Cryptographic communication method, encryption method, and cryptographic communication system

[0002] This application relates to two applications concurrently filed herewith entitled (1) “ENCRYPTION METHOD, DECRYPTION METHOD, ENCRYPTION/DECRYPTION METHOD, CRYPTOGRAPHIC COMMUNICATIONS SYSTEM, AND COMPUTER USABLE MEDIUM” that claims foreign priority based on Japanese patent applications 10-262036, filed Sep. 16, 1998 and 11-105815, filed Apr. 13, 1999 (inventors: Masao Kasahara and Yasuyuki Murakami; Express Mail EL 446 156 827 US) and (2) “ENCRYPTION METHOD, DECRYPTION METHOD, ENCRYPTION/DECRYPTION METHOD, CRYPTOGRAPHIC COMMUNICATIONS SYSTEM, AND COMPUTER USABLE MEDIUM” that claims foreign priority based on Japanese patent applications 10-262037, filed Sep. 16, 1998 and 11-105814, filed Apr. 13, 1999 (inventors: Masao Kasahara and Yasuyuki Murakami; Express Mail EL 446 156 835 US), which applications are hereby incorporated by reference in their entirety.

[0003] 1. Field of the Invention

[0004] The present invention relates to a cryptographic communication method, encryption method, and cryptographic communication system that afford a high level of safety in encryption and communication of information by utilizing information encrypted in such a manner not to be understood by anyone but the intended persons.

[0005] 2. Description of the Related Art

[0006] In today's society, sometimes called the advanced information society, documents and graphic information that are important for business are transmitted and processed in the form of electronic information, using a computer network as a platform. By its nature, this electronic information is easy to duplicate, making it hard to tell an original from a copy, and information security has therefore become an important issue. In particular, building a computer network that satisfies the requirements of “shared computer resources,” “multi-access,” and “broad area networking” is essential to the establishment of a true highly sophisticated information society, but this includes factors that are in conflict with the goal of information security between involved parties. Encryption technology, which has been used primarily for military and diplomatic purposes in past human history, is attracting attention as an effective means for resolving this conflict.

[0007] Cryptography deals with exchanging information in such a way that the meaning thereof cannot be understood by anyone but the intended recipient. In cryptography, converting original text that can be understood by anyone (plaintext) into text that is meaningless to a third party (ciphertext) is called encryption, while returning the ciphertext to the original plaintext is called decryption, and the overall process of this encryption and decryption is called a cryptosystem. Secret encryption and decryption keys are used in the process of encryption and decryption, respectively. Since a secret decryption key is necessary during decryption, only someone who knows this decryption key can decrypt a ciphertext, so encryption allows the confidentiality of information to be preserved.

[0008] An encryption key may be the same as or different from a decryption key. A cryptosystem in which the two keys are the same is called a common or shared key cryptosystem, and the DES (Data Encryption Standards) employed by the Bureau of Standards of the US Department of Commerce is a typical example of this. Public key cryptosystems are proposed as an example of a cryptosystem in which the two keys are different. With a public key cryptosystem, each user (entity) that utilizes the cryptosystem generates a pair of keys, i.e., an encryption key and a decryption key, discloses the encryption key on a public key list, and keeps just the decryption key secret. An advantage of a public key cryptosystem is that the paired encryption key and decryption key are different and a one-way function is utilized, which makes it impossible for someone to deduce the decryption key from the encryption key.

[0009] A public key cryptosystem is a revolutionary cryptosystem in which the encryption key is disclosed, and satisfies the above three requirements necessary for the establishment of a sophisticated information society. A great deal of research has gone into these systems in an effort to utilize them in such fields as information communication technology, and a typical public key cryptosystem that has been proposed is the RSA cryptosystem. This RSA cryptosystem makes use of the difficulty of factoring large prime numbers using one-way functions. There are also public key cryptosystems that make use of the difficulty of solving discrete logarithm problems.

[0010] There is also a cryptosystem that makes use of personal ID (identity) information, such as the name or address of each entity. With this cryptosystem, a shared encryption key is generated between the sender and recipient on the basis of ID information. Cryptographic methods based on this ID information include (1) those that require pre-communication between the sender and recipient prior to the transmission of the ciphertext, and (2) those that do not require pre-communication between the sender and recipient prior to the transmission of the ciphertext. Since the second type of method does not require any pre-communication, it is very convenient for an entity, and is expected to become a mainstay of cryptosystems in the future.

[0011] The second type of scheme is called ID-NIKS (an ID-based non-interactive key sharing scheme), in which an encryption key is shared, without any pre-communication being performed, by using ID information about the communicating party. ID-NIKS does not require that a public key and secret key be exchanged between the sender and recipient, nor does it require a key list or service by a third party, allowing secure communication between any entities.

[0012] FIG. 9 of the accompanying drawings is a diagram showing the principle behind this ID-NIKS system The existence of a center that can be trusted is assumed, and a common key generation system is built around this center. In this diagram, ID information such as the name, address, and telephone number of an entity X, which is personal information of entity X, is expressed as h(ID_x) using a hash function h(·). The center calculates secret information S_xias follows on the basis of public center information {PC_i}, secret center information (SC_i}, and entity X ID information h(ID_x), and secretly distributes this calculated information to entity X.

[0013] S_xi=F_i({SC_i}, {PC_i}, h(ID_x))

[0014] For communication with any other entity Y, entity X uses his own secret information {S_Xi}, public center information {PC_i}, and the other entity Y ID information h(ID_Y) to generate a common key K_XYfor encryption and decryption as follows.

[0015] K_XY=f({S_Xi}, {PC_i}, h(ID_Y))

[0016] Similarly, entity Y also generates a common key K_YXfor entity X. As long as the relationship K_XY=K_YXholds true, then these keys K_XYand K_YXcan be used as an encryption key and decryption key between entities X and Y.

[0017] With the public key cryptosystems discussed above, in the case of an RSA cryptosystem, for instance, this public key is over ten times as long as current telephone numbers, and is therefore far from simple. In contrast, with an ID-NIKS system, if each set of ID information is registered in the form of a roster, then a public key can be generated between itself and any other entity by referring to this roster. Therefore, if the ID-NIKS illustrated in FIG. 9 could be safely implemented, it would be possible to construct a convenient cryptosystem over a computer network to which many entities subscribe. It is for this reason that an ID-NIKS system is expected to be at the forefront of future cryptosystems.

[0018] It is preferable for ID-NIKS, in which common keys that serve as encryption and decryption keys are mutually shared by using the ID information of the communicating parties without any pre-communication being performed, to be sufficiently secure against attack involving a collusion of a plurality of entities, for example. However, this ID-NIKS has the problem that the secret parameters of the center can be revealed if enough people (entities) are in collusion since such attack method has been studied. Whether a cryptologically safe ID-NIKS system can be constructed is an important question for a sophisticated information society, and a search is underway for a more ideal encryption system.

[0019] An object of the present invention is to provide a novel cryptographic communication method and cryptographic communication system involving ID-NIKS system, with which secret key generation functions and key sharing functions are not separable, key sharing is probabilistically possible, and high degree of security is realized.

[0020] According to the first aspect of the present invention, there is provided a cryptographic communication method for communication of information between entities, in which a center generates and sends an entity-specific secret key to each entity, one entity uses a common key determined from its own entity-specific secret key sent from the center and a publicly known public key of the other entity to encrypt a plaintext into a ciphertext and transmits it to the other entity, and the other entity uses the same common key as the above-mentioned common key, determined from the second entity-specific secret key sent from the center to the other entity (the-other-entity-specific secret key) and the disclosed public key of the one entity, to decrypt the ciphertext back into a plaintext, characterized in that each-entity-specific secret key includes plural types of secret keys in which each of a plurality of numbers serves as a modulus, said plural types of secret keys being generated using respective entities' public keys and entity-specific random numbers, one entity uses a plural types of its own secret keys and the public key of the other entity or entities to generate the common key, and the other entity or entities use a plurality of its own secret keys and the public key of the one entity to generate said common key.

[0021] In the generation of the common key, the random number may be eliminated by addition over integer ring.

[0022] The random number may be a multidimensional random number vector.

[0023] The computational formulas for generating two types of the secret keys at the center may be as follows.

[0024] {right arrow over (sA{right arrow over (v_i)})mod P

[0025] {right arrow over (tB{right arrow over (v_i)})mod Q

[0026] Where

[0027] Vector s_i: one secret key of entity i

[0028] Vector t_i: the other secret key of entity i

[0029] P and Q: publicly known prime numbers

[0030] A and B: symmetric matrices composed of random numbers known only to the center

[0031] Vector γ_i: a personal random number vector composed of random numbers

[0032] The computational formulas for generating the common key for the entities may be as follows.

[0033] A_ij′=^t{right arrow over (s{right arrow over (vP

[0034] B_ij′=^t{right arrow over (t{right arrow over (vQ

[0035] K_ij=A_ij′+B_ij′

[0036] Where

[0037] K_ij: common key generated by one entity i for another entity j

[0038] Vector v_j: public key of entity j

[0039] A_ij′ and B_ij′: intermediate values for generating the common key K_ij

[0040] k bits of the P and Q may be employed to satisfy the following formulas.

[0041] P≡δ(mod R)

[0042] Q≡ε(mod R)

[0043] Where

[0044] R: prime number of d bits

[0045] δ and ε: number of e bits

[0046] k>d>e

[0047] The computational formulas for generating the common key for the entities may be given by the following equations.

[0048] K_ij′=A_ij′+B_ij′

[0049] K_ij″=K_ij′mod R

[0050] $A_{\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_i}\overrightarrow{{\upsilon }_j}\mathrm{mod}P$ $B_{\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{t_i}\overrightarrow{{\upsilon }_j}\mathrm{mod}Q$ $K_{\mathrm{ij}}=\lfloor \frac{K_{\mathrm{ij}}^{″}}{2^{e+c}}\rfloor$

[0051] Where

[0052] K_ij: common key generated by one entity i for another entity j

[0053] K_ij′: an intermediate value for generating the common key K_ij

[0054] K_ij″: an intermediate value for generating the common key K_ij, i.e., remainder of dividing K_ij′ by R

[0055] The public key for each entity may be determined by utilizing a hash function to calculate the each entity-specific information.

[0056] According to the second aspect of the present invention, there is provided an encryption method, in which secrets keys are sent to a plurality of entities, said secret keys being specific to the entities to which they are sent, and said entities use their own said entity specific keys to encrypt a plaintext into a ciphertext, characterized in that each-entity-specific secret key includes plural types of secret keys in which each of a plurality of numbers serves as a modulus, said plural types of secret keys being generated using its own entity-specific public key and its own entity-specific secret random number, and one entity encrypts a plaintext into a ciphertext using a common key generated using its own plural types of secret keys and a public key of the other entity, who is the intended recipient of the ciphertext.

[0057] According to the third aspect of the present invention, there is provided a cryptographic communication system including a plurality of entities, in which encryption into a ciphertext of a plaintext to be transmitted and decryption of the transmitted ciphertext back into an original plaintext are performed mutually between the plurality of entities, characterized in that the cryptographic communication system includes: a center that uses each-entity-specific public key and each-entity-specific secret random number to generate for each entity a plurality of types of secret keys in which each of a plurality of numbers serves as a modulus in relation to the plurality of types of secret keys, and sends these secret keys to each entity; and a plurality of entities, one of which uses a plurality of its own type of secret keys sent from the center and another-entity-specific public key to generate a common key for performing the encryption, and another of which uses a plurality of its own type of secret keys sent from the cente and an entity-specific key of the one entity for performing the decryption. The “another entity” is an entity which receives the ciphertext and decrypts it into the original plaintext.

