5  are realized by the program executed on a computer. In this case, the present invention can be executed by loading the program on a main memory of the computer from the recording medium which has recorded the program, such as CD-ROM, DVD (digital versatile disc), floppy disk medium, a hard disk medium, magnetic tape medium or a semiconductor memory etc. 
   EMBODIMENT 2 
   Referring to  FIGS. 6 and 7 , the shuffle method with proof and the verifying method therefor according to Embodiment (2) of the present invention are explained. In the following, it is assumed that ┌=0, 1. 
   As the shuffle information  601 , the shuffle matrix  602 , element coefficients  603 , coefficient basis  604 ,  605  and sub-equation coefficient  606  are prepared as follows: 
   As for the shuffle matrix  602 , it is generated in the same way as in the Embodiment (1) described above. 
   As for the element coefficient  603  ρ′, ρ″, coefficient basis  604  v, coefficient basis  605  v, sub-equation coefficient  606 , λ[μ]; μ=0, . . . , n, a number other than 1, 0 on F q  is generated for ρ′, ρ″λ[μ]; μ=0, . . . , n. while an element of F p  of an order number q is generated for the coefficient basis u, v:
 
ρ′∈ R   F   q , ≠0, 1
 
ρ″∈ R   F   q , ≠0, 1
 
 v∈   R   F*   p , ≠1 , s.t. v   q =1 /F*   p 
 
λ[μ]∈ R   F   q , ≠0, 1, μ=0 , . . . , n 
 
 u∈   R   F*   p ≠1 , s.t.u   q =1 /F*   p 
 
   From the input encrypted message sequence and from the public key, an input message sequence  600  g[μ┌]; μ=0, . . . , n; ┌=0, 1 are generated in the same way as in Embodiment (1). 
   In the following, the shuffle method with proof is used. 
   The transformation information retention commitment processing  623  is performed, as in Embodiment (1), to generate an output message sequence  603  g″[μ, ┌]; μ=0, . . . , n; ┌=0, 1, where g″[i, ┌]; i=1, . . . , n; ┌=0, 1 is an output encrypted message sequence  604 , and g″[0, ┌]; ┌=0, 1 is a transformation information retention commitment  605 . 
   By the identity coefficient computation in the transformation condition commitment generating processing  625 , the element coefficient  603  ρ′, ρ″ and the identity coefficient  607  ψ[i], φ[i], φ[0], ρ′, ρ″; i=1, . . . , n are generated as
 
ρ′=ρ′
 
ρ″=ρ″
 
 ψ[i]=Σ   j=1   n (3 A[j,  0 ]+ρ″λ[j] ) A[j, i]/F   q   i= 1 , . . . , n 
 
 φ[i]=Σ   j=1   n (3 A[j,  0 ]A[j,  0 ]A[j, i]+ 2 ρ″λ[j]A[j,  0 ]A[j, i] )+ρ′ A[ 0 , i]/F   q   i= 1 , . . . n 
 
φ[0]=Σ j=1   n ( A[j,  0 ]A[j,  0 ]A[j,  0 ]+ρ″λ[j]A[j,  0] A[j,  0])+ρ″λ[0 ]+ρ′A[ 0, 0 ]/F   q 
 
using the element coefficients  603  ρ′, ρ″ and the shuffle matrix  602 .
 
   Moreover, using the coefficient basis  604  v. the identity coefficient  607  ρ′, ρ″, φ[0] is committed  609  to
 
 ω=v   φ[0]   /F*   p 
 
 v″=v   ρ″   /F*   p 
 
 v′=v   ρ′   /F*   p 
 
by the hiding processing  608 . φ[i], . . . might be hidden as v^ φ[i], . . . also. In addition, using the coefficient basis  605  u, the quasi-element coefficients  606  λ[μ]; μ=0, . . . , n, are committed  612  to
 
 u[ 0 ]=u   λ[0]   /F*   p 
 
 u[i]=u   λ[i]   /F*   p   i= 1 , . . . , n. 
 
   From the foregoing, ψ[i], φ[i], ω, v″, v′, v, u, u[0], u[i]; i=1, . . . , n, are the transformation condition commitment  613 . 
   Here, the commitment  614  is set to transformation information retention commitment  605  and to transformation condition commitment  613 . 
   By the response generating processing  624 , the above input message sequence  600 , output encrypted message sequence  604  and the commitment  614  are set as an argument of the challenge value generating function  615  to generate a challenge value  616  as
 
 c[ 0]=1
 
 c[i ]=Hash[ i ]( g[ν, ┌], g″[ν, ┌], u, u[ν], v, φ[j], ψ[j], ω, v′, v″; ┌= 0, 1, 2; ν=0 , . . . , n; j= 1 , . . . , n ) i= 1 , . . . , n 
 
and, from this challenge value  616 , the response  618  is generated  617  as
 
 r[μ]=Σ   ν=0   n   A[μ, ν]c[ν]/F   q μ=0 , . . . , n 
 
using th shuffle matrix  602 .
 
   Moreover, from the sub-equation coefficient  606  λ[μ]; μ=0, . . . , n, and from the response  618 , the sub-response  620  is generated  619  as
 
 r′=λ[ 0]+Σ i=1   n   λ[i]r[i]r[i]/F   q .
 
   The commitment  614 , response  618  and the sub-response  620  are output as the shuffle proof message  622  and, as a result of the shuffle, an output encrypted message sequence  604  is output. The verifying method is explained hereinafter with reference to  FIGS. 6 and 7 . 
   By the shuffle verifying method, the input message sequence  600  g[μ, ┌]; μ=0, . . . , n; ┌=0, 1, the output encrypted message sequence  604  g″[i, ┌]; i=1, . . . , n; ┌=0, 1, the transformation information retention commitment  605  as the commitment  614  in the shuffle proof message (text)  622  g″[0, ┌]; ┌=0, 1 and the transformation condition commitment  609 ,  912  ψ[i], φ[i], ω, v″, v′, v, u, u[0], u[i]; i=1, . . . , n are substituted into the challenge value generating function  704  to generate the challenge value  705  as
 
 c[ 0]=1
 
 c[i ]=Hash [ i ]( g[ν, ┌], g″[ν, ┌], u, u[ν], v, φ[j], ψ[j], ω, v′, v″; ┌= 0, 1, 2; ν=0 , . . . , n; j= 1 , . . . , n ) i= 1 , . . . , n. 
 
   By using this challenge value  705 , input message sequence  600 , transformation information retention commitment  605 , output message sequence  603  as the output encrypted message sequence  604  and the response  618 , it is verified  706 , by the transformation information retention commitment processing  710 , that the verification equation
 
             μ=0   n   g[μ, ┌]   r[μ] =           μ=0   n   g″[μ, ┌]   c[μ]   /F*   p ┌=0, 1
 
holds.

   Using the challenge value  705 , response  618  and the transformation condition commitments  609 ,  612 , it is verified  708  from the transformation condition verifying processing  711  that the verifying equation
 
 v″   r′   v′   r[0]   v^{Σ   i=1   n   r[i]r[i]r[i]}=ωv^{Σ   i=1   n ( c[i]c[i]c[i]+ψ[i]c[i]c[i]+φ[i]c[i] )}/ F*   p 
 
and the verifying equation  707 
 
 u   r′ =             i=1   n   u[i]   r[i]r[i]   /F*   p 
 
hold.

   If all of the above verifying equations hold, the proof message is accented  709 . 
   The above-described shuffle method with proof has the effect of assuring that the shuffle matrix transformation for the input message sequence has been carried out by the shuffle matrix having a “permutation matrix” at least belonging to the quasi-permutation matrix. At this time, the possibility that the output encrypted message sequence g″[i ┌]; i=1, . . . , n; ┌=0, 1 has output
 
 g″[j,  0 ]=g[i,  0] w[i]   g[ 0, 0] A[0, j]   /F*   p 
 
 g″[j,  1 ]=g[i,  1] w[i]   g[ 0, 1] A[0, j]   /F*   p 
 
cannot be excluded. If w[i] is all 1, the shuffle holds. If w[i]; i=1, . . . , n assumes one of the cubic roots of 1 on F q .
 