[0058] According to the fourth aspect of the present invention, there is provided a cryptographic communication method for the communication of information between entities, in which an entity-specific secret key is generated by a center and sent to each entity from the center, one entity utilizes a common key determined from its own entity-specific secret key sent from the center and a publicly known public key of the other entity to encrypt a plaintext into a ciphertext and transmits it to the other entity, and the other entity utilizes the same common key as the above-mentioned common key, said common key being determined from its own entity-specific secret key sent from the center and the disclosed public key of the other entity, to decrypt the ciphertext back into a plaintext, characterized in that each entity-specific secret key includes a plurality of secret keys in which each of a plurality of numbers serves as a modulus, said plurality of secret keys being generated using a plurality of public keys of each entity and a plurality of each entity-specific random numbers, and each entity uses its own multiple secret keys and the plurality of public keys of the other entity to generate the common key.

[0059] When the common key is produced, the plurality of random numbers may be eliminated by addition over integer ring.

[0060] The plurality of random numbers may be a plurality of multidimensional random number vectors.

[0061] The computational formula for generating four secret keys at the center may be an equation (A) below, and the computational formula for generating the common key at each entity may be an equation (B) below. $\begin{array}{cc}\begin{array}{c}\overrightarrow{s_{11,i}}=\left(A_1\overrightarrow{{\upsilon }_{1,i}}+\overrightarrow{{\gamma }_{1,i}}\right)\mathrm{mod}{P}_1\\ \overrightarrow{s_{22,i}}=\left(A_2\overrightarrow{{\upsilon }_{2,i}}+\overrightarrow{{\gamma }_{2,i}}\right)\mathrm{mod}{P}_2\\ \overrightarrow{s_{21,i}}=\left(A_3\overrightarrow{{\upsilon }_{1,i}}-\overrightarrow{{\gamma }_{2,i}}\right)\mathrm{mod}{P}_3\\ \overrightarrow{s_{12,i}}={(}^t{A}_3\overrightarrow{{\upsilon }_{2,i}}-\overrightarrow{{\gamma }_{1,i}})\mathrm{mod}{P}_3\end{array}\}& \left(A\right)\end{array}$

[0062] Where

[0063] Vector S_11,i: first secret key of entity i

[0064] Vector S_22,i: second secret key of entity i

[0065] Vector S_21,i: third secret key of entity i

[0066] Vector S_12,i: fourth secret key of entity i

[0067] Vector v_1,i: first public key of entity i

[0068] Vector v_2,i: second public key of entity i

[0069] P₁, P₂, and P₃: publicly known prime numbers

[0070] A₁and A₂: secret symmetric matrices composed of random numbers known only to the center

[0071] A₃: secret matrix composed of random numbers known only to the center

[0072] Vector γ_1,i: first personal random number vector composed of random numbers

[0073] Vector γ_2,i: second personal random number vector composed of random numbers

[0074] K_ij=A_11,ij′+A_22,ij′+A_21,ij′+A_12,ij′  (B)

[0075] $\left(\begin{array}{c}{A}_{11,\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_{11,i}}\overrightarrow{{\upsilon }_{1,j}}\mathrm{mod}{P}_1\\ A_{22,\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_{22,i}}\overrightarrow{{\upsilon }_{2,j}}\mathrm{mod}{P}_2\\ A_{21,\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_{21,i}}\overrightarrow{{\upsilon }_{2,j}}\mathrm{mod}{P}_3\\ A_{12,\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_{12,i}}\overrightarrow{{\upsilon }_{1,j}}\mathrm{mod}{P}_3\end{array}\right)$

[0076] Where

[0077] K_ij: common key generated by one entity i for another entity j

[0078] Vector v_1,j: first public key of entity j

[0079] Vector v_2,j: second public key of entity j

[0080] A_11,ij′, A_22,ij′, A_21,ij′, and A_12,ij′: intermediate values for generating the common key K_ij

[0081] There may be d number of public keys for each entity, the computational formula for generating d²number of the secret keys at the center is as follows (C), and the computational formula for generating the common key at each entity is as follows (D). $\begin{array}{cc}\overrightarrow{s_{\mathrm{yz},i}}=\left(A_{\mathrm{yz}}\overrightarrow{{\upsilon }_{z,i}}+\overrightarrow{{\gamma }_{\mathrm{yz},i}}\right)\mathrm{mod}{P}_{\mathrm{yz}}& \left(C\right)\end{array}$

[0082] Where

[0083] A_yz(y and z=1, 2, . . . , d): a sub-matrix generated by partitioning an n×n symmetric matrix A in both the row and column directions at n₁, n₂, . . . , n_d(where n=n₁+n₂+ . . . +n_d)

[0084] Vector S_yz,i: d²number of secret keys of entity i

[0085] Vector v_z,i: a vector generated by dividing a public key column vector v_iof entity i in the column direction into sizes of n₁, n₂, . . . , n_d

[0086] Vector γ_yz,i: a vector generated by dividing a personal random number column vector γ_z,iof entity i in the column direction into sizes of n₁, n₂, . . . , n_d

[0087] P_yz: publicly known prime number $\begin{array}{cc}{K}_{\mathrm{ij}}=\sum _{y=1}^d\sum _{z=1}^d{A}_{\mathrm{yz},\mathrm{ij}}^{\prime }\left(A_{\mathrm{yz},\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_{\mathrm{yz},i}}\overrightarrow{{\upsilon }_{y,j}}\mathrm{mod}{P}_{\mathrm{yz}}\right)& \left(D\right)\end{array}$

[0088] Where

[0089] K_ij: common key generated by one entity i for another entity j

[0090] Vector v_y,j: a vector generated by dividing a public key row vector v_jof entity i in the row direction into sizes of n₁, n₂, . . . , n_d

[0091] A_yz,ij′: d²number of intermediate values for generating the common key K_ij

[0092] A plurality of combinations may be used in which each combination includes d number of the public keys for each entity, and d²number of the secret keys for each entity.

[0093] The computational formula for generating 2d number of the secret keys at the center is as follows (E), and the computational formula for generating the common key at each entity is as follows (F). $\begin{array}{cc}\begin{array}{c}\overrightarrow{s_{r,i}}=\left(A_r\overrightarrow{{\upsilon }_{r+1,i}}+\overrightarrow{{\gamma }_{r,i}}\right)\mathrm{mod}{P}_r\\ \overrightarrow{t_{r,i}}=\left(B_r\overrightarrow{{\upsilon }_{r,i}}-\overrightarrow{{\gamma }_{r+1,i}}\right)\mathrm{mod}{P}_r\end{array}\}& \left(E\right)\end{array}$

[0094] Where

[0095] Vector S_r,i: d number of secret keys of entity i

[0096] Vector t_r,i: d number of secret keys of entity i

[0097] Ar: a secret matrix composed of random numbers known only to the center

[0098] B_r=^tA_r

[0099] Vector V_r,i: d number of public keys of entity i

[0100] Vector γ_r,i: d number of personal random number vectors composed of random numbers

[0101] Pr: publicly known prime number $\begin{array}{cc}{K}_{\mathrm{ij}}=\sum _{r=1}^d\left(A_{r,\mathrm{ij}}^{\prime }+B_{r,\mathrm{ij}}^{\prime }\right)\left(\begin{array}{c}{A}_{r,\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_{r,i}}\overrightarrow{{\upsilon }_{r,j}}\mathrm{mod}{P}_r\\ B_{r,\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{t_{r,i}}\overrightarrow{{\upsilon }_{r+1,j}}\mathrm{mod}{P}_r\end{array}\right)& \left(F\right)\end{array}$

[0102] Where

[0103] K_ij: common key generated by one entity i for another entity j

[0104] Vector v_r,j: d number of public keys of entity j

[0105] A_r,ij′ and B_r,ij′: an intermediate value for generating the common key K_ij

[0106] The plurality of public keys for each entity may be determined by utilizing a hash function to calculate the specific information of each entity.

[0107] According to the fifth aspect of the present invention, there is provided an encryption method, in which secret keys are generated at a center and sent to a plurality of entities from the center, said secret keys being specific to the entities to which they are sent, and said entities use their own said entities' specific keys to encrypt a plaintext into a ciphertext, characterized in that each-entity-specific secret key of each entity includes a plurality of secret keys in which each of a plurality of numbers serves as a modulus, said plurality of secret keys being generated using a plurality of its own entity-specific public keys and a plurality of its own entity-specific secret random numbers, and one entity encrypts a plaintext into a ciphertext using a common key generated using these plurality of its own secret keys and a plurality of public keys of the other entity, who is the intended recipient of the ciphertext.

[0108] According to the sixth aspect of the present invention, there is provided a cryptographic communication system, in which encryption into a ciphertext of a plaintext to be transmitted, and decryption of the transmitted ciphertext back into a plaintext, are performed mutually between a plurality of entities, characterized in that the cryptographic communication system includes a center for using a plurality of each-entity-specific public keys and a plurality of each-entity-specific secret random numbers to generate for each entity a plurality of secret keys in which each of a plurality of numbers serves as a modulus, and for sending these plural secret keys to each entity, and a plurality of entities each of which uses its own plural secret keys sent from the center and a plurality of public keys specific to another entity, who is an entity to receive a ciphertext or an entity to send a ciphertext, to generate a common key for performing the encryption processing and decryption processing.

[0109] First, let us discuss the conditions for realizing an ID-NIKS system, and the conditions for secure ID-NIKS. Here, i, j, y, and z express entities, v_iis the public key of entity i which in most cases is the ID hash value, s_iis the secret key of entity i, and K_ijis the common key with entity j determined by entity i.

[0110] The following three conditions must be met for an ID-NIKS system to be realized.

[0111] Condition 1 (Secret Key Generation Condition)

[0112] The center is able to determine the corresponding secret key s_ifrom the public key v_iof entity i using a secret key generation function f(·).

[0113] s_i=f(v_i)

[0114] Condition 2 (Common Key Generation Condition)

[0115] The common key K_ijcan be determined from the secret key s_iof entity i and the public key v_jof entity j using a common key generation function g(·).

[0116] K_ij=g(s_i, v_j)

[0117] Condition 3 (Key Sharing Condition)

[0118] The common key K_ijgenerated by entity i for entity j is equal to the common key K_jigenerated by entity j for entity i.

[0119] K_ij=K_ji

[0120] Therefore, a key sharing function F(·) in which the variables are v_iand v_j, obtained by substituting the secret key generation function f(·) into the common key generation function g(·), is a symmetric function.

[0121] F(v_i, v_j)=F(v_j, v_i)

[0122] Where F(v_i, v_j)=g(f(v_i), (v_j)=g(s_i, v_j)

[0123] The term “separable” here is defined as follows.

[0124] Definition: If we let a suitable commutative method be ◯, when the following formula is always satisfied, then the function f is separable by ◯.

[0125] f(x+y)=f(x)◯f(y)

[0126] For example, f(x)=ax and f(x)=a^xare separable as shown below.

[0127] f(x+y)=a(x+y)=ax+ay=f(x)+f(y)

[0128] f(x+y)=a^x+y=a^x·a^y=f(x)·f(y)

[0129] Meanwhile, f(x)=ax+b is not separable as shown below.

[0130] f(x+y)=a(x+y)+b=ax+ay+b

[0131] f(x)+f(y)=ax+b+ay+b=ax+ay+2b

[0132] Thus, f(x+y)≠f(x)+f(y)

[0133] When the public key v_zof an entity z that is the object of attack is expressed by linear combination of public keys v_iof conspirators, and furthermore when either one of the secret key generation function or key sharing function is separable in polynomial time, then the secret key and common key of the entity can be forged without finding the center's secrets. This method of attack has in the past been called a linear attack.

[0134] It used to be thought that it was easy to express a public key v_zby linear combination, but methods have also been developed with which it is not always easy to express the public key v_zof a target entity by linear combination. In view of this, linear attack should be considered as two parts: a first-stage attack portion in which the public key v_zis expressed as a linear combination, and a second-stage attack portion in which the function is separated and a key is forged. In the following description, this first stage of linear attack will be distinguished by being called combination attack, while the second stage will be called separation attack. The linear attack will refer to this combination attack and separation attack jointly.

[0135] The following theorems apply for separation attack.

[0136] Theorem 1 (separation attack against secret key):

[0137] A secret key s_zcan be forged in polynomial time by separation attack using a secret key when a public key v_zis subjected to combination attack using integer coefficients, when the secret key generation function is separable in polynomial time by an operation ◯, and when the inverse with respect to the operation ◯ is found in polynomial time.

[0138] Theorem 2 (separation attack against common key):

[0139] A common key K_y2can be forged in polynomial time by separation attack using only a common key when a public key v_zis subjected to combination attack using integer coefficients, when the secret key generation function is separable in polynomial time by an operation ◯, and when the inverse with respect to the operation ◯ is found in polynomial time.

[0140] The following conditions 4 and 5 should be met in order to build an ID-NIKS system that is secure against the separation attacks discussed above.

[0141] Condition 4 (Security of Secret Key Against Separation Attack)

[0142] It is difficult to separate the secret key generation function f in polynomial time.

[0143] Condition 5 (Security of Common Key Against Separation Attack)

[0144] It is difficult to separate the key sharing function F in polynomial time.

[0145] This condition 5 is extremely stringent, and simply means that if the functional form at the key sharing stage is separable, i.e., the key sharing function is separable□C then the system is not secure, regardless of intermediate calculations. For instance, a product-sum type of ID-NIKS and a power product type of ID-NIKS do not satisfy this condition.

[0146] With the present invention, the secret key generation function and the key sharing function are inseparable, and key sharing is probabilistically possible. Inseparability is accomplished by calculation over finite fields in which a large prime number serves as the modulus, and then eliminating the random numbers by addition over integer ring. The basis of security with the present invention is that when the size of a coefficient is limited, it is difficult to express any vector by the linear combination of the vectors of the conspirators. The collusion threshold is also raised by the use of random number vectors specific to each entity.

[0147] Also, the present invention involves the use of a plurality of public keys for each entity, and these plurality of public keys are combined to eliminate random numbers, so the complexity of the random number elimination is increased and higher security is achieved. Furthermore, because the random numbers are divided, the problem of digits being carried up is ameliorated.

[0148] FIG. 1 schematically illustrates the structure of the cryptographic communication system according to the present invention;

[0149] FIG. 2 illustrates communication of information between two entities in the cryptographic communication system shown in FIG. 1;

[0150] FIGS. 3 and 4 illustrate numerical examples according to the first embodiment;

[0151] FIGS. 5 and 6 illustrate numerical examples of the second embodiment according to the present invention;

[0152] FIG. 7 is a diagram illustrating the structure of another cryptographic communication system according to the present invention;

[0153] FIG. 8 is a diagram illustrating communication of information between two entities in the cryptographic communication system shown in FIG. 7; and

[0154] FIG. 9 illustrates the principle behind ID-NIKS.

[0155] Referring to FIG. 1, illustrated is a schematic structure of the cryptographic communication system according to the present invention. A center 1 that can be trusted to conceal information is set up. A public institution is an example of this center 1. The center 1 is connected to a plurality of entities a, b, . . . , z, which are the users of this cryptosystem, by private channels 2a, 2b, . . . , 2z, and secret key information from the center 1 is transmitted to the various entities a, b, . . . , z via these private channels 2a, 2b, . . . , 2z. Channels 3ab, 3az, 3bz, . . . are provided between each two entities, and a ciphertext produced by the encryption of information is transferred back and forth between the entities via these channels 3ab, 3az, 3bz, . . .

[0156] A first embodiment of the ID-NIKS according to the present invention will now be described. This embodiment deals with a basic scheme in which a personal random number is added to the secret key, and the random number portion is eliminated during sharing.

[0157] Preprocessing at the Center 1

[0158] The center 1 prepares the following public keys and secret keys, and discloses the public keys.

[0159] Public keys P and Q: large prime numbers of k bits

[0160] Secret keys A and B: n×n symmetric matrices composed of k-bit random numbers

[0161] Vector γ_i: personal random number vector composed of 1-bit random numbers

[0162] Also disclosed at the same time from the center 1 is a one-way function vector h(·) that generates an n-dimensional public key vector composed of m-bit positive integers from the ID of an entity concerned, which is specific information about that entity. Here, the inner product of the random number vector γ_iand the public key vector v_jof any entity should not exceed P and Q. Further, to simplify the discussion, parameter b will be set as in the following formula 1, and we will consider a case in which b>0. This b is a parameter related to the probability of key sharing and security, as will be discussed below. Approximate suitable values are k=1240, l=200, m=1000, n=1000, and b=30.

[0163] b=k−(l+m+log₂n)  (1)

[0164] Registration Processing of an Entity

[0165] Upon receiving a request for registration from entity i, the center 1 determines the vector s_iand vector t_iaccording to the following formulas 2 and 3 using the prepared keys and the public key vector v_i(=vector h(ID_i)) of entity i, privately sends the determined vector s_iand vector t_ito entity i, and completes the registration. The “mod” here is the modulus of a binomial operator. Specifically, “a mod P” expresses an operation for finding the remainder of dividing a by P.

[0166] {right arrow over (sA{right arrow over (v_i)})mod P  (2)

[0167] {right arrow over (tB{right arrow over (v_i)})mod Q  (3)

[0168] It should be noted that the above formula 3 may instead be expressed as in the following formula 3′, involving subtraction over integer ring in the final stage of key sharing. In this case, however, the common key K_ij(discussed below) may end up being a negative value. In the description that follows, the vector t_iwill be determined by formula 3.

[0169] {right arrow over (tB{right arrow over (v_i)})mod Q  (3′)

[0170] Processing for Generation of Common Key Between Entities

[0171] An entity i performs the following calculations to determine a common key K_ijwith an entity j. First, A_ij′ and B_ij′ are determined according to formulas 4 and 5 using modulus P and modulus Q, respectively, and then K_ijis obtained by the calculation in formula 6 over integer ring.

[0172] A_ij′=^t{right arrow over (s{right arrow over (vP  (4)

[0173] B_ij′=^t{right arrow over (t{right arrow over (vQ  (5)

[0174] K_ij=A_ij′+B_ij′  (6)

[0175] The communication of information between entities in the cryptosystem discussed above will now be described. Referring to FIG. 2, illustrated is communication of information between two entities a and b. In this example, the entity a encrypts a plaintext (message) M into a ciphertext C and sends it to the entity b, and the entity b decrypts this ciphertext C back into the original plaintext M.

[0176] The entity a side is equipped with a public key generator 11 that inputs personal identity information ID_bof the entity b and uses a hash function to obtain a vector v_b(public key), a common key generator 12 that generates a common key K_abwith the entity b, which the entity a asks for, on the basis of the secret vectors s_aand t_asent from the center 1 and the vector v_b, which is the public key provided from the public key generator 11, and an encryptor 13 that uses the common key K_abto encrypt the plaintext M into the ciphertext C and outputs it to a channel 30.

[0177] The entity b side is equipped with a public key generator 21 that inputs personal identity information ID_aof the entity a and uses a hash function to obtain a vector v_a(public key), a common key generator 22 that generates a common key K_bawith the entity a, which the entity b asks for, on the basis of the secret vectors s_band t_bsent from the center 1 and the vector v_a, which is the public key provided from the public key generator 21, and a decryptor 23 that uses the common key K_bato decrypt the ciphertext C into the plaintext M inputted from the channel 30 and outputs the result.

[0178] The operation will now be described. When the entity a attempts to send information to the entity b, first, the personal identity information ID_bof the entity b is inputted to the public key generator 11 to obtain the vector v_b(public key), and the vector v_bthus obtained is sent to the common key generator 12. The vectors s_aand t_adetermined according to formulas 2 and 3 are inputted to the common key generator 12 from the center 1. The common key K_abis then determined according to formulas 4 to 6, and sent to the encryptor 13. At the encryptor 13, this common key K_abis used to encrypt the plaintext M into the ciphertext C, and the ciphertext C is sent via the channel 30.

[0179] The ciphertext C that has been sent through the channel 30 is inputted to the decryptor 23 of the entity b. The personal identity information ID_aof the entity a is inputted to the public key generator 21 to obtain the vector v_a(public key), and the vector v_athus obtained is sent to the common key generator 22. The vectors s_band t_bdetermined by according to formulas 2 and 3 are inputted to the common key generator 22 from the center 1. The common key K_bais then determined according to formulas 4 to 6, and sent to the decryptor 23. At the decryptor 23, this common key K_bais used to decrypt the ciphertext C into the original plaintext M.

[0180] The feasibility of this method will now be proven.

[0181] It is shown below that if b in the above formula 1 is increased, K_ij=K_jiat a high probability. This is given as formulas 7 and 8. $\begin{array}{cc}{A}_{\mathrm{ij}}^{″}={\hspace{0.17em}}^t\overrightarrow{{\upsilon }_i}A\overrightarrow{{\upsilon }_j}\mathrm{mod}P& \left(7\right)\\ B_{\mathrm{ij}}^{″}={\hspace{0.17em}}^t\overrightarrow{{\upsilon }_i}B\overrightarrow{{\upsilon }_j}\mathrm{mod}Q& \left(8\right)\end{array}$

[0182] Given this, since A_ij′ is as shown in formula 9, either formula 10-1 or formula 10-2 holds true over integer ring. $\begin{array}{c}\begin{array}{cc}{A}_{\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_i}\overrightarrow{{\upsilon }_j}\mathrm{mod}P& \left(9\right)\\ ={\hspace{0.17em}}^t\left(A\overrightarrow{{\upsilon }_i}+\overrightarrow{{\gamma }_i}\right)\overrightarrow{{\upsilon }_j}\mathrm{mod}P& \\ =\left({\hspace{0.17em}}^t\overrightarrow{{\upsilon }_i}A\overrightarrow{{\upsilon }_j}+\hspace{0.17em}{\hspace{0.17em}}^t\overrightarrow{{\gamma }_i}\overrightarrow{{\upsilon }_j}\right)\mathrm{mod}P& \end{array}\\ A_{\mathrm{ij}}^{\prime }=\{\begin{array}{ccc}{A}_{\mathrm{ij}}^{″}+{\hspace{0.17em}}^t\overrightarrow{{\gamma }_i}\overrightarrow{{\upsilon }_j}& \left(A_{\mathrm{ij}}^{″}+{\hspace{0.17em}}^t\overrightarrow{{\gamma }_i}\overrightarrow{{\upsilon }_j}<P\right)& \left(10\text{-}1\right)\\ A_{\mathrm{ij}}^{″}+{\hspace{0.17em}}^t\overrightarrow{{\gamma }_i}\overrightarrow{{\upsilon }_j}-P& \left(A_{\mathrm{ij}}^{″}+{\hspace{0.17em}}^t\overrightarrow{{\gamma }_i}\overrightarrow{{\upsilon }_j}\ge P\right)& \left(10\text{-}2\right)\end{array}\end{array}$

[0183] Here, the inner product of the transposed vector of the random number vector γi and the public key vector v_j(the second term on the right side in formula 9) has added a (l+m)-bit number n times, and is a number whose magnitude is less than 2^l+m·n, and since A_ij″ can be k bits, adding the inner product of the transposed vector of γ_iand the vector v_jallows the probability that the whole will be greater than the modulus P to be estimated at roughly (2^l+m·n)/2^k=1/2^b. Therefore, the probability that formula 10-1 will hold true is approximately 1−1/2^b.