   If the degree of freedom equal to the cubic root power of 1 on F q  is allowed as a decoded message, or if the symbol as set on the plain text is entered to cancel the degree of freedom of the cubic root power, the shuffle with proof can be established by this embodiment. 
   Meanwhile, the processing and the function of the transformation information retention commitment processing  623 , the transformation condition commitment generating processing  625  and the response generating processing  624  of the shuffle device with proof are realized by a program executed on the computer. Also, the processing and the function of the transformation information retention verifying processing  710  and the transformation condition verifying processing  711  of the shuffle device with proof are realized by a program executed on the computer. In this case the present invention can be executed by loading the program to the main memory of the computer from a recording medium having recorded the program, such as a CD-ROM, DVD (digital versatile disc), floppy disc, magnetic tape medium or a semiconductor memory, and by executing the so-loaded program. 
   EMBODIMENT 3 
   As an Embodiment (3) of the present invention, the shuffle method with proof and the corresponding verifying method are explained with reference to  FIGS. 8 and 9 . It is assumed that ┌=0, 1, and that there are two sets of the public keys, namely η[−1, ┌], η[0, ┌]; ┌=0, 1, both having the same secret key. 
   As the shuffle information  801 , the shuffle matrix  802 , element coefficients  803 ,  805 , coefficient basis  804 ,  806  and the sub-equation coefficient  807  are prepared as follows: 
   The shuffle matrix, used in the preferred embodiment, differs in size from those of the Embodiment (1) and (2), and is a n+2 rows by n+1 column matrix. 
   The permutation matrix, constituting this shuffle matrix  802 , is A[i, j]; i, j=1, . . . , n, with the re-encryption secret random number being 2×n components of A[−1, j], A[0, j]; j=1, . . . , n, with the knowledge hiding factor being n+2 components of A[μ, 0]; μ=−1, . . . , n. These components are generated in a similar manner to Embodiment (1). 
   As for the element function  803  r′[−1], r′[0], element coefficients  805  ρ, ρ′, ρ″, coefficient basis  804  v, coefficient basis  806  u, sub-equation coefficient  807  λ[μ]; μ=0, . . . ,n, a number on F q  other than 1, 0 is generated for r′[−1], r′[0], ρ, ρ′, ρ″, λ[μ]; μ=0, . . . , n, whilst an element of F* p  of the number of orders q is generated for the coefficient basis u, v, by the technique similar to that of Embodiment (1).
 
 r′[ −1]∈ R   F   q , ≠0, 1
 
 r′[ 0], ∈ R   F   q , ≠0, 1
 
ρ∈ R   F   q , ≠0, 1
 
ρ′∈ R   F   q , ≠0, 1
 
ρ″∈ R   F   q , ≠0, 1
 
 v∈   R   F*   p , ≠0, 1 , s.t. v   q =1 /F*   p 
 
 λ[μ]∈   R   F   q , ≠0, 1μ=0 , . . . , n 
 
 u∈   R   F*   p , ≠0, 1 , s.t. u   q =1 /F*   p 
 
   From the input message sequence η[i, 0], η[i, 1]; i=1, . . . , n and the public key η[−1, ┌], η[0, ┌]; ┌=0, 1, the input message sequence  800  g[μ┌]; μ=−1, . . . , n; ┌=0, 1 is set to
 
 g[ −1, ┌]=η[−1, ┌]┌=0, 1
 
 g[ 0, ┌]=η[0, ┌]┌=0, 1
 
 g[i, ┌]=η[i, ┌]/F*   p   i= 1 , . . . , n, ┌= 0, 1
 
   In the following, the shuffle method with proof is used. 
   By the shuffle matrix operation in the transformation information retention commitment generating processing  832 , the shuffle matrix  802  is made to act on the input message sequence  800  as now explained to generate an output message sequence  809  g″[μ, ┌]; μ=0, . . . , n; ┌=0, 1 as
 
 g″[μ, ┌]=               ν=−1   n   g[ν, ┌]   A[ν, μ]   /F*   p μ=0 , . . . , n, ┌= 0, 1
 
where g″[i, ┌]; i=1, . . . , n; ┌=0, 1 is the output encrypted message sequences  810 , and g″[0, ┌]; ┌=0, 1 is the transformation information retention commitment  811 .

   By the identity coefficient computation  812 ,  816  in the transformation condition commitment generating processing  833 ,  834 , the identity coefficients  817  ψ[i], φ[i], φ[0], ρ, ρ′, ρ″; i=1, . . . , n and the identity coefficients  813  φ [ν], r′[0], r′[−1]; ν=0, . . . ,n are computed  818 ,  812 , using the element coefficients  803 ,  805  r′[−1], r′[0], ρ, ρ′, ρ″ and the shuffle matrix  802 :
 
ρ=ρ
 
ρ′=ρ′
 
ρ″=ρ″
 
 ψ[i]=Σ   j=1   n (3 A[j,  0 ]+ρ″λ[j] ) A[j, i]/F   q   i= 1 , . . . , n 
 
 φ[i]=Σ   j=1   n (3 A[j,  0 ]A[j,  0 ]A[j, i]+ 2 ρ″λ[j]A[j,  0 ]A[j, i ])+ρ′ A[ 0 , i]+ρA[− 1 , i]/F   q   i= 1 ,. . . , n 
 
φ[0]=Σ j=1   n ( A[j,  0 ]A[j,  0 ]A[j,  0 ]+ρ″λ[j]A[j,  0 ]A[j,  0])+ρ″λ[0 ]+ρ′A[ 0, 0 ]+ρA[− 1, 0 ]/F   q 
 
 r′[− 1 ]=r′[− 1]
 
 r′[ 0 ]=r′[ 0]
 
Φ[0]=Σ j=1   n   A[j,  0 ]A[j,  0 ]+r′[ 0 ]A[ 0, 0 ]+r′[− 1 ]A[− 1, 0 ]/F   q 
 
 Φ[i]= 2Σ j=1   n   A[j,  0 ]A[j, i]+r′[ 0 ]A[ 0 , i]+r′[− 1 ]A[− 1 , i]/F   q   i= 1 , . . . , n 
 
   Moreover, using the coefficient basis  804  v, the identity coefficients  813 ,  817  r′[−1], r′[0], Φ[0], φ[0], ρ, ρ′, ρ″ are committed  819 , by the hiding Processing  814 ,  818 , to
 
 ω=v   φ[0]   /F*   p 
 
 v″=v   ρ″   /F*   p 
 
 v′=v   ρ′   /F*   p 
 
 ω′=v   ρ   /F*   p 
 
   Φ[i], . . . φ[i], . . . might be hidden as v^ φ[i], . . . v^ Φ[i], . . . also and committed  815  to
 
 V=v   r′[−1]   /F*   p 
 
 V′=v   r′[0]   /F*   p 
 
 Ω=v   φ[0]   /F*   p .
 
   Moreover, using the coefficient basis  806  u, the sub-equation coefficient  807  λ[μ]; μ=0, . . . , n is committed  821 ,  820  to
 
 u[ 0 ]=u   λ[0]   /F*   p 
 
 u[i]=u   λ[i]   /F*   p   i= 1 , . . . n. 
 
   From the foregoing, Φ[i], V′, V, Ω, ψ[i], φ[i], ω, v″, v′, ω′, v, u, u[0], u[i]; i=1, . . . , n is to be the transformation condition commitment  822 . 
   The commitment  823  is to be the transformation information retention commitment  811  and the transformation condition commitment  822 . 
   By the response generating processing  835 , with the input message sequence  800 , output encrypted message sequences  810  and the commitment  823  as the argument of the challenge value generating function  824 , the challenge value  825  is generated as
 
 c[ 0]=1
 
 c[i ]=Hash [ i ]( g[μ, ┌], g″[ν, ┌], u[ν], u, φ[j], ψ[j], ω, ω′, v′, v″, v, Φ[j], Ω, V′, V; μ=− 1 , . . . , n; ν= 0 , . . . , n; j= 1 , . . . n; ┌= 0, 1, 2)  i= 1 , . . . , n 
 
and, from this challenge value  825 , the response  827  is generated  826  as
 
 r[μ]=Σ   ν=0   n   A[μ, ν]c[ν]/F   q μ=1 , . . . , n 
 
using the shuffle matrix  802 .
 
   By the sub-equation coefficient  807  λ[μ]; μ=0, . . . , n and by the response  827 , the sub-response  829  is generated  828  as
 
 r′=λ[ 0]+Σ i=1   n   λ[i]r[i]r[i]/F   q .
 