[0184] Similarly, B_ij′ is expressed by formula 11, either formula 12-1 or formula 12-2 holds true over integer ring, and formula 12-1 holds true at a probability of approximately 1−1/2^b. $\begin{array}{c}\begin{array}{cc}{B}_{\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{t_i}\overrightarrow{{\upsilon }_j}\mathrm{mod}Q& \left(11\right)\\ ={\hspace{0.17em}}^t\left(B\overrightarrow{{\upsilon }_i}-\overrightarrow{{\gamma }_i}\right)\overrightarrow{{\upsilon }_j}\mathrm{mod}Q& \\ =\left({\hspace{0.17em}}^t\overrightarrow{{\upsilon }_i}B\overrightarrow{{\upsilon }_j}+\hspace{0.17em}{\hspace{0.17em}}^t\overrightarrow{{\gamma }_i}\overrightarrow{{\upsilon }_j}\right)\mathrm{mod}Q& \end{array}\\ B_{\mathrm{ij}}^{\prime }=\{\begin{array}{ccc}{B}_{\mathrm{ij}}^{″}-{\hspace{0.17em}}^t\overrightarrow{{\gamma }_i}\overrightarrow{{\upsilon }_j}& \left(B_{\mathrm{ij}}^{″}+{\hspace{0.17em}}^t\overrightarrow{{\gamma }_i}\overrightarrow{{\upsilon }_j}\ge 0\right)& \left(12\text{-}1\right)\\ B_{\mathrm{ij}}^{″}-{\hspace{0.17em}}^t\overrightarrow{{\gamma }_i}\overrightarrow{{\upsilon }_j}+Q& \left(B_{\mathrm{ij}}^{″}+{\hspace{0.17em}}^t\overrightarrow{{\gamma }_i}\overrightarrow{{\upsilon }_j}<0\right)& \left(12\text{-}2\right)\end{array}\end{array}$

[0185] Therefore, K_ijis expressed by one of the following formulas 13-1, 13-2, 13-3, and 13-4.

[0186] K_ij=A_ij″+B_ij″  (13-1)

[0187] K_ij=A_ij″+B_ij″−P  (13-2)

[0188] K_ij=A_ij″+B_ij″+Q  (13-3)

[0189] K_ij=A_ij″+B_ij″−P+Q  (13-4)

[0190] The probability that formula 13-1 will hold true is approximately (1−1/2^b)²≈1−1/2^b−1, so formula 13-1 will hold true at an extremely high probability if b is made large enough. The probability that formulas 13-2 and 13-3 will hold true is approximately (1/2^b)·(1−1/2^b) for both, and the probability that formula 13-4 will hold true is approximately 1/2^2b, and all of these are extremely low probabilities.

[0191] This is exactly the same with K_ji, where formula 13-1 holds true at an extremely high probability of approximately (1−1/2^b)². Therefore, the probability that the two entities i and j can share a key, that is, the probability that K_ij=K_ji, is approximately (1−1/2^b)⁴≈1−1/2^b-2. Thus, it can be seen that a key can be shared at an extremely high probability in this first embodiment. In the case of the above-mentioned suitable values, b=30 is sufficiently large, so the probability of failure to share a key is only about 1/2²⁸.

[0192] Here, numerical examples of the first embodiment, in which we let n=1 (both the matrix and the vector are scalar), are shown in FIGS. 3 and 4. P, Q, A, and B are decimal and ten-digit, while the vectors v_i, v_j, γ_i, and γ_j(actually scalar) are each decimal and three-digit. Setting the number of digits in this way keeps the inner product of the vectors from reaching the number of digits of P and Q. In this example, the inner product has 6 digits, i.e., 3 digits×3 digits, so the first four digits are the same for A_ijand A_jiand for B_ijand B_ji.

[0193] Next, let us consider the security of the encryption scheme in this first embodiment.

[0194] Security of Secret Key Against Separation Attack

[0195] Because there are two secret keys with this scheme, there are two secret key generation functions f_Pand f_Q, as shown in formulas 14 to 16. However, we will let the right side of formula 14 be a function that generates an n-dimensional vector having a random number less than 2^las a component. $\begin{array}{cc}\overrightarrow{\gamma }=\overrightarrow{\varrho }\left(2^l\right)& \left(14\right)\\ f_P\left(\overrightarrow{x}\right)\equiv A\overrightarrow{x}+\overrightarrow{\gamma }\left(\mathrm{mod}P\right)& \left(15\right)\\ f_Q\left(\overrightarrow{x}\right)\equiv B\overrightarrow{x}-\overrightarrow{\gamma }\left(\mathrm{mod}Q\right)& \left(16\right)\end{array}$

[0196] We obtain the following formula 17 by substituting the vector x+vector y into the secret key generation function f_P. Meanwhile, formula 18 holds true. $\begin{array}{cc}{f}_P\left(\overrightarrow{x}+\overrightarrow{y}\right)\equiv A\left(\overrightarrow{x}+\overrightarrow{y}\right)+\overrightarrow{\varrho }\left(2^l\right)\left(\mathrm{mod}P\right)& \left(17\right)\\ \begin{array}{c}{f}_P\left(\overrightarrow{x}\right)+f_P\left(\overrightarrow{y}\right)\equiv A\overrightarrow{x}+\overrightarrow{\varrho }\left(2^l\right)+A\overrightarrow{y}+\overrightarrow{\varrho }\left(2^l\right)\\ \equiv A\left(\overrightarrow{x}+\overrightarrow{y}\right)+\overrightarrow{\varrho }\left(2^l\right)+\overrightarrow{\varrho }\left(2^l\right)\left(\mathrm{mod}P\right)\end{array}& \left(18\right)\end{array}$

[0197] Thus, we can see that with formula 18 the range over which the random number portion can be taken is twice as large as compared to formula 17. The effect of the random numbers becomes greater if the variables are further increased, so the secret key generation function f_Pcannot be separated. Similarly, the secret key generation function f_Qis also inseparable. Specifically, the situation is as shown in the following formulas 19 and 20, and condition 4 is met. $\begin{array}{cc}{f}_P\left(\overrightarrow{x}+\overrightarrow{y}\right)\not\equiv f_P\left(\overrightarrow{x}\right)+f_P\left(\overrightarrow{y}\right)\left(\mathrm{mod}P\right)& \left(19\right)\\ f_Q\left(\overrightarrow{x}+\overrightarrow{y}\right)\not\equiv f_Q\left(\overrightarrow{x}\right)+f_Q\left(\overrightarrow{y}\right)\left(\mathrm{mod}Q\right)& \left(20\right)\end{array}$

[0198] Security of Common Key Against Separation Attack

[0199] The key sharing function F is expressed by the following formulas 21 to 23. $\begin{array}{cc}\begin{array}{c}ℱ\left(\overrightarrow{x},\overrightarrow{y}\right)=\left({\hspace{0.17em}}^t\overrightarrow{x}A\overrightarrow{y}\mathrm{mod}P\right)+\left({\hspace{0.17em}}^t\overrightarrow{x}B\overrightarrow{y}\mathrm{mod}Q\right)\\ ℱ\left(\overrightarrow{x},\overrightarrow{y}\right)={\hspace{0.17em}}^t\overrightarrow{x}A\overrightarrow{y}-\mu \left(\overrightarrow{x},\overrightarrow{y}\right)P+{\hspace{0.17em}}^t\overrightarrow{x}B\overrightarrow{y}-\upsilon \left(\overrightarrow{x},\overrightarrow{y}\right)Q\\ ={\hspace{0.17em}}^t\overrightarrow{x}\left(A+B\right)\overrightarrow{y}-\mu \left(\overrightarrow{x},\overrightarrow{y}\right)P-\upsilon \left(\overrightarrow{x},\overrightarrow{y}\right)Q\end{array}& \left(21\right)\\ \mu \left(\overrightarrow{x},\overrightarrow{y}\right)=\lfloor \frac{{\hspace{0.17em}}^t\overrightarrow{x}A\overrightarrow{y}}{P}\rfloor & \left(22\right)\\ \upsilon \left(\overrightarrow{x},\overrightarrow{y}\right)=\lfloor \frac{{\hspace{0.17em}}^t\overrightarrow{x}B\overrightarrow{y}}{Q}\rfloor & \left(23\right)\end{array}$

[0200] The first term in formula 21 is a separable function, but the second and third terms are inseparable functions (with the round-off functions expressed in formulas 22 and 23, either f(x+y)=f(x)+f(y) or f(x)+f(y)+1 holds true, but separation is generally impossible if we consider multi-variable combination, and thus the function μ and the function ν that make use of a round-off function are inseparable), so this key sharing function F is inseparable. Specifically, the relationship of formula 24 is exhibited, and condition 5 is met. $\begin{array}{cc}ℱ\left(\overrightarrow{a},\overrightarrow{x}+\overrightarrow{y}\right)\ne ℱ\left(\overrightarrow{a},\overrightarrow{x}\right)+ℱ\left(\overrightarrow{a},\overrightarrow{y}\right)& \left(24\right)\end{array}$

[0201] Security Against Combination Attack Over Finite Fields

[0202] If we let the following formula 25 express a matrix V containing the public key vectors v_iof n number of linearly independent conspirators, then formulas 26 and 27 can be determined using P and Q as moduli for any vector v_z. $\begin{array}{cc}V=\left[\overrightarrow{{\upsilon }_1}\left|\overrightarrow{{\upsilon }_2}\right|\dots |\overrightarrow{{\upsilon }_{n^{\prime }}}\right]& \left(25\right)\\ \overrightarrow{u_{z\left(P\right)}}\equiv V^{-1}\overrightarrow{{\upsilon }_z}\left(\mathrm{mod}P\right)& \left(26\right)\\ \overrightarrow{u_{z\left(Q\right)}}\equiv V^{-1}\overrightarrow{{\upsilon }_z}\left(\mathrm{mod}Q\right)& \left(27\right)\end{array}$