   The above commitment  823 , response  827  and the sub-response  829  are output as the shuffle Proof message  831  and an output encrypted message sequences  810  is output as the result of the shuffle. 
   The verifying method is now explained with reference to  FIG. 9 . 
   By the shuffle verifying method, the input message sequence  800  g[μ, ┌]; μ=−1, . . . , n; ┌=0, 1 an output encrypted message sequence  810  g″[i, ┌]; i=1, . . . , n; ┌=0, 1 the transformation information retention commitment  811  g″[0, ┌]; ┌=0, 1 of the commitment  823  in the shuffle proof message  831  and the transformation condition commitments  815 ,  819 ,  821  Φ[i], V′, V, Ω, ψ[i], φ[i], ω, v″, v′, ω′, v, u, u[0], u[i]; i=1, . . . , n are substituted into a challenge value generating function  900  to generate the challenge value  901  as
 
 c[ 0]=1
 
 c[i ]=Hash [ i ]( g[μ, ┌], g″[ν, ┌], u[ν], u, φ[j], ψ[j], ω, ω′, v′, v″, v, Φ[j], Ω, V′, V; μ=− 1 , . . . , n; ν= 0 , . . . , n; j= 1 , . . . , n; ┌= 0, 1, 2) i= 1 , . . . n. 
 
   By the transformation information retention verifying processing  907 , it is verified  902  that the verifying equation
 
             μ=−1   n   g[μ, ┌]   r[μ] =           μ=0   n   g″[μ, ┌]   c[u]   /F*   p  ┌=0, 1
 
holds, by employing the challenge value  901 , using the input message sequence  800 , transformation information retention commitment  811 , an output message sequence  809  as an output encrypted message sequences  810 , and the response  827 .

   By the transformation condition verifying processing  908 ,  909 , it is verified  904  that the verifying equation (identity)
 
 v″   r′   v′   r[0] ω′ r[−1]   v^{Σ   i=1   n   r[i]r[i]r[i]}=ωv^{Σ   i=1   n ( c[i]c[i]c[i]+ψ[i]c[i]c[i]+φ[i]c[i] )}/ F*   p 
 
holds, while it is verified  905  that the verifying equation
 
 u   r′   =u[ 0]             i=1   n   u[i]   r[i]r[i]   /F*   p 
 
holds, and also it is verified  903  that the verifying equation
 
 V′   r[0]   V   r[−1]   v^{Σ   i=1   n   r[i]r[i]}=Ωv^{Σ   i=1   n ( c[i]c[i]+Φ[i]c[i ])}/ F*   p 
 
holds, using the challenge value  901 , response  827  sub-response  829  and the transformation condition commitments  815 ,  819  and  821 .

   If all of the above verifying equations hold, the proof text is accepted. 
   The above-described shuffle method with proof has the effect of assuring that the shuffle matrix transformation for the input message sequence has been carried out by the “permutation matrix”, at least having the shuffle matrix belonging to the permutation matrix. This means that the shuffle has been carried out, with the present embodiment being the shuffle with proof. 
   Meanwhile, the processing and the function of the transformation information retention commitment processing  832 , the transformation condition commitment generating processing  833 ,  834  and the response generating processing  835  of the shuffle device with proof are realized by a program executed on the computer. Also, the processing and the function of the transformation information retention verifying processing  907  and the transformation condition verifying processing  908 ,  909  of the shuffle device with proof are realized by a program executed on the computer. In this case the present invention can be executed by loading the program to the main memory of the computer from a recording medium having recorded the program, such as a CD-ROM, DVD (digital versatile disc), floppy disc, magnetic tape medium or a semiconductor memory, and by executing the so-loaded program. 
   EMBODIMENT 4 
   Referring to  FIGS. 10 and 11 , the shuffle method with proof and the corresponding verifying method of Embodiment (4) of the present invention are now explained. In the following, it is assumed that ┌=0, 1, while the public key is one set of η[0, ┌]; ┌=0, 1. 
   As the shuffle information  1006 , the shuffle matrix  1001 , a second information-hiding factor  1004 , element coefficients  1002 ,  1005 , coefficient basis  1003 ,  1008  and sub-equation coefficients  1007  are prepared as follows: 
   The shuffle matrix  1001  is generated as in Embodiment (1) described above and is represented by  A [μ, ν]; μ, ν=0, . . . , n. 
   The second information hiding factor  1004  A[ν, 0]; ν=0, . . . , n is generated in a similar manner. 
   As for the element coefficient  1005  ρ′, ρ″, element coefficient  1002  r′[0], coefficient basis  1003  v, coefficient basis  1008  u, and a sub-equation coefficient  1007  λ[μ]; μ=0, . . . , n, a number on F q  other than 1, 0 is generated for r′[0], ρ′, ρ″, λ[μ]; μ=0, . . . , n and an element on F p  of a number of orders of q is generated for the coefficient basis u, v.
 
ρ′∈ R   F   q , ≠0, 1
 
ρ″∈ R   F   q , ≠0, 1
 
 r′[ 0], ∈ R   F   q , ≠0, 1
 
 v∈   R   F*   p , ≠0, 1 , s.t. v   q =1 /F*   p 
 
λ[μ]∈ R   F   q , ≠0, 1μ=0 , . . . , n 
 
 u∈   R   F*   p , ≠0, 1 , s.t. u   q =1 /F*   p 
 
   From the input encrypted message sequence η[i, 0], η[i, 1]; i=1, . . . , n and the public key η[0, ┌]; ┌=0, 1, the input message sequence  1000  g[μ┌]; μ=0, . . . , n; ┌=0, 1 is represented by
 
 g[ 0, ┌]=η[0, ┌]┌0, 1
 
 g[i, ┌]=η[i, ┌]i= 1 , . . . , n, ┌= 0, 1.
 
   In the following, the shuffle method with proof is used. 
   By the shuffle matrix operation  1009  in the transformation information retention commitment generating processing  1042 , the shuffle matrix  1001  is made to act on the input message sequence  1000  in the following manner to generate an output message sequence  1010  g″[μ, ┌]; μ=0, . . . , n; ┌=0, 1 as
 
 g″[μ, ┌]=               ν=0   n   g[ν, ┌]   A[ν, μ]   /F*   p μ=0 , . . . , n, ┌= 0, 1.

   Here, g″[i, ┌]; i=1, . . . , n; ┌=0, 1 and the output message sequence  1011 , g″[0, ┌]; ┌=0, 1 are set to the first transformation information retention commitment  1012 . 
   By the second transformation information retention commitment generating processing  1044 , selection is made  1018  from the input message sequence  1000  to represent the second input message sequence  1019  as g[μ, ┌′]. Here, ┌′=0. 
   The second transformation information retention commitment  1021  G″[0, ┌′] is generated  1020  as
 
 G″[ 0, ┌′]=             ν=0   n   g[ν, ┌′]   B[ν, 0]   /F*   p ┌=0or 1.

   By the identity coefficient calculations  1022  in the transformation condition commitment generating processing  1045 , the identity coefficients  1023  ψ[i], φ[i], φ[0], ρ′, ρ″; i=1, . . . , n is generated, using the element coefficient  1005  ρ′, ρ″ and the shuffle matrix  1001 , by
 
ρ′=ρ′
 
ρ″=ρ″
 
 ψ[i]=Σ   j=1   n (3 A[j,  0 ]+ρ″λ[j ]) A[j, i]/F   q   i= 1 , . . . , n 
 
 φ[i]=Σ   j=1   n (3 A[j,  0 ]A[j,  0 ]A[j, i]+ 2 ρ″λ[j]A[j,  0 ]A[j, i ])+ρ′ A[ 0 , i]/F   q   i= 1 ,. . . , n 
 
φ[0]=Σ j=1   n ( A[j,  0 ]A[j,  0 ]A[j,  0 ]+ρ″λ[j]A[j,  0 ]A[j,  0])+ρ″λ[0 ]+ρ′A[ 0, 0 ]/F   q .
 
   Using the coefficient basis  1003  v, the identity coefficients  1023  φ[0], ρ′, ρ″ is committed  1025 , by the hiding processing  1024 , by
 
 ω=v   φ[0]   /F*   p 
 
 v″=v   ρ″   /F*   p 
 
 v′=v   ρ′   /F*   p .
 
   φ[i], . . . might be hidden as v^ φ[i], . . . also. Moreover, using the coefficient basis  1008  u, the sub-equation coefficients  1007  λ[μ]; μ=0, . . . , n are committed  1027  by
 
 u[ 0 ]=u   λ[0]   /F*   p 
 
 u[i]=u   λ[i]   /F*   p   i= 1 , . . . , n. 
 
   By the identity coefficient calculation  1013  in the transformation condition commitment generating processing  1043 , and using the element coefficient  1002 r′[0], shuffle matrix  1001  and the second information hiding factor  1004 , the identity coefficients  1014  Φ[ν], r′[0]; ν=0, . . . , n are generated as
 
 r′[ 0 ]=r′[ 0]
 
Φ[0]=Σ j=1   n   B[j,  0 ]B[j,  0 ]+r′[ 0 ]B[ 0, 0 ]/F   q 
 
 Φ[i]= 2Σ j=1   n   B[j,  0 ]A[j, i]+r′[ 0 ]A[ 0 , i]/F   q   i= 1 , . . . , n. 
 
   Also, using the coefficient basis  1003  v, and by the hiding processing  1015 , the identity coefficients  1014  r′[0], Φ[0] are committed  1016  by
 
 V′=v   r′[0]   /F*   p 
 
 Ω=v   Φ[0]   /F*   p .
 
   Φ[i], . . . might be hidden as v^Φ2[i], . . . also. By the above, the first transformation condition commitment  1028  is expressed to ψ[i], Φ[i], ω, v″, v′, v, u, u[0], u[i]; i=1, . . . , n. The second transformation condition commitment  1016  is represented by Φ[i], V′, Ω, v; i=1, . . . n. 
   The first commitment  1017  is represented by the first transformation information retention commitment  1012  and the fist transformation condition commitment  1028 , whilst the second commitment  1029  is represented as the second transformation information retention commitment  1021  and the second transformation condition commitment  1016 . 
   By the response generating processing  1046 , the first challenge value  1031  is generated as
 
 c[ 0]=1
 
 c[i ]=Hash [ i ]( g[ν, ┌], g″[ν, ┌], u[ν], u, φ[j], ψ[j], ω, v′, v″, v; ν= 0 , . . . , n; j= 1 , . . . , n; ┌= 0, 1) i= 1 , . . . , n, 
 
with the above input message sequence  1000 , output encrypted message sequence  1011  and with the first commitment  1017  as an argument of a challenge value generating function  1030 . From this challenge value  1031 , and using the shuffle matrix  1001 , the first response  1033  is generated  1033  is generated  1032  as
 
 r[μ]=Σ   ν=0   n   A[μ, ν]c[ν]/F   q μ=0 , . . . , n. 
 
   The, the sub-response  1039  is generated  1038  as
 
 r′=λ[ 0]+Σ i=1   n   λ[i]r[i]r[i]/F   q 
 
from the sub-equation coefficient  1007  λ[μ]; μ=0, . . . n and the response  1033 .
 
   By the response generating processing  1047 , the second challenge value  1035  is generated as
 
 C[ 0]=1
 
 C[i ]=Hash[ i ]( g[ν, ┌′], G″[ 0 , ┌′], g″[j, ┌′], Φ[j], Ω, V′; ν= 0 , . . . , n; j= 1 , . . . , n; ┌′= 0) i= 1 , . . . , n 
 
with the second input message sequence  1019 , output encrypted message sequence  1011  and the second commitment  1029  and with the second challenge value  1035  as an argument of the challenge value generating function  1034 . From this challenge value  1035 , and using the shuffle matrix  1001  and the second information hiding factor  1004 , the second response  1037  is generated  1036  as
 
 R[μ]=B[μ,  0]+Σ i=1   n   A[μ, i]C[i]/F   q μ=0 , . . . n. 
 
   The aforementioned commitments  1017  and  1029 , the responses  1033 ,  1037  and the sub-response  1039  are output as shuffle proof  1040  to output an output encrypted message sequence  1011  as the result of the shuffle. 
   The verifying method is explained with reference to  FIG. 11 . 
   By the shuffle verifying method, the input message sequence  1000 , output encrypted message sequence  1011  and the first commitments  1012 ,  1025  and  1027  of the shuffle proof message  1040  are substituted into the challenge value generating function  1100  to generate a first challenge value  1101  as
 
 c[ 0]=1
 
 c[i ]=Hash[i](input message sequence, output encrypted message sequence and first commitment),  i= 1 , . . . , n. 
 
   Then, a second input message sequence  1019 , second commitments  1016 ,  1021  of the shuffle processing message  1040  and the output encrypted message sequence  1011  are substituted into the challenge value generating function  1108  to generate a second challenge value  1109  as
 
 C[ 0]=1
 
 C[i ]=Hash[i](second input message sequence, output encrypted message sequence and second commitment)  i= 1 , . . . , n. 
 
   By the transformation information retention verifying processing  1112 , it is verified  1103  that the verifying equation
 
             μ=0   n   g[μ, ┌]   r[μ]             μ=0   n   g″[μ, ┌]   c[μ]   /F*   p ┌=0, 1
 
holds, using the first challenge value  1101 , input message sequence  1000 , first transformation information retention commitment  1012 , output encrypted message sequence  1011  and the first response  1033 .

   By the transformation information retention verifying processing  1113 , and using the second challenge value  1109 , second input message sequence  1019 , second transformation information retention commitment  1021 , output encrypted message sequence  1011  and the second response  1037 , it is certified  1105  that the second knowledge verifying equation
 
             μ=0   n   g[μ, ┌′]   R[μ]   =G″[ 0, ┌′]           i=1   n   g″[i, ┌′]   c[l]   /F*   p ┌′=0
 
holds.

   By the transformation condition verifying processing  1111  and using the first challenge value  1101 , first response  1033  and the first transformation condition commitment  1025 , it is verified that a verifying equation  1102 
 
 v″   r′   v′   r[0]   v^{Σ   i=1   n   r[i]r[i]r[i]}=ωv^{Σ   i=1   n ( c[i]c[i]c[i]+ψ[i]c[i]c[i]+φ[i]c[i ])}/ F*   p ,
 
sub-response  1039 , sub-response commitment  1027 , first response  1033  and the verifying equation  1107 
 
 u   r′   =u[ 0]             i=1   n   u[i]   r[i]r[i]   /F*   p 
 
hold.

   By the transformation condition verifying processing  1114 , second challenge value  1109 , second response  1037  and the second transformation condition commitment  1016 , it is verified that the verifying equation  1106 
 
 V′   r[0]   v^{Σ   i=1   n   R[i]R[i]}=Ωv^{Σ   i=1   n ( C[i]C[i]+Φ[i]C[i ])}/ F*   p 
 
holds.
 
   If all of the above verifying equations hold, the proof message (text) is accepted  1110 . 
   The above shuffle method with proof is effective in assuring that the shuffle matrix transformation for the input message sequence has been carried out by a “permutation matrix” having at least a shuffle matrix belonging to a permutation matrix. This means that the shuffle has been carried out. So, the present embodiment is a shuffle with proof. 
   Meanwhile, the processing and the function of the transformation information retention commitment processing  1042 , transformation condition commitment generating processing  1043 ,  1045  and the response generating processing  1046 ,  1047  are realized by a program executed on a computer. Moreover, the processing and the function of the transformation information retention commitment processing  1112 ,  1113  and the transformation condition verifying processing  1111 ,  1114  of the shuffle verifying device are realized by a program executed on a computer. In this case, the present invention can be carried out by loading the program on a main memory of a computer from a recording medium having the program recorded thereon, such as a floppy disk medium, a hard disk medium, a magnetic tape medium or a semiconductor memory, and by executing the so-loaded program. 
   EMBODIMENT 5 
   The input message sequence generating method according to Embodiment (5) of the present invention is now explained by referring to  FIG. 12 . It is noted that ┌ assumes the values of 0 , 1 or 2. 
   The secret key x corresponding to the public key  302  g[0, 0] and g[0, 1] is owned in a distributed manner by t provers. 
   With the secret key x[^]; ^=1, . . . , t, the public key of each prover is g[0, 0], g[0, 1, ^]=g[0, 0] x [^] ; ^=1, . . . , t and the entire public key is g[0,0], g[0, 1]=             k=1   t g[0, 1, ^].
   The input encrypted message sequence  301  η[i, 0], η[i, 1]; i=1, . . . , n and the public key  302  η[0, 0], η[0, 1] are input, an input vector  1201  is generated by the basis generating function  1200  by the public key  302  and the ElGamal domain parameters p, q, with respect to the basis generating function  1200 , and the input message sequence  300  g[μ, ┌]; μ=0, . . . , n; ┌=0, 1, 2 is represented by
 
 g[ 0, ┌]=η[0, ┌]┌0, 1
 
 g[i, ┌]=η[i, ┌]/F*   p   i= 1 , . . . , n, ┌= 0, 1
 
 g[μ,  2]=Hash′[μ]( p, q, η[ 0, 0 ], g[ 0, 1, ^]; ^=1 , . . . , t ) μ0 , . . . n. 
 