[0203] As shown by formulas 28 and 29, the vector v_zcan be expressed by linear combination by n number of conspirators. $\begin{array}{cc}\overrightarrow{{\upsilon }_z}\equiv V\overrightarrow{u_{z\left(P\right)}}\left(\mathrm{mod}P\right)& \left(28\right)\\ \overrightarrow{{\upsilon }_z}\equiv V\overrightarrow{u_{z\left(Q\right)}}\left(\mathrm{mod}Q\right)& \left(29\right)\end{array}$

[0204] Here, if we give the following formula 30, the vector s_z′ can be determined as shown in formula 31 using P as the modulus. $\begin{array}{cc}\overrightarrow{u_{z\left(P\right)}}={\hspace{0.17em}}^t\left(u_{\mathrm{z1}\left(P\right)},u_{\mathrm{z2}\left(P\right)},\dots ,u_{\mathrm{zn}\left(P\right)}\right)& \left(30\right)\\ \begin{array}{c}{\overrightarrow{s_z}}^{\prime }\equiv u_{\mathrm{z1}\left(P\right)}\overrightarrow{s_1}+\dots +u_{\mathrm{zn}\left(P\right)}\overrightarrow{s_n}\\ \equiv u_{\mathrm{z1}\left(P\right)}A\overrightarrow{{\upsilon }_1}+u_{\mathrm{z1}\left(P\right)}\overrightarrow{{\gamma }_x}+\dots +u_{\mathrm{zn}\left(P\right)}A\overrightarrow{{\upsilon }_n}+u_{\mathrm{zn}\left(P\right)}\overrightarrow{{\gamma }_n}\\ \equiv A\left(u_{\mathrm{z1}\left(P\right)}\overrightarrow{{\upsilon }_x}+\dots +u_{\mathrm{zn}\left(P\right)}\overrightarrow{{\upsilon }_n}\right)+u_{\mathrm{z1}\left(P\right)}\overrightarrow{{\gamma }_x}+\dots +u_{\mathrm{zn}\left(P\right)}\overrightarrow{{\gamma }_n}\\ \equiv A\overrightarrow{{\upsilon }_z}+\Gamma \overrightarrow{u_{z\left(P\right)}}\left(\mathrm{mod}P\right)\end{array}& \left(31\right)\end{array}$

[0205] Where, $\Gamma =\left[\overrightarrow{{\gamma }_1}\left|\overrightarrow{{\gamma }_2}\right|\dots |\overrightarrow{{\gamma }_n}\right]$

[0206] Here, if we provide the following formula 32, this is expressed as in formula 33. $\begin{array}{cc}{\overrightarrow{{\gamma }_{z\left(P\right)}}}^{\prime }=\Gamma \overrightarrow{u_{z\left(P\right)}}& \left(32\right)\\ {\overrightarrow{s_z}}^{\prime }\equiv A\overrightarrow{{\upsilon }_z}+{\overrightarrow{{\gamma }_{z\left(P\right)}}}^{\prime }\left(\mathrm{mod}P\right)& \left(33\right)\end{array}$

[0207] Similarly, if. we provide the following formula 34, the vector t_z′ can be determined as shown in formula 35 using Q as the modulus. $\begin{array}{cc}{\overrightarrow{{\gamma }_{z\left(Q\right)}}}^{\prime }=\Gamma \overrightarrow{u_{z\left(Q\right)}}& \left(34\right)\\ \begin{array}{c}{\overrightarrow{t_z}}^{\prime }\equiv u_{\mathrm{z1}\left(Q\right)}\overrightarrow{t_1}+\dots +u_{\mathrm{zn}\left(Q\right)}\overrightarrow{t_n}\\ \equiv B\overrightarrow{{\upsilon }_z}-\Gamma \overrightarrow{u_{z\left(Q\right)}}\\ \equiv B\overrightarrow{{\upsilon }_z}-{\overrightarrow{{\gamma }_{z\left(Q\right)}}}^{\prime }\left(\mathrm{mod}Q\right)\end{array}& \left(35\right)\end{array}$

[0208] The random number portion can be eliminated when the random number term in formula 33 is equal to the random number term in formula 35, and its value is less than (1+b) bits. Accordingly, it is conceivable that the vector s_z′ determined through collusion would be used in place of the true secret key vector s_zof entity z. Nevertheless, it cannot be substituted for the true secret key vector s_zfor the following two reasons.

[0209] Reason 1: The left side of formula 26 and the left side of formula 27 generally consist of k bits, so the random number term of formula 33 and the random number term of formula 35 are sufficiently large.

[0210] Reason 2: The left side of formula 26 is not equal to the left side of formula 27, so the random number term of formula 33 and the random number term of formula 35 are not equal, and the random number portion cannot be eliminated.

[0211] Therefore, this first embodiment can be considered secure against combination attack over finite fields by n number of conspirators.

[0212] Security Against Combination Attack Over Integer Ring

[0213] It is conceivable that a combination attack over integer ring would be possible if a Euclidean algorithm were applied by n′ number of conspirators equal to or over (n+1) in number. Specifically, a coefficient vector u_z′ (component expressed by formula 38) that satisfies formula 37 over integer ring can be determined if a matrix V containing the public key vectors v_iof n′ number of conspirators is given as the following formula 36. $\begin{array}{cc}V=\left[\overrightarrow{{\upsilon }_1}\left|\overrightarrow{{\upsilon }_2}\right|\dots |\overrightarrow{{\upsilon }_{n^{\prime }}}\right]& \left(36\right)\\ \overrightarrow{{\upsilon }_z}=V\overrightarrow{u_z}& \left(37\right)\\ \overrightarrow{u_z}={\hspace{0.17em}}^t\left(u_{\mathrm{z1}},u_{\mathrm{z2}},\dots ,u_{{\mathrm{zn}}^{\prime }}\right)& \left(38\right)\end{array}$

[0214] Next, the vector s_z′ and the vector t_z′ are determined according to formulas 39 and 40. $\begin{array}{cc}\begin{array}{c}{\overrightarrow{s_z}}^{\prime }\equiv u_{\mathrm{z1}}\overrightarrow{s_1}+\dots +u_{{\mathrm{zn}}^{\prime }}\overrightarrow{s_{n^{\prime }}}\\ \equiv A\overrightarrow{{\upsilon }_z}+{\overrightarrow{{\gamma }_z}}^{\prime }\left(\mathrm{mod}P\right)\end{array}& \left(39\right)\\ \begin{array}{c}{\overrightarrow{t_z}}^{\prime }\equiv u_{\mathrm{z1}}\overrightarrow{t_1}+\dots +u_{{\mathrm{zn}}^{\prime }}\overrightarrow{t_{n^{\prime }}}\\ \equiv B\overrightarrow{{\upsilon }_z}-{\overrightarrow{{\gamma }_z}}^{\prime }\left(\mathrm{mod}Q\right)\end{array}& \left(40\right)\end{array}$

[0215] Where, ${\overrightarrow{{\gamma }_z}}^{\prime }=\Gamma \overrightarrow{u_z}$

[0216] If the vector s_z′ and vector t_z′ thus determined are compared to the true secret key vector s_zand secret key vector t_z, the random number term vectors γ_z′ are both larger, but when the values of |u_zi| are all small, and the size of the vectors γ_z′ is less than (1+b) bits, the relationship of the following formula 41 holds true with respect to any public key vector v_j, and the vectors s_z′ and t_z′ play the same role as the true secret key vectors s_zand t_z. $\begin{array}{cc}{}^t\overrightarrow{{\gamma }_z}^{\prime }\overrightarrow{{\upsilon }_j}<P& \left(41\right)\end{array}$

[0217] If the conspirators decide an entity z to be attacked and are (fortunately) able to find their vectors v_iand coefficient vector u_zso that the vector v_zcan be expressed by combination of small coefficients, then this system is not always secure. Nevertheless, finding this coefficient vector u_zis generally very difficult. Thus, when the size of the coefficient is limited, the basis of security is the fact that it is difficult to express any public key vector by linear combination of the conspirators public key vectors.

[0218] Probability that Combination Attack Over Integer Ring will Work

[0219] As discussed above, an attack by conspirators will work if the value of |u_zi| is small in relation to all the components u_ziof the coefficient vector u_z. Since it is not easy to consider the probability that the entity z to be attacked will be decided on and that the vector v_zcan be expressed by the linear combination of a public key using a small coefficient, herein we will shift our focus and discuss the probability of successful attack by concentrating on the total number of public keys with which an attack will succeed when N number of conspirators gather together. We will assume that the attack will work if the size of the coefficient can be limited to w (≦b) bits. In this case, the probability that a common key produced using a forged secret key will coincide with the correct common key is 1−1/2^b-w.

[0220] When the attack is attempted by one person, the public keys of 2^2wpeople can be expressed by the vector v₁of one person if a coefficient whose size is limited to w bits is used. If we let a gathering of these public. keys be V₁, then V₁is expressed by the following formula 42.

[0221] V₁={u₁{right arrow over (vu₁=0, 1, . . . , 2^w−1}  (42)

[0222] When the attack is colluded by two people, similarly, the public keys of 2^2wpeople can be expressed using the vector v₁and the vector v₂. If we let a gathering of these public keys be V₂, then V₂is expressed by the following formula 43.

[0223] V₂={u₁{right arrow over (vu₂{right arrow over (vu₁, u₂=0, 1, . . . , 2^w−1}  (43)

[0224] Similarly, when the attack is planed by N people, the public keys of a maximum of 2^Nwpeople can be expressed by the linear combination of the vector v₁, vector v₂, . . . , vector V_N. In order to choose conspirators most efficiently, the vector v_ishould be selected such that the condition of formula 44 is met.

[0225] {right arrow over (vV_i−1  (44)

[0226] The total number of public keys is 2^mn, while the maximum number of public keys with which an attack by N conspirators will work is 2^Nw, so the probability that a combination attack will work against an arbitrary entity is 2^Nw/2^mn. Furthermore, the probability that a true common key can be forged for use with an arbitrarily selected other entity is (2^Nw/2^mn)·(1−1/2^b-w) If we let N be the collusion threshold n of a conventional product-sum type of NIKS, then even if w=b, the probability that a combination attack will work is 1/2^(m-b)n, and is only 1/2⁹⁷⁰⁰⁰⁰when the proper value is used.

[0227] Lower Limit to Collusion Threshold at Which a Combination Attack Over Integer Ring Will Work

[0228] With a combination attack against a conventional product-sum type of NIKS, there is no limit to the size of the coefficient vector, so if n number of linearly independent public key vectors are chosen, all 2^mnof the public key can be expressed through linear combination by these n conspirators. However, if the size of the coefficient is limited to w bits, then the total number of gathered vectors V_Nexpressed by linear combination using the public keys of N people is just 2^Nwat most.

[0229] Assuming that there are N people in collusion, N must satisfy the condition of the following formula 45, that is, formula 46, in order for all of the 2^mnpublic keys of these N people to be expressed by linear combination.

[0230] 2^Nw≧2^mn  (45)

[0231] N≧mn/w  (46)

[0232] Thus, it can be.concluded that this first embodiment is an encryption system with which the lower limit of the collusion threshold is m/w times larger than that of a power product type of NIKS system.

[0233] Actually, the vector u_zthat satisfies the above formula 37 will not be unique even though it is known to be the vector v_zεV_N, and finding among them a vector whose coefficients are all smaller than m bits is believed to be exceedingly difficult.