   When the input message sequence generating method of the present embodiment is applied to Embodiments 1 to 4, the gamut of the value of ┌ is all changed from 0, 1 to 0, 1, 2. The newly-introduced component of ┌=2, which is neither an input message sequence nor a public key, represents a component of the input message sequence not envisaged by a person who produced an input message sequence, and acts for imposing limitations on the response that can be generated by the prover, thus preventing the person who prepared the input message sequence and the person who produced the shuffle proof message (text) from acting together in falsifying the re-encryption proof text. 
   When the input message sequence generating method of the present embodiment is to be applied to Embodiment 3, the input message sequence is expanded to g[−1, ┌] to give
 
 g[− 1, ┌]=η[−1, ┌]┌=0, 1
 
 g[ 0, ┌]=η[0, ┌]┌=0, 1
 
 g[i, ┌]=η[i, ┌]/F*   p   i= 1 , . . . , n, ┌= 0, 1
 
 g[μ,  2]=Hash ′[μ]( p, q, η[ 0, 0 ], g[ 0, 1, ^]; ^=1 , . . . , t ) μ=−1 , . . . , n 
 
from the public key g[−1, 0], g[−11], η[0, 0], η[0,1].
 
   When the input message sequence generating method of the present embodiment is to be applied to Embodiment 4, ┌′=2 and the second transformation information retention commitment is changed by the second information hiding factor to
 
 G″[ 02]=             ν=−1   n   g[ν,  2] A′[ν, 0]   /F*   p 
 
 G″[i 2]=           ν=−1   n   g[ν,  2] A[ν, i]   /F*   p   i= 1 , . . . , n. 

   Moreover, in the shuffle method with proof or the shuffle verifying method, the second challenge value is changed to
 
 C[ 0]=1
 
 C[i ]=Hash[ i ]( g[ν,  2 ], G″[ν,  2 ], Φ[j], Ω, V′;; ν= 0 , . . . , n; j= 1 , . . . , n; ┌= 0, 1) i= 1 , . . . , n 
 
with the second input message sequence g[μ, ┌′]; ┌=2 and the second commitment as an argument of the challenge value generating function.
 
   In addition, the second knowledge verifying equation in the transformation information retention verification processing is changed to
 
             μ=0   n   g[μ,  2] r[μ] =           μ=0   n   G″[μ, ┌′]   c[μ]   /F*   p .

   EMBODIMENT 6 
   Referring to  FIGS. 13 and 14 , the input message sequence generating method according to Embodiment 6 of the present invention is explained. It is noted that ┌ assumes the value of 0, 1. 
   The secret key is owned in a distributed manner by t provers, as in embodiment 5. By a method for a public key sequence method with proof  1304 , each prover ^; ^=1, . . . , t inputs a secret key  1301 x[^] and a pseudo-secret key  1302 α[^] as a public key sequence information  1300 , with the input encrypted message sequence  301  η[i, 0], η[i, 1]; i=1, . . . , n and the public key  302  η[0,0], η[0,1] as a common initial value  1310 , to acquire a dispersed public key sequence pair  1305  g′[μ, 1, ^]; μ=0, . . . , n and the public key sequence proof message (text)  1306 . 
   If, by the public key sequence verifying method  1307 , the authenticity of the dispersed public key sequence pair  1305  has been verified from the dispersed public key sequence pair  1305  output by each prover, public key sequence proving text and the common initial value  1310 , the dispersed public key sequence pairs  1305  of the provers g′[μ, 1, ^]; μ=0, . . . , n; ^=1, . . . , t are combined to change the public key sequence pair  1404  3g′[μ, 1, ^]; μ=0, . . . , n to g′[μ, 1]=             ^=1   t g′[μ, 1, ^]/F* p μ=0, . . . , n, where exchange is made such that g′[0, 1]=η[0, 1].
   From the input message sequence  301  η[i, 0], η[i, 1]; i=1, . . . , n as the common initial value and from the public key  302  η[0, 0], η[0, 1], a public key sequence basis  1401  g′[μ0]; μ=0, . . , n is generated  1400  as
 
 g′[ 0,0]=η[0,0]
 
 g′[i,  0]=Hash′[ i ](η[0,0], η[0, 1 , ^], η[j, ┌]; ^= 1 , . . . , t; ┌= 0, 1 ; j= 1 , . . . , n ;) i= 1 , . . . , n  where g′[0,0] is exchanged as in the public key sequence pair  1403 .
 
   The public key sequence basis  1401  and the public key sequence pair  1403  are combined to form a public key sequence  1404  g′[μ, ┌]; μ=0, . . . , n, ┌=0, 1. 
   From the public key sequence  1404 , input encrypted message sequence  301  and the public key  302 , the input message sequence  300  g[μ, ┌]; μ=0, . . . , n; ┌=0, 1 is set to
 
 g[ 0, ┌]=η[0, ┌]┌=0, 1
 
 g[i, ┌]=η[i, ┌]g′[i, ┌]/F*   p   i= 1 , . . . , n, ┌ =0, 1
 
(at pre-processing  1402 ).
 
   When the input message sequence generating method of the present embodiment is applied to Embodiment 3, a public key sequence g′[μ, ┌]; μ=−1, . . . , n; ┌=0, 1 is generated for the input encrypted message sequence η[i, ┌]; i=1, . . . , n; ┌=0, 1 and the public key η[0, ┌]; ┌=0, 1, where g′[0, ┌]; ┌=0, 1 is equal to the public key. The input message sequence g[μ, ┌]; μ=−1, . . . , n; ┌=0, 1 is set to
 
 g[− 1, ┌]=η[0, ┌]┌=0, 1
 
 g[i, ┌]=η[i, ┌]g′[i, ┌]/F*   p   i= 0 , . . . , n, ┌= 0, 1.
 
   In the present embodiment, since the newly generated public key sequence is not envisaged even by a person who prepared an input message sequence, the components of an input message sequence obtained on multiplying them by the input encrypted message cannot be envisaged. So. the operation of imposing limitations on a response that can be generated by a prover is produced to prevent the person who prepared an input encrypted message and the person who prepared a shuffle proof text (message) acting together in falsification of the shuffle proof text. 
   The processing and the function of the public key sequence method with proof  1304 , pre-processor  1309  and the public key sequence verifying device  1307  are realized by a program executed on a computer. The present invention can be executed by loading the program on a main memory of a computer and running the loaded program from a recording medium having the program recorded thereon (such as one of a CD-ROM, a DVD (digital versatile disc), a floppy disk medium, a hard disk medium, a magnetic tape medium or a semiconductor memory). 
   EMBODIMENT 7 
   The input message sequence generating method according to Embodiment 7 of the present invention is explained with reference to  FIGS. 15 to 18 . It is noted that ┌ assumes the values of 0, 1 and, as in the previous Embodiment 5, the secret key  1502 x is owned in a scattered manner by t provers. 
   Each prover ^; ^=1, . . . ,t inputs the secret key  1502  x[^] and a pseudo secret key  1503  α[^] as the public key sequence information  1501 , by the Public key sequence method with proof  1504 , with the ElGamal area variable set to a common initial value  1500 , to acquire the scattered public key sequence pairs  1505  g′[μ, 1, ^]; μ=0, . . . , and the public key sequence proof text (message)  1506 . 
   If, by the public key verifying method  1507 , the scattered public key sequence pair  1505  has been proved to be authentic from the scattered public key sequence pair  1505  output by each prover, public key sequence proof text  1506  and from the common initial value  1500 , the scattered public key sequence pairs owned by the provers  1505  g′[μ, 1, ^]; μ=0, . . . , n; ^=1, . . . , t are combined to set (change) the Public key sequence pair  1509  g′[μ, 1, ^]; μ=0, . . . , n to
 
 g′[μ,  1]=             ^=1   t   g′[μ,  1 , ^]/F*   p μ=0 , . . . , n. 

   From the common initial value  1500 , the public key sequence basis g′[μ, 0]; μ=0, . . . , n is generated as
 
 g′[μ,  0]=Hash′[μ]( p, q )μ=0 , . . . , n. 
 
   The common initial value  150 A basis and the public key sequence pair  1509  are combined to give a public key sequence  1611  g′[μ┌]; μ=0, . . . , n, ┌=0, 1. ( FIG. 16 ) 
   Each person generating an input encrypted message i=1, . . . , n generates an input encrypted message  1607  η[i, ┌]; ┌=0, 1, from the plain text  1602 m [i], private public key  1601  g′[i, ┌]; ┌=0, 1 a secret random number  1604 s [i] and from a pseudo secret random number  1605 s′[i], by the encryption method with proof  1606 , as
 
 η[i,  0 ]=g′[i,  0] s[i]   /F*   p 
 
 η[i,  1 ]=m[i]g′[i,  1] s[i]   /F*   p .
 
   The commitment (pseudo encrypted message basis  1704 ), challenge value  1707  and the response  1709  are generated in the order of
 
 η[i,  2 ]=g′[i,  0] s′[i]   /F*   p 
 
 c′[i ]=Hash[0](η[ i,  0 ], η[i,  1 ], η[i 2])
 
 θ′[i]=c′[i]s[i]+s′[i]/F*   q 
 
   with the pseudo encrypted message basis  1704  and the response  1709  being set to an encrypted proof message  1608 . 
   By the encryption verifying device,
 
 c′[i ]=Hash[0](η[ i,  0 ], η[i,  1 ], η[i,  2])
 
and the challenge value  1801  are found for all of the input encrypted messages  1607  and the encrypted proof messages  1608  and, using the response  1709 , it is verified  1610  that the verifying equation  1802 
 
 η[i,  0] θ′[i]   =η[i,  1] c′[i]   η[i,  2 ]/F*   p 
 
holds. If the authenticity of all of the input encrypted messages  1607  is verified, the input message sequence  300  is set to
 
 g[ 0, ┌]= g′[ 0, ┌]
 
 g[i, ┌]=η[i, ┌]i= 1 , . . . , n 
 
from the input encrypted message  323 , η[i, ┌]; ┌=0, 1 and from the co-owned public key  1600  g′[0, ┌]; ┌=0, 1.
 
   If the input message sequence-generating method of the present embodiment is applied to the above-described Embodiment 3, the following:
 
 g[− 1 ┌]=g′[− 1, ┌]
 
 g[ 0 , ┌]=g′[ 0, ┌]
 
 g[i, ┌]=η[i, ┌]i= 1 , . . . , n 
 
is set.
 
   In the present embodiment, since the initially generated public key sequence cannot be envisaged even by a person who prepared the input encrypted message, the components of the input encrypted message shown to have been encrypted based on this public key sequence cannot be envisaged. This imposes limitations on the response that can be generated by the prover to prevent the person who prepared the input encrypted message and the person who prepared the shuffle proof text (message) from acting in concert to falsify the shuffle proof text. 
   Meanwhile, the processing and the function of a public key sequence device with proof  1504  and a public key sequence verifying device  1507  as shown in  FIG. 15  are realized by a program run on a computer. The processing and the function of an encrypting device with proof  1606  and an encryption verifying device  1609  are realized by a program run on a computer. In this case, the program is loaded on a main memory of a computer from a recording medium having the program recorded thereon, such as a CD-ROM, a DVD (digital versatile disk), a floppy disk medium, a hard disk medium, a magnetic tape medium or a semiconductor memory, and run to execute the present invention. 
   EMBODIMENT 8 
   The method for public key sequence with proof, according to Embodiment 8 of the present invention is explained with reference to  FIGS. 19 and 20 . 
   A common initial value e, a secret key  1902 x and a pseudo secret key  1903  α, are input as the public key sequence information  1901 . 
   From a common initial value  1900 , a public key sequence basis  1905  g′[μ, 0]; μ=0, . . . , n is generated  1904  as
 
 g′[μ,  0]=Hash′[μ]( e )μ=0 , . . . , n. 
 
   From this, and by the secret key  1902 x and the pseudo secret key  1903  α, the (dispersed) public key sequence pair  1907  g′[μ, 1]; μ=0, . . . , n is generated  1906  as
 
 g′[μ,  1 ]=g′[μ,  0] x   /F*   p μ=0 , . . . , n 
 
whilst the pseudo public key sequence pair  1909  is generated  1908  as
 
 g′[μ,  2 ]=g′[μ,  0] α   /F*   p μ=0 , . . . , n. 
 
   A challenge value  1912  and a response  1914  are sequentially generated as
 
 c ″=Hash[0]( g′[μ,  0 ], g′[μ,  2]; μ=0 , . . . , n )
 
 θ=c″x+α/F   q 
 
with the pseudo public key sequence pair  1909  and a response  1914  constituting a public key sequence proof text  1915 .
 
   By the public key sequence verifying method, a challenge value  2003  is generated  2000  as
 
 c ″=Hash[0]( g′[μ,  0 ], g′[μ,  2]; μ=0 , . . . , n ) and, using a response  1914 , a verifying equation
 
 g′[μ,  0] θ   =g′[μ,  0] c″   g′[μ,  2 ]/F*   p μ=0 , . . . , n 
 
is verified  2004 .
 
   In the present embodiment, since no one can envisage the initially generated public key sequence basis, no one can envisage the components of the public key sequence prepared based thereon. 
   Meanwhile, the processing and the function of a public key sequence device with proof and a public key sequence verifying device are realized by a program run on a computer. In this case, the program is loaded on a main memory of a computer from a recording medium having the program recorded thereon, such as a CD-ROM, a DVD (digital versatile disk), a floppy disk medium, a hard disk medium, a magnetic tape medium or a semiconductor memory, and run to execute the present invention. 
   EMBODIMENT 9 
   As Embodiment 9 of the present invention, decoding with proof is explained. As in Embodiment 5, described above, the secret key x is owned in a scattered fashion by t provers. 
   ^; ^=1, . . . , t′th prover inputs the result of partial decoding by a ^−1st prover and partially decodes it. The result of partial decoding by the ^th prover constitutes a decoded text. It is noted that the result of partial decoding by the 0th prover means the output of the above-mentioned ultimate shuffle. 
   The partial decoding with proof, performed by the ^th prover (partial decoding and submission of the corresponding proof text) is explained. 
   By a pseudo random number generator, a number β[^] on Fq other than 1, 0 is prepared.
 
β[^]∈ R   F   q , ≠0, 1.
 
   The own public key g[0, 0], g′[0, 1, ^] is set to g[0,0], g[0, 1], and the input encrypted message sequence is set to g[i, ┌]; i=1, . . . , n, ┌=0, 1. From the own public key and the secret key x[^], the partial decoding basis G[μ, 0, ^]; μ=0, . . . , n and the pseudo partial decoding basis G[μ, 1, ^]; μ=0, . . . , n are generated as
 
 G[μ,  0 , ^]=g[μ,  0] x[^]   /F*   p μ=0 , . . . , n 
 
 G[μ,  1 , ^]=g[μ,  0] β[^]   /F*   p μ=0 , . . . , n . As commitments,  g[μ, ┌^]; μ= 0 , . . . , n, ┌= 0, 1, ^=0 , . . . , t  is output.
 
   Although g[0, 1, ^]=g[0, 0] x[^] G[0, 0, ^] is overlapped with the public key, the same key is computed. 
   A challenge value is generated as
 
 c [^]=Hash[0]( g[μ,  0 ], G[μ, ┌, ^]; μ= 0 , . . . , n; ┌= 0, 1) and,
 
using this challenge value, a response r[^] is generated as r[^]=β[^]+c[^]x[^]/F q  and output. The partial decoding basis, pseudo partial decoding basis and the response are output as proof text for the partial decoding with proof.
 
   The partial decoding is output as
 
 g[i,  0 ]→g[i,  0 ]i= 1 , . . . , n 
 
 g[i,  1 ]→g[i,  1 ]/G[i,  0 , ^]/F*   p   i= 1 , . . . , n. 
 
   In the verifying processing, a challenge value is generated from the input encrypted message sequence and the proof text, as
 
 c [^]=Hash[0]( g[μ,  0 ], G[μ, ┌, ^]; μ= 0 , . . . , n; ┌ =0, 1)
 
and, using the response in the proof text, input encrypted message sequence, partial decoding basis and pseudo partial decoding basis,
 
 g[μ,  0] r[^]   =G[μ,  0, ^] c[^]   G[μ,  1 , ^]/F*   p μ=0 , . . . , n 
 
is confirmed. It is then verified that the partial decoding has been made using this G[μ, 0, ^] before acceptance.
 
   The results of the foregoing for all of t provers are made into the decoded text. 
   [Authenticity] 
   The authenticity of the above-described embodiment is now explained. 
   [Completeness] 
   That the input message sequence, the output message sequence comprised of an output encrypted message sequence and a transformation information retention commitment, the accompanying response and challenge value meet the verifying equation of the transformation information retention verifying processing may be understood from
 
             μ=1   n+m   g[μ, ┌]   r[μ] =           μ=1   n+m   g[μ, ┌]^{Σ   ν=1   n+m′   A[μ, ν]c[ν]}/F*   p =           ν=1   n+m′ (           μ=1   n+m   g[μ, ┌]   A[μ, ν] ) c[ν]   /F*   p =           ν=1   n+m′   g″[ν, ┌]   c[ν]   /F*   p .