[0234] When the proper value is used, if the conspirators are anticipating that the forgery of the common key with an arbitrary entity will be successful at a probability of at least (1−1/2¹⁰), then they can only take a size up to w=20, and it can be seen that the collusion threshold will be m/w=1000/20=50, which is about 50 times higher than that of an ordinary product-sum type of NIKS system.

[0235] A second embodiment of the ID-NIKS according to the present invention will now be described. This is a modification to the first embodiment.

[0236] It has been shown in the first embodiment that attack will be difficult if b is set low. On the other hand, if b is set low, there is a lower probability that the proper entities can correctly share keys. Specifically, with the first embodiment, feasibility and security are in a trade-off relationship in which b is a parameter.

[0237] The second embodiment is an encryption system that allows keys to be correctly shared even if b is made further smaller, which is an improvement over the first embodiment. In this second embodiment, a practical ID-NIKS system that is extremely powerful against combination attack can be constructed by lowering b to the limit of zero.

[0238] Preprocessing at the Center 1

[0239] The center 1 prepares the following public keys and secret keys, and discloses the public keys.

[0240] Public keys R: prime number of d bits

[0241] P and Q: large prime numbers of k bits

[0242] Secret keys A and B: n×n symmetric matrices composed, of k-bit random numbers

[0243] Vector γ_i: personal random number vector composed of 1-bit random numbers

[0244] The center 1 generates a relatively small prime number R of d bits in the above-mentioned public keys, and generates prime numbers P and Q of k bits so as to satisfy the following formulas 47 and 48.

[0245] P≡δ(mod R)  (47)

[0246] Q≡ε(mod R)  (48)

[0247] Here, δ and ε are numbers of e bits, and k>d>e.

[0248] Also disclosed at the same time is a one-way function vector h(·) that generates an n-dimensional public key vector composed of m-bit positive integers from the ID of an entity. Here, b=k−(l+m+log₂n) (≧0) is set to be extremely small. Approximate suitable values are k=1210, l=200, m=1000, n=1000, b=0, d=500, e=50, and c=50 (the meaning of c will be discussed below).

[0249] Registration Processing of an Entity

[0250] Upon receiving a request for registration from an entity i, the center 1 determines the vector s_iand vector t_iaccording to the following formulas 49 and 50 using the prepared keys and the public key vector v_i(=vector h(ID_i)) of the entity i, privately sends the determined vector s_iand vector t_ito the entity i, and completes the registration.

[0251] {right arrow over (sA{right arrow over (v_i)})mod P  (49)

[0252] {right arrow over (tB{right arrow over (v_i)})mod Q  (50)

[0253] Processing for Generation of Common Key Between Entities

[0254] The entity i performs the following calculations to determine a common key K_ijwith an entity j. First, A_ij′ and B_ij′ are determined according to formulas 51 and 52 using modulus P and modulus Q, respectively, and then K_ij′ is obtained by the calculation in formula 53 over integer ring. The K_ij′ thus determined is divided by R and the remainder K_ij″ is determined as shown in formula 54. The remainder K_ij″ is then rounded off as shown in formula 55 to obtain K_ij. $\begin{array}{cc}{A}_{\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_i}\overrightarrow{{\upsilon }_j}\mathrm{mod}P& \left(51\right)\\ B_{\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{t_i}\overrightarrow{{\upsilon }_j}\mathrm{mod}Q& \left(52\right)\\ K_{\mathrm{ij}}^{\prime }=A_{\mathrm{ij}}^{\prime }+B_{\mathrm{ij}}^{\prime }& \left(53\right)\\ K_{\mathrm{ij}}^{″}=K_{\mathrm{ij}}^{\prime }\mathrm{mod}R& \left(54\right)\\ K_{\mathrm{ij}}=\lfloor \frac{K_{\mathrm{ij}}^{″}}{2^{e+c}}\rfloor & \left(55\right)\end{array}$

[0255] The operation in which information is communicated between entities is the same as in the first embodiment, and so will not be described here.

[0256] The feasibility of this scheme will now be proven.

[0257] It is shown below that if c is large enough, K_ij=K_jiat a high probability. Just as in the first embodiment, K_ij′ is expressed by one of the following formulas 56-1, 56-2, 56-3, and 56-4.

[0258] K_ij′=A_ij″+B_ij″  (56-1)

[0259] K_ij′=A_ij″+B_ij″−P  (56-2)

[0260] K_ij′=A_ij″+B_ij″+Q  (56-3)

[0261] K_ij′=A_ij″+B_ij″−P+Q  (56-4)

[0262] K_ij″ is expressed by one of the following formulas 57-1, 57-2, 57-3, and 57-4.

[0263] K_ij″≡A_ij″+B_ij″(mod R)  (57-1)

[0264] K_ij″≡A_ij″+B_ij″−δ(mod R)  (57-2)

[0265] K_ij″≡A_ij″+B_ij″+ε(mod R)  (57-3)

[0266] K_ij″≡A_ij″+B_ij″−δ+ε(mod R)  (57-4)

[0267] δ and ε are e bits, so by rounding off the last e+c bits, K_ij=K_jiat a probability of 1−1/2^cor higher. Strictly speaking, the probability that carrying down will result from a subtraction of δ and the probability that carrying up will result from an addition of ε must be taken into account, but in both cases this probability is 1/2^d-e, and is extremely small if d is made large enough. For instance, when the proper value is used, this value is no more than 1/2⁴⁵⁰. Thus, it can be seen that keys can be shared at an extremely high probability with the second embodiment by increasing c.

[0268] FIGS. 5 and 6 show numerical examples of the second embodiment, where n=1 (both the matrix-and the vector are scalar). P, Q, A, and B are decimal and ten-digit, while R is decimal and eight-digit, and the vectors v_i, v_j, γ_i, and γ_j(actually scalar) are each decimal and five-digit. δ and ε are decimal and one-digit, and in the final stage they are decimal numbers with the lower two digits-rounded off.

[0269] Next, let us consider the security of the encryption system in this second embodiment.

[0270] The remainder operation in which R is the modulus contributes only to increasing the key sharing probability without a loss of security, and the second embodiment is based on exactly the same security as the first embodiment. Also, with the second embodiment, an inseparable rounding-off operation is further used in the final stage of the common key generation processing, so security against attack using a common key is enhanced.

[0271] Furthermore, key sharing is possible with any of formulas 56-1 to 56-4, regardless of the size of b, if c is set large enough, and b can be set as low as zero. For example, when the suitable values are used, b=0, and therefore w=0, in which case the value of the right side of formula 46 becomes infinitely large, and attack by collusion becomes unfeasible. In this case, the inner product of the public key vector v_jand the random number vector γ_iis k bits, which covers all the digits.

[0272] FIG. 7 is a diagram illustrating the structure of another cryptographic communication system according to the present invention. A center 101 and a plurality of entities a, b, . . . , z are connected to each other by channels 102a, 102b, . . . , 102z, as well as channels 103ab, 103bz, . . . and 103az, like the system shown in FIG. 1. These channels are substantially the same as FIG. 1 so that description thereof is omitted here.

[0273] A third embodiment of the ID-NIKS of the present invention will now be described. In this embodiment, two public keys are used for each entity.

[0274] Preprocessing at the Center 101

[0275] A center 101 prepares the following public keys and secret keys, and discloses the public keys.

[0276] Public keys P₁, P₂, and P₃: large prime numbers of k bits

[0277] Secret keys

[0278] A₁: n×n symmetric matrices composed of random numbers belonging to GF (P₁)

[0279] A₂: n×n symmetric matrices composed of random numbers belonging to GF (P₂)

[0280] A₃: n×n matrices composed of random numbers belonging to GF (P₃)

[0281] Vectors γ_1,iand γ_2,i: personal random number vector composed of 1-bit random numbers

[0282] Also disclosed at the same time are two one-way function vectors h₁(·) and h₂(·) that each generate an n-dimensional public key vector composed of m-bit positive integers from the ID of an entity, which is specific information about the entity. The parameter b here is set as in the following formula, and we will consider a case in which b>0.

[0283] b=k−(l+m+log₂n)

[0284] Registration Processing of an Entity

[0285] Upon receiving a request for registration from an entity i, the center 101 determines the first secret key vector s_11,i, second secret key vector s_12,i, third secret key vector s_21,i, and fourth secret key vector s_22,ifor the entity i according to the following formulas 58 to 61 using the prepared keys and the first public key vector v_1,i(=vector h₁(ID_i)) and second public key vector v_2,i(=vector h₂(ID_i)) of the entity i. The center 101 then privately sends the determined vectors to the entity i, and completes the registration. The “mod” here is the modulus of a binomial operator. Specifically, “a mod P” expresses an operation for finding the remainder of dividing a by P.

[0286] {right arrow over (sA₁{right arrow over (v_1,i)})mod P₁  (58)

[0287] {right arrow over (sA₂{right arrow over (v_2,i)})mod P₂  (59)

[0288] {right arrow over (sA₃{right arrow over (v_2,i)})mod P₃  (60)

[0289] {right arrow over (s^tA₃{right arrow over (v_1,i)})mod P₃  (61)

[0290] Processing for Generation of Common Key Between Entities

[0291] The entity i performs the following calculations to determine a common key K_ijwith an entity j. First, A_11,ij′ and A_22,ij′ are determined according to formulas 62 and 63 using modulus P₁and modulus P₂, respectively, then A_21,ij′ and A_12,ij′ are determined according to formulas 64 and 65, respectively, using modulus P₃, and then K_ijis obtained by the calculation in formula 66 over integer ring.

[0292] A_11,ij′=^t{right arrow over (s{right arrow over (vP₁  (62)

[0293] A_22,ij′=^t{right arrow over (s{right arrow over (vP₂  (63)

[0294] A_21,ij′=^t{right arrow over (s{right arrow over (vP₃  (64)

[0295] A_12,ij′=^t{right arrow over (s{right arrow over (vP₃  (65)

[0296] K_ij=A_11,ij′+A_22,ij′+A_21,ij′+A_12,ij′  (66)

[0297] Here, if b is sufficiently large, K_ij=K_jiat an overwhelmingly high probability.

[0298] The communication of information between entities in the cryptosystem discussed above will now be described. FIG. 8 illustrates communication of information between particular two entities a and b. In this example, the entity a encrypts a plaintext M into a ciphertext C and sends it to the entity b, and the entity b decrypts this ciphertext C back into the original plaintext M.

[0299] The entity a side is equipped with a public key generator 111 that inputs personal identity information ID_bof the entity b and uses a hash function to obtain vectors v_1,band v_2,b(public keys), a common key generator 112 that generates a common key K_abwith the entity b, which the entity a asks for, on the basis of the secret vectors s_11,a, s_22,a, s_21,a, and s_12,asent from the center 101 and the vectors v_1,band v_2,b, which are the public keys provided from the public key generator 111, and an encryptor 113 that uses the common key K_abto encrypt the .plaintext M into the ciphertext C and outputs it to a channel 130.

[0300] The entity b side is equipped with a public key generator 121 that inputs personal identity information ID_aof the entity a and uses a hash function to obtain vectors v_1,aand v_2,a(public keys), a common key generator 122 that generates a common key K_bawith the entity a, which the entity b asks for, on the basis of the secret vectors s_11,b, s_22,b, s_21,b, and s_12,bsent from the center 101 and the vectors v_1,aand v_2,a, which are the public keys from the public key generator 121, and a decryptor 123 that uses the common key K_bato decrypt the ciphertext C into the plaintext M inputted from the channel 130 and outputs the result.