   That the sub-equation coefficients (generator) committed, the accompanying response and the sub-response meet the verifying equation may be seen from
 
 u   r′   =u^{λ[ 0]+Σ i=1   n   λ[i]r[i]r[i]}/F*   p   =u   λ[0]               i=1   n ( u   λ[i] ) r[i]r[i]   /F*   p   =u[ 0]           i=1   n   u[i]   r[i]r[i]   /F*   p .

   That the coefficients of an identity output by the transformation condition commitment generating processing, the accompanying response and the sub-response meet the verifying equation of the knowledge verifying processing can be seen by the following: 
   That the coefficients of the identity of Embodiment 1 hold can be seen from
 
 v′   r[0]               i=1   n   v   r[i]r[i]   /F*   p =( v   r′[0] ) r[0]   v^{Σ   i=1   n Σ μ=0   n Σ ν=0   n   A[i, μ]A[i, ν]c[μ]c[ν]}/F*   p   =v^{r′[ 0]Σ μ=0   n [0 , μ]c[μ]+ 2Σ i=1   n Σ j=1   n   A[i,  0 ]A[i, j]c[j]+Σ   i=1   n   A[i,  0 ]A[i,  0]+Σ i=1   n Σ j=1   n Σ k=1   n   A[i, j]A[i, k]c[j]c[k]}/F*   p   =v^{Σ   i=1   n   φ[i]c[i]+φ[ 0]+Σ i=1   n   c[i]c[i]}/F*   p   =ωv^{Σ   i=1   n ( c[i]c[i]+φ[i]c[i ])}/ F*   p .

   In the foregoing, the fact that A[i, j] is a permutation matrix is used. 
   As for the coefficients of the identity of Embodiment 2, described above, the index part for v of
 
 v″   r′   v′   r[0]               i=1   n   v   r[i]r[i]r[i]   /F*   p 
 
is
 
Σ i=1   n   r[i]r[i]r[i]+Σ   i=1   n   ρ″λ[i]r[i]r[i]+ρ′r[ 0 ]/F*   p =Σ h=1   n Σ i=1   n Σ j=1   n Σ k=1   n   A[h, i]A[h, j]A[h, k]c[i]c[j]c[k]+Σ   h=1   n Σ i=1   n Σ j=1   n (3 A[h,  0 ]A[h, i]A[h, j]+ρ″λ[h]A[h, i]A[h, j ]) c[i]c[j]+Σ   h=1   n Σ i=1   n (3 A[h,  0 ]A[h,  0 ]A[h, i]+ 2 ρ″λ[h]A[h,  0 ]A[h, i]+ρ′A[ 0 , i ]) c[i]+Σ   h=1   n ( A[h,  0 ]A[h,  0 ]A[h,  0 ]+ρ″λ[h]A[h,  0 ]A[h,  0]) +ρ″λ[0 ]+ρ′A[ 0, 0 ]/F   q =Σ h=1   n ( c[h]c[h]c[h]+ψ[h]c[h]c[h]+φ[i]c[i]+φ[ 0]) / F   q 
 
which is equal to an index part of
 
 V^{Σ   h=1   n ( c[h]c[h]c[h]+ψ[h]c[h]c[h]+φ[i]c[i ])}ω[0 ]/F   p .

   For deriving the last equation, the fact that A[i, j] is a permutation matrix has been used (relied on). 
   The same discussion holds for the aforementioned Embodiments 3 and 4. 
   That the public key sequence basis, output by the method for public key sequence with proof of the aforementioned Embodiment 8, the public key sequence pair, pseudo public key sequence pair, the accompanying response and the challenge value meet the verifying equation of the verification processing may be seen from
 
 g′[μ,  0] r   =g′[μ,  0] c x+α   /F*   p   =g′[μ,  0] x c   g′[μ,  0] α   /F*   p   =g′[μ,  1] c   g′[μ,  2 ]/F*   p .
 
[Soundness]
 
   For finding the response r[μ]; μ=1, . . . , n+m satisfying the verifying equation in the transformation information retention verification processing for a given challenge value c[ν]; ν=1, . . . , n+m′, it is necessary to know A[μ, ν]; μ=1, . . . , n+m; ν=1, . . . , n+m′. 
   It is because finding a response satisfying the verifying equation in the equivalence detection processing without knowing A[μ, ν]; μ=1, . . . , n+m; ν=1, . . . , n+m′ for given g[μ, ┌], g″[ν┌]; μ=1, . . . , n+m; ν=1, . . . , n+m′ is tantamount to solving the discrete logarithmic problem. 
   The reason is that being unaware of A[μ, ν] means that, as for at least one g″[ν, ┌], the representation having g[μ, ┌]; μ=1, . . . , n+m as the basis is not known, and that, if a response satisfying the verifying equation for an optional c can be found, the discrete logarithm can be solved by selecting such c[ν] as will give c[ξ]=1, c[ν]=0; ν=0, . . . , ξ−1, ξ+1, . . . , n+m′. 
   Also, since the challenge value c[ν] has a commitment g[μ, ┌], g″[μ, ┌] as an argument, the commitment cannot be adjusted after deciding the challenge value (the challenge value generating function requests this property to be had). Therefore, a prover may take the challenge value as a random number given after commitment decision. 
   If, for any component of g[ν, ┌], its representation having another component as the basis is not known, forming plural responses satisfying the verifying equation is tantamount to solving the problem of discrete logarithm. The reason is that, if the verifying equation holds for different r[μ] and r′[μ], non-obvious representation of “1” having g[μ, ┌] as the basis may be obtained on dividing both sides by each other, which is equivalent to solving the problem of discrete logarithm. 
   As for the input message sequence g[μ, ┌]; μ=1, . . . , n+m; ┌=0, . . . generated by the input message sequence generating method, since the vector g[μ, ┌]; μ=1, . . . , n+m for any ┌ is evidently generated by the Hash function or by the operation e.g., of multiplying the vector generated by a Hash function, it is felt to be number-theoretically difficult to express one using the other as the basis, vice versa. 
   From the foregoing, a prover cannot calculate except generating r[μ]=Σ v=1   n+m′ A[μ, ν]c[ν]/F q ; μ=1, . . . , n+m using g″[ν, ┌]=             μ=1   n+m g[μ, ┌] A[μν] /F* p ; ν=1, . . . , n+m′ as r[μ]; μ=1, . . . , n+m satisfying the verifying equation. Th same applies for a method employing an individual public key.
   If the relation
 
 g″[ν, ┌]=               μ=1   n+m   g[μ, ┌]   A[μ, ν]   /F*   p ν=1 , . . . , n+m′ 
 
is proved for given ┌, as described above, similar proof may be given for other ┌ as follows:

   If the verifying equation holds for g[μ, ┌], g″[ν, ┌] included in an argument of the challenge value generating function,
 
 g″[ν, ┌]=               μ=1   n+m   g[μ, ┌]   A[μ, ν]   /F*   p ν=1 , . . . , n+m′. 

   The reason is as follows: If the verifying equation holds for a representation
 
 g″[ν, ┌]=               μ=1   n+m   g[μ, ┌]   A′[μν]   /F*   p ν=1 , . . . , n+m′, 
 
then
 
=           μ=1   n+m   g[μ, ┌]^{Σ   ν=1   n+m′ ( A[μ, ν]−A′[μ, ν] ) c[ν]}= 1 /F*   p 
 
holds.

   However, it is only when
 
             μ=1   n+m   g[μ, ┌]   A[μ, ν] =           μ=1   n+m   g[μ, ┌]   A′[μ, ν]   /F*   p ν=1 , . . . , n+m′ 
 
that the above equation holds for c[ν] selected at random.

   In the above-described Embodiment 2, if, given u, u[μ]; μ=0, . . . , n, obtained on committing the quasi-element (generator) coefficients by the transformation condition commitment generating processing, the response r[i]; i=1, . . . , n and the sub-response r′ meet the verifying equation, the sub-response r′ is unique, such that the sub-response r′ is represented by the above equation by
 
 r′=λ[ 0]+Σ i=1   n   λ[i]r[i]r[i]/F   q 
 
satisfying the verifying equation.
 
   By expanding the index part of v of the left side of the verifying equation of the identity of Embodiment 2, we obtain:
 
Σ h=1   n Σ i=1   n Σ j=1   n Σ k=1   n   A[h, i]A[h, j]A[h, k]c[i]c[j]c[k]+Σ   h=1   n Σ i=1   n Σ j=1   n (3 A[h,  0 ]A[h, i]A[h, j]+ρ″λ[h]A[h, i]A[h, j ]) c[i]c[j]+Σ   i=1   n (Σ h=1   n (3 A[h,  0 ]A[h,  0 ]A[h, i]+ 2 ρ″λ[h]A[h,  0 ]A[h, i ])+ρ′ A[ 0 , i ]) c[i]+Σ   h=1   n ( A[h,  0 ]A[h,  0 ]A[h,  0 ]+ρ″λ[h]A[h,  0 ]A[h,  0])+ρ″λ[0 ]+ρ′A[ 0,0 ]/F   q .
 
   The index part of v of the right side is
 
Σ i=1   n ( c[i]c[i]c[i]+ψ[i]c[i]c[i]+φ[i]c[i ])+φ[0 ]/F*   q .
 
   Therefore, if the verifying equation is to hold for any c[μ]; μ=0, . . . , n, the coefficients of c[μ]c[ν]c[ξ]; μ, ν, ξ=0, . . . , n must be the same. Otherwise, the possibility of the verifying equation retention for an arbitrarily given c[μ] may be disregarded. 
   This assures
 
Σ h=1   n   A[h, i]A[h, j]A[h, k]=δ′[i, j, k]/F   q   i, j, k= 1 , . . . , n 
 
Σ h=1   n (3 A[h,  0 ]A[h, i]A[h, j]+ρ″λ[h]A[h, i]A[h, j ])=δ[ i, j]ψ[i]/F   q   i, j= 1 , . . . , n 
 
Σ h=1   n (3 A[h,  0 ]A[h,  0 ]A[h, i]+ 2 ρ″λ[h]A[h,  0 ]A[h, i ]) +ρ′ A[ 0 , i]=φ[i]/F   q   i= 1 ,. . . , n 
 
Σ h=1   n ( A[h,  0 ]A[h,  0 ]A[h,  0 ]+ρ″λ[h]A[h,  0 ]A[h,  0]) +ρ″λ[0 ]+ρ′A[ 0, 0]=φ[0 ]/F   q 
 
   using the relation that 
   for δ[i, j]=1 i=j 
   =0 and others and 
   for δ′[i, j, k]=1 i=j=k 
   =0 and others. From this, the following may be found for A[i, j]; i, j=1, . . . , n. 
   An n-dimensional vector A[h, j]A[h, k]; h=1, . . . , n having a h′th element A[h, j]A[h, k] for given j, k; j≠k and an n-dimensional vector A[h, i]; h=1, . . . , n having a h′th element A[h, i] for given i are considered. It is assumed that n vectors A[h,i]; i=1, . . . , n span (lie in) a n-dimensional space, that is, the entire vectors may be represented by linear combination of A[h, i]; i=1, . . . , n. Then, from the above equation, the vector A[h, j]A[h, k]; h=1, . . . , n has an inner product of 0 with respect to the entire vectors A[h, i], the following equation holds:
 
 A[h, j]A[h, k]= 0 /F   q   h= 1 , . . . , n. 
 
   It is seen from above that, among the n vectors A [h, i]; h=1, . . . , n; i=1, . . . , n, only one is a vector the respective h generators of which are not zero. 
   It is also seen from above that A[h, i]A[h, j]A[h, k]≠0 for i=j=k and hence the vector A[h, i]; h=1, . . . , n has at least one non-zero element. Therefore, the entire vectors A[h, i]; h=1, . . . , n have only one non-zero element which, from the above equation, is 1 1/3 . 
   It is now shown that n vectors A[h, i]; h=1, . . . , n; i=1, . . . , n span (are in) a n-dimensional space. 
   Using n scalars κ[i]; i=1, . . . , n, the vector a[h]; h=1, . . . , n is represented by
 
 a[h]=Σ   i=1   n   κ[i]A[h, i]h= 1 , . . . , n /F   q .
 
   If it is shown that κ[i]=0 for a[h]=/F q , it can be shown that n vectors A[h, i]; h=1, . . . n; i=1, . . . , n lie in the n-dimensional space. If, with a[h]=0 /F q , both sides of the above equation are multiplied by a n-dimensional vector A[h, i]A[h, i] whose h′th element is A[h, i]A[h, i],
 
0 =κ[i]/F   q   i= 1 ,. . . , n 
 
from the above two equations. It has been shown from above that A[i, j] is a permutation matrix or a quasi-permutation matrix obtained on multiplying certain generators of the permutation matrix with 1 1/3 .
 
   By expanding an index part of v of the left side of the verifying equation of an equation of Embodiment 1, we obtain
 
 r[ 0 ]r[ 0]+Σ i=1   n   r[i]r[i]/F   q =Σ i=1   n Σ j=1   n Σ k=1   n   A[i, j]A[i, k]c[j]c[k]+Σ   j=1   n (Σ i=1   n 2 A[i,  0 ]A[i, j]+r′[ 0 ]A[ 0 , j ]) c[j]+Σ   i=1   n   A[i,  0 ]A[i,  0 ]+r′[ 0 ]A[ 0, 0 ]/F   q .
 
   The index part of v on the right side is
 
Σ i=1   n ( c[i]c[i]+φ[i]c[i ])+φ[0 ]/F   q .
 
   So, in order for the verifying equation to hold for any c[μ]; μ=0, . . . n, the coefficients of c[μ]cν; μ, ν=0, . . . , n must be the same. The possibility that the verifying equation holds for arbitrarily given responses otherwise can be neglected. 
   This assures
 
Σ h=1   n   A[h, i]A[h, j]=δ[i, j]/F   q 
 
φ[i]=Σ h=1   n 2 A[h,  0 ]A[h, i]+r′[ 0 ]A[ 0,  i]/F   q 
 
φ[0]=Σ i=1   n   A[i,  0 ]A[i,  0 ]+r′[ 0 ]A[ 0, 0] /F   q 
 
and hence the possibility that the verifying equation holds can be neglected if A[i, j]; i, j=1, . . . , n is not an orthonormal matrix.
 
   For the above-described Embodiments 3 and 4, similar discussion holds, such that A[i, j]; i, j=1, . . . , n is a permutation matrix and simultaneously an orthonormal matrix. This indicates that the matrix is a permutation matrix. 
   [Witness Indistinguishability] 
   It is shown that, in the shuffle proof text, the shuffle information is hidden number-theoretically. 
   As a result of the shuffle, such values as r[μ], r′, φ[i],
 
 ψ[i], ω, v′, v″, v, u, u[i] 
 
 R[μ], Φ[i], V′, Ω, v, 
 
become apparent in addition to g″[ν], m″[μ]. These afford the information pertinent to shuffle. However, if the identity coefficients are committed and hidden so that the number of unknowns pertinent to the shuffle matrix processing is larger than the number of conditions other than the results of the exponential calculations, solution becomes impossible unless the problem of discrete logarithm is solved to increase the number of the conditions. However, certain minor adjustments may be needed since the solution may become possible depending on the manner of appearance in the conditions of the unknowns without dependency on the number of variables.
 
   The meritorious effects of the present invent ion are summarized as follows. 
   According to the present invention, as described above, the computational resources for shuffle with proof may be decreased as compared to that in the prior-art technique. 
   In particular, it may be contemplated that a number of practical applications of verifying processing cannot be computed beforehand. So, if the computational resources for verification is compared, 320n+2n times of modular exponentiation processing operations are needed for a safety variable of 160, in the prior-art technique (1), whilst 8 (n log n−n+1) modular exponentiation processing operations are needed in the prior-art technique (2). According to the present invention, 7n+14 times of modular exponentiation processing operations suffice, such that, for n&gt;4, the volume of the modular exponentiation processing operations is smaller than the case of any prior-art techniques. 
   Moreover, according to the present invention, the modular exponentiation processed in the course of the verification is not the individual modular exponentiation processing operations, but the processing for finding the product of the modular exponentiation processing operations, and hence calculations may be carried out with a smaller computational resources than in case of individual modular exponentiation processing operations. So, a prospect for a higher processing speed may result. 
   It should be noted that other objects, features and aspects of the present invention will become apparent in the entire disclosure and that modifications may be done without departing the gist and scope of the present invention as disclosed herein and claimed as appended herewith. 
   Also it should be noted that any combination of the disclosed and/or claimed generators, matters and/or items may fall under the modifications aforementioned.