[0301] The operation will now be described. When the entity a attempts to send information to the entity b, first, the personal identity information ID_bof the entity b is inputted to the public key generator 111 to obtain the vectors v_1,band v_2,b(public keys), and the vectors v_1,band v_2,band thus obtained are sent to the common key generator 112. The vectors s_11,a, s_22,a, s_21,a, and s_12,adetermined by the center 101 according to formulas 58 to 61 are inputted to the common key generator 112. The common key K_abis then determined according to formulas 62 to 66, and sent to the encryptor 113. At the encryptor 113, this common key K_abis used to encrypt the plaintext M into the ciphertext C, and the ciphertext C is sent via the channel 130.

[0302] The ciphertext C that has been sent through the channel 130 is inputted to the decryptor 123 of the entity b. The personal identity information ID_aof the entity a is inputted to the public key generator 121 to obtain the vectors v_1,aand v_2,a(public keys), and the vectors v_1,aand v_2,athus obtained are sent to the common key generator 122. The vectors s_11,b, s_22,b, s_21,b, and s_12,bdetermined by the center 101 according to formulas 58 to 61 are inputted to the common key generator 122. The common key K_bais then determined according to formulas 62 to 66, and sent to the decryptor 123. At the decryptor 123, this common key K_bais used to decrypt the ciphertext C into the plaintext M.

[0303] The formal expression of this encryption system will now be described. In order to facilitate an understanding of the encryption system in this third embodiment, the formulas 58 to 61 given above are defined by corresponding sub-matrices as shown below. It should be noted that a multiplying operation applied to the matrices and vectors in this formal expression is different from an ordinary multiplying operation applied to matrices in that addition is not carried out over partitions between sub-matrices. The symbols and/or numerals in ( ) designate a module. $\left(\begin{array}{cc}\overrightarrow{s_{11,i}}& \overrightarrow{s_{12,i}}\\ \overrightarrow{s_{21,i}}& \overrightarrow{s_{22,1}}\end{array}\right)=\left(\begin{array}{cc}{A}_1\left(P_1\right)& {}^{t}A_3\left(P_3\right)\\ A_3\left(P_3\right)& A_2\left(P_2\right)\end{array}\right)\left(\frac{\overrightarrow{{\upsilon }_{1,i}}}{\overrightarrow{{\upsilon }_{2,i}}}\right)+\left(\begin{array}{cc}\overrightarrow{{\gamma }_{1,i}}& -\overrightarrow{{\gamma }_{1,i}}\\ -\overrightarrow{{\gamma }_{2,i}}& \overrightarrow{{\gamma }_{2,i}}\end{array}\right)$

[0304] The above-mentioned formulas 62 to 65 are expressed as follows. $\left(\begin{array}{cc}{A}_{11,\mathrm{ij}}^{\prime }& A_{12,\mathrm{ij}}^{\prime }\\ A_{21,\mathrm{ij}}^{\prime }& A_{22,\mathrm{ij}}^{\prime }\end{array}\right)={\hspace{0.17em}}^t\left(\begin{array}{cc}\overrightarrow{s_{11,i}}\left(P_1\right)& \overrightarrow{s_{12,i}}\left(P_3\right)\\ \overrightarrow{s_{21,i}}\left(P_3\right)& \overrightarrow{s_{22,i}}\left(P_2\right)\end{array}\right)\left(\frac{\overrightarrow{{\upsilon }_{1,i}}}{\overrightarrow{{\upsilon }_{2,i}}}\right)$

[0305] For matrix A, if the operation of determining the sum of all the sub-matrices of A is expressed as ΣA, then the common key K_ijcan be expressed as follows. $K_{\mathrm{ij}}=\sum \left(\begin{array}{cc}{A}_{11,\mathrm{ij}}^{\prime }& A_{12,\mathrm{ij}}^{\prime }\\ A_{21,\mathrm{ij}}^{\prime }& A_{22,\mathrm{ij}}^{\prime }\end{array}\right)$

[0306] According to this formal expression, the third embodiment is considered to be a system that makes use of a symmetric matrix with the modulus being also included.

[0307] A fourth embodiment of the ID-NIKS according to the present invention will now be described. This embodiment deals with a generalized system in which d number of public keys are used for each entity. Specifically, in this fourth embodiment, d number of public keys are used, with their size being expressed by n₁, n₂, . . . , n_d, respectively. An n×n symmetric matrix A is readied, which is divided in the row and column directions into n₁, n₂, . . . , n_dto express sub-matrices A_yz(y and z are 1, 2, . . . , d). Here, we will let n be n₁+n₂+ . . . +n_d.

[0308] Because A is a symmetric matrix, A_yz=^tA_zyholds true with respect to each (y, z) sub-matrix of the matrix A. We will assume that the moduli are the same within each sub-matrix, and P_yz=P_yzis satisfied in relation to the modulus. Furthermore, since the random number term must be eliminated during common key generation, we will let the vector ∘ be a zero vector that satisfies the following formula 67. In this case, the secret key of the entity i is expressed by d²number of formulas. $\begin{array}{cc}\sum _{z=1}^d\overrightarrow{{\gamma }_{z,i}}=\overrightarrow{o}& \left(67\right)\end{array}$

[0309] Preprocessing at the Center 101

[0310] The center 101 prepares the following public keys and secret keys, and discloses the public keys.

[0311] Public keys P_yz: large prime numbers of k bits

[0312] Secret keys A_yz: n_y×n_zmatrices composed of random numbers belonging to GF (P_yz)

[0313] Vectors γ_z,i: n-dimensional personal random number vectors composed of 1-bit random numbers (y and z=1, 2, . . . , d)

[0314] Also disclosed at the same time is a one-way function vector h(·) that generates an n-dimensional public key vector composed of m-bit positive integers from the ID of an entity. The parameter b_r(r=1, 2, . . . , d) is set as shown by the following formula, and we will consider a case when b_r>0.

[0315] b_r=k−(l+m+log₂n_r)

[0316] Registration Processing of an Entity

[0317] Upon receiving a request for registration from an entity i, the center 101 determines d²number of secret key vectors s_yz,ifor the entity i according to the following formula 68 using the prepared keys and the public key vector v_i(=vector h(ID_i)) of the entity i, privately sends these vectors to the entity i, and completes the registration. The vector v_z,iand vector γ_yz,iexpress vectors obtained by dividing the column vector v_iand vector γ_z,iin the column direction, respectively, at a size of n₁, n₂, . . . , n_d. The “mod” here is the modulus of a binomial operator.

[0318] {right arrow over (sA_yz{right arrow over (v_yz,i)})mod P_yz  (68)

[0319] Processing for Generation of Common Key Between Entities

[0320] The entity i performs the following calculations to determine a common key K_ijwith an entity j. First, A_yz,ij′ is determined according to formula 69 using the respective moduli P_yz, and K_ijis then obtained by the calculation of formula 70 over integer ring. $\begin{array}{cc}{A}_{\mathrm{yz},\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_{\mathrm{yz},i}}\overrightarrow{{\upsilon }_{y,j}}\mathrm{mod}{P}_{\mathrm{yz}}& \left(69\right)\\ K_{\mathrm{ij}}=\sum _{y=1}^d\sum _{z=1}^d{A}_{\mathrm{yz},\mathrm{ij}}^{\prime }& \left(70\right)\end{array}$

[0321] Proof will now be given for the key sharing in the fourth embodiment (K_ij=K_jiholds true). If we substitute formula 68 into formula 69, we obtain the following formulas 71 and 72. When all of the b_rvalues are large enough, the first formula of formula 72 holds true at a high probability. In this case, the random number term will be eliminated from formula 67 by calculation of formula 70, so K_ij=K_jiholds true at a high probability. $\begin{array}{cc}{A}_{\mathrm{yz},\mathrm{ij}}^{\prime }=\left({\hspace{0.17em}}^t\overrightarrow{{\upsilon }_{z,i}}{A}_{\mathrm{zy}}\overrightarrow{{\upsilon }_{y,j}}+{\hspace{0.17em}}^t\overrightarrow{{\gamma }_{\mathrm{yz},i}}\overrightarrow{{\upsilon }_{y,j}}\right)\mathrm{mod}{P}_{\mathrm{yz}}& \left(71\right)\\ =\{\begin{array}{c}{\hspace{0.17em}}^t\overrightarrow{{\upsilon }_{z,i}}{A}_{\mathrm{zy}}\overrightarrow{{\upsilon }_{y,j}}\mathrm{mod}{P}_{\mathrm{yz}}+{\hspace{0.17em}}^t\overrightarrow{{\gamma }_{\mathrm{yz},i}}\overrightarrow{{\upsilon }_{y,j}}\\ {\hspace{0.17em}}^t\overrightarrow{{\upsilon }_{z,i}}{A}_{\mathrm{zy}}\overrightarrow{{\upsilon }_{y,j}}\mathrm{mod}{P}_{\mathrm{yz}}+{\hspace{0.17em}}^t\overrightarrow{{\gamma }_{\mathrm{yz},i}}\overrightarrow{{\upsilon }_{y,j}}-P_{\mathrm{yz}}\end{array}& \left(72\right)\end{array}$

[0322] The operation in which information is communicated between entities is the same as in the third embodiment, and so will not be described here.

[0323] If we use the above-mentioned formal expression for this fourth embodiment, the d²number of secret keys of the entity i can be all be expressed together as follows. $\left(\begin{array}{ccc}\overrightarrow{s_{11,i}}& \dots & \overrightarrow{s_{1d,i}}\\ ⋮& ⋰& ⋮\\ \overrightarrow{s_{\mathrm{d1},i}}& \dots & \overrightarrow{s_{\mathrm{dd},i}}\end{array}\right)=\left(\begin{array}{ccc}{A}_{11}\left(P_{11}\right)& \dots & A_{1d}\left(P_{1d}\right)\\ ⋮& ⋰& ⋮\\ A_{\mathrm{d1}}\left(P_{\mathrm{d1}}\right)& \dots & A_{\mathrm{dd}}\left(P_{\mathrm{dd}}\right)\end{array}\right)\left(\begin{array}{c}\overrightarrow{{\upsilon }_{1,i}}\\ ⋮\\ \overrightarrow{{\upsilon }_{d,i}}\end{array}\right)+\left(\begin{array}{ccc}\overrightarrow{{\gamma }_{11,i}}& \dots & \overrightarrow{{\gamma }_{1d,i}}\\ ⋮& ⋰& ⋮\\ \overrightarrow{{\gamma }_{\mathrm{d1},i}}& \dots & \overrightarrow{{\gamma }_{\mathrm{dd},i}}\end{array}\right)$

[0324] The common key K_ijcan be expressed as follows. $K_{\mathrm{ij}}=\sum {\hspace{0.17em}}^t\left(\begin{array}{ccc}\overrightarrow{s_{11,i}}& \dots & \overrightarrow{s_{1d,i}}\\ ⋮& ⋰& ⋮\\ \overrightarrow{s_{\mathrm{d1},i}}& \dots & \overrightarrow{s_{\mathrm{dd},i}}\end{array}\right)\left(\begin{array}{c}\overrightarrow{{\upsilon }_{1,j}}\\ ⋮\\ \overrightarrow{{\upsilon }_{d,j}}\end{array}\right)$

[0325] A fifth embodiment of the ID-NIKS of the present invention will now be described. This embodiment relates to a generalized system in which a plurality of sets of the public keys in the fourth embodiment are used. Specifically, the fifth embodiment is a generalized system in which the random numbers are eliminated only when a plurality of sets (c sets) of the public keys from the fourth embodiment are readied and the sum of all of these c sets is taken. In the following example, we will describe a case in which c=2, d number of public keys are used, with their size being expressed by n₁, n₂, . . . , n_d, and e number of public keys are used with their size being expressed by n₁″, n₂″, . . . , n_e″.

[0326] Preprocessing at the Center 101

[0327] The center 101 prepares the following public keys and secret keys, and discloses the public keys.

[0328] Public keys P_yzQ_yz: large prime numbers of k bits

[0329] Secret keys A_yz: n_y×n_zmatrices composed of random numbers belonging to GF (P_yz)

[0330] B_yz: n_y×n_zmatrices composed of random numbers belonging to GF (Q_yz)

[0331] Vectors γ_z,i: n-dimensional personal random number vectors composed of 1-bit random numbers

[0332] Vectors δ_z,i: n-dimensional personal random number vectors composed of 1-bit random numbers

[0333] Registration Processing of an Entity

[0334] Upon receiving a request for registration from an entity i, the center 101 determines d²number of secret key vectors s_yz,iand e²number of secret key vectors t_yz,ifor the entity i according to the following formulas 73 and 74 using the prepared keys and the public key vector v_i(=vector h(ID_i)) of the entity i, privately sends these vectors to the entity i, and completes the registration. The vector v_z,iand vector γ_yz,iexpress vectors obtained by dividing the column vector v_iand vector γ_z,i, respectively, in the column direction at a size of n₁, n₂, . . . , n_d, the vector v_z,i″ and vector δ_yz,iexpress vectors obtained by dividing the column vector v_iand vector δ_z,i, respectively, in the column direction at a size of n₁″, n₂″, . . . , n_e″, and the “mod” here is the modulus of a binomial operator. Here, the vector v_iis different only in the method of division, and the same vectors are used for both formulas 73 and 74.

[0335] {right arrow over (sA_yz{right arrow over (v_yz,i)})mod P_yz  (73)

[0336] (y, z=1, 2, . . . , d)

[0337] $\begin{array}{cc}{t}_{\mathrm{yz},i}=\left(B_{\mathrm{yz}}{\overrightarrow{{\upsilon }_{z,i}}}^{″}+\overrightarrow{{\delta }_{\mathrm{yz},i}}\right)\mathrm{mod}{Q}_{\mathrm{yz}}\left(y,z=1,2,\dots ,e\right)& \left(74\right)\end{array}$

[0338] The random number terms should be determined in such a manner that the following formula 75 is satisfied. By doing so, the random number terms will be eliminated only when the above two sets of keys are used. $\begin{array}{cc}\sum _{z=1}^d\overrightarrow{{\gamma }_{z,i}}+\sum _{z=1}^e\overrightarrow{{\delta }_{z,i}}=\overrightarrow{o}& \left(75\right)\end{array}$

[0339] Processing for Generation of Common Key Between Entities

[0340] The entity i performs the following calculations to determine a common key K_ijwith an entity j. First, A_yz,ij′ and B_yz,ij′ are determined according to formulas 76 and 77 using the respective moduli P_yzand Q_yz, and K_ijis then obtained by the calculation of formula 78 over integer ring. $\begin{array}{cc}{A}_{\mathrm{yz},\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_{\mathrm{yz},i}}\overrightarrow{{\upsilon }_{y,j}}\mathrm{mod}{P}_{\mathrm{yz}}\left(y,z=1,2,\dots ,d\right)& \left(76\right)\\ B_{\mathrm{yz},\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{t_{\mathrm{yz},i}}{\overrightarrow{{\upsilon }_{y,j}}}^{\prime }\mathrm{mod}{Q}_{\mathrm{yz}}\left(y,z=1,2,\dots ,e\right)& \left(77\right)\\ K_{\mathrm{ij}}=\sum _{y=1}^d\sum _{z=1}^d{A}_{\mathrm{yz},\mathrm{ij}}^{\prime }+\sum _{y=1}^e\sum _{z=1}^e{B}_{\mathrm{yz},\mathrm{ij}}^{\prime }& \left(78\right)\end{array}$

[0341] Because the random number term is eliminated if a sum of all the terms is taken from formula 75, K_ij=K_jiholds true at a high probability when all of the b_rvalues are sufficiently large, just as in the fourth embodiment.

[0342] The operation in which information is communicated between entities is the same as in the third embodiment, and so will not be described here.

[0343] A sixth embodiment of the ID-NIKS according to the present invention will now be described. This embodiment relates to a system in which the usable number of secret keys is limited to 2d. The fourth embodiment requires a large number (d(d+1)/2) of secret keys, and the fifth embodiment requires approximately c times more secret keys. Therefore, it is impractical to use these systems as they are. In view of this, a practical system that keeps the number of secret keys used down to 2d will be described as the sixth embodiment.

[0344] Preprocessing at the Center 101

[0345] The center 101 prepares the following public key and secret key, and discloses the public key.

[0346] Public key P_r: large prime number of k bits

[0347] Secret key A_r: n×n matrix composed of random numbers belonging to GF (P_r)

[0348] Vector γ_r,i: personal random number vector composed of 1-bit random numbers

[0349] The center 101 also discloses at the same time d number of one-way function vectors h₁(·), h₂(·), . . . , h_d(·) that generate d number of n-dimensional public key vectors v_r,i=vectors h_r(ID_i) (r=1, 2, . . . , d) composed of m-bit positive integers from the ID of an entity. The parameter b here is set as in the following formula, and we will consider a case in which b>0.

[0350] b=k−(l+m+log₂n

[0351] Registration Processing of an Entity

[0352] Upon receiving a request for registration from an entity i, the center 101 determines d number of secret key vectors s_r,iand t_r,ifor the entity i according to the following formulas 79 and 80 using the prepared keys and the d number of public key vectors v_1,i, v_2,i, . . . , v_d,iof the entity i, privately sends these vectors to the entity i, and completes the registration. We will let B_r=^tA_r, and when the index becomes d+1, it will be converted to 1 and treated accordingly. $\begin{array}{cc}\overrightarrow{s_{r,i}}=\left(A_r\overrightarrow{{\upsilon }_{r+1,i}}+\overrightarrow{{\gamma }_{r,i}}\right)\mathrm{mod}{P}_r& \left(79\right)\\ \overrightarrow{t_{r,i}}=\left(B_r\overrightarrow{{\upsilon }_{r,i}}-\overrightarrow{{\gamma }_{r+1,i}}\right)\mathrm{mod}{P}_r& \left(80\right)\end{array}$

[0353] Processing for Generation of Common Key Between Entities

[0354] The entity i performs the following calculations to determine a common key K_ijwith an entity j. First, A_r,ij′ and B_r,ij′ are determined according to formulas 81 and 82 using modulus P_r, and then K_ijis obtained by the calculation in formula 83 over integer ring. $\begin{array}{cc}{A}_{r,\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{s_{r,i}}\overrightarrow{{\upsilon }_{r,j}}\mathrm{mod}{P}_r& \left(81\right)\\ B_{r,\mathrm{ij}}^{\prime }={\hspace{0.17em}}^t\overrightarrow{t_{r,i}}\overrightarrow{{\upsilon }_{r+1,j}}\mathrm{mod}{P}_r& \left(82\right)\\ K_{\mathrm{ij}}=\sum _{r=1}^d\left(A_{r,\mathrm{ij}}^{\prime }+B_{r,\mathrm{ij}}^{\prime }\right)& \left(83\right)\end{array}$

[0355] The operation in which information is communicated between entities is the same as in the third embodiment, and so will not be described here.

[0356] The formal expression in this sixth embodiment is expressed as follows.

[0357] When d=2 (corresponds to the fifth embodiment) $\left(\begin{array}{cc}O& \overrightarrow{s_{1,i}}\\ \overrightarrow{t_{1,i}}& O\end{array}\right)=\left(\begin{array}{cc}O& A_1\left(P_1\right)\\ B_1\left(P_1\right)& O\end{array}\right)\left(\frac{\overrightarrow{{\upsilon }_{1,i}}}{\overrightarrow{{\upsilon }_{2,i}}}\right)+\left(\begin{array}{cc}O& \overrightarrow{{\gamma }_{1,i}}\\ -\overrightarrow{{\gamma }_{2,i}}& O\end{array}\right)$ $\left(\begin{array}{cc}O& \overrightarrow{s_{2,i}}\\ \overrightarrow{t_{2,i}}& O\end{array}\right)=\left(\begin{array}{cc}O& A_2\left(P_2\right)\\ B_2\left(P_2\right)& O\end{array}\right)\left(\frac{\overrightarrow{{\upsilon }_{1,i}}}{\overrightarrow{{\upsilon }_{2,i}}}\right)+\left(\begin{array}{cc}O& \overrightarrow{{\gamma }_{1,i}}\\ -\overrightarrow{{\gamma }_{2,i}}& O\end{array}\right)$

[0358] When d≧3 (corresponds to the fourth embodiment) $\left(\begin{array}{ccccc}O& \overrightarrow{s_{1,i}}& O& \dots & \overrightarrow{t_{d,i}}\\ \overrightarrow{t_{1,i}}& O& \overrightarrow{s_{2,i}}& ⋰& ⋮\\ O& \overrightarrow{t_{2,i}}& O& ⋰& O\\ ⋮& ⋰& ⋰& ⋰& \overrightarrow{s_{d-1,i}}\\ \overrightarrow{s_{d,i}}& \dots & O& \overrightarrow{t_{d-1,i}}& O\end{array}\right)=\left(\begin{array}{ccccc}O& A_1\left(P_1\right)& O& \dots & B_d\left(P_d\right)\\ B_1\left(P_1\right)& O& A_2\left(P_2\right)& ⋰& ⋮\\ O& B_2\left(P_2\right)& O& ⋰& ⋰\\ ⋮& ⋰& ⋰& ⋰& ⋰\\ A_d\left(P_d\right)& \dots & ⋰& ⋰& O\end{array}\right)\left(\begin{array}{c}\overrightarrow{{\upsilon }_{1,i}}\\ \overrightarrow{{\upsilon }_{2,i}}\\ ⋮\\ ⋮\\ \overrightarrow{{\upsilon }_{d,i}}\end{array}\right)+\left(\begin{array}{ccccc}O& \overrightarrow{{\gamma }_{1,i}}& O& \dots & -\overrightarrow{{\gamma }_{1,i}}\\ -\overrightarrow{{\gamma }_{2,i}}& O& \overrightarrow{{\gamma }_{2,i}}& ⋰& ⋮\\ O& -\overrightarrow{{\gamma }_{3,i}}& O& ⋰& O\\ ⋮& ⋰& ⋰& ⋰& \overrightarrow{{\gamma }_{d-1,i}}\\ \overrightarrow{{\gamma }_{d,i}}& \dots & O& -\overrightarrow{{\gamma }_{d,i}}& O\end{array}\right)$