Patent Grant
8458478
The present invention relates to an application of an information security technique. In particular, it relates to a message-recoverable signature from which a message can be recovered.
Non-patent literature 1 discloses a conventional message-recoverable signature technique. This technique uses the random oracle model to guarantee security. In the following, this technique will be generally described.
According to this technique, the following conditions are assumed.
message m∈{0,1}k2
function F1: {0,1}k2→{0,1}k1
function F2: {0,1}k1→{0,1}k2
function H: {0,1}k1+k2→{0,1}k
E: elliptic curve defined on a finite field Fq
p: prime number that satisfies p·R=O, where R represents a point on the elliptic curve E, and O represents a point at infinity
G1: points in a subset of the elliptic curve E, where order of the subset is p
W∈Z/pZ
secret key: x∈Z/pZ
public key: (Fq,E,G1,Y)(Y=−x·G1(∈E))
{0, 1}δ represents δ-bit data, and {0, 1}δ→{0, 1}ε represents a function of mapping of δ-bit data to ε-bit data.
<Signature Generation>
Signature generation is performed as follows. Note that Rx represents the x coordinate of the point R∈E, and (+) represents an exclusive OR operator.
m′=F1(m)|(F2(F1(m))(+)m) (1)
Rx=(w·G1)x
r=R(+)m′ (2)
c=H(r)
z=w+c·x mod p
signature σ=(r,z)
<Signature Verification>
Signature verification is performed as follows. Note that [m′]k1 represents the leading k1 bits of m′, and [m]k2 represents the remaining k2 bits of m′.
m′=r(+)(z·G1+H(r)·Y)x
m=[m′]k2(+)F2([m′]k1)
If [m′]k1=F1(m), the verification is passed.
However, according to the method described in the non-patent literature 1, (F2(F1(m)) in the expression (1) and the Rx in the expression (2) have a fixed bit length, and the message m has to have a fixed bit length.
As a result, there is a problem of inefficiency that even when the message m is shorter than the fixed length, the bit length of a part r of the signature σ cannot be accordingly shortened. In addition, when the bit length of the message m is longer than the fixed length, only a part of the message m can be substituted into the expression (1), and thus, a message-recoverable signature intended for all the bits of the message m cannot be generated.
A signature generating apparatus according to the present invention performs signature generation as described below.
Note that a secret key x is an integer of the signature generating apparatus, and an M-bit recovery message mrec∈{0, 1}M is at least a part of the target of the signature. The signature generating apparatus generates an arbitrary value k which is an integer, calculates R=gk∈G, where G is a cyclic group of order q having a generator g, to genterate the calculation result R. Note that “gk∈G” means k times calculations defined on the cyclic group G to the generator g (as described in detail later). The signature generating apparatus applies a hash function H1: {0, 1}*→{0, 1}L to a value α, to generate an L-bit hash value h=H1(α)∈{0, 1}L. Note that the hash function H1 outputs an L-bit value in response to an input value, the value α depends on the calculation result R and the recovery message msec, L is a positive integer shared with a signature verifying apparatus. The expression “apply a function ε to δ” means that δ or a value that identifies δ is substituted into the function ε. The signature generating apparatus applies a hash function H2: {0, 1}*→{0, 1}M to a value β, to generate an M-bit hash value u=H2(β) ∈{0, 1}M. Note that an output bit length M of the hash function H2 is determined according to the bit length M of the recovery message mrec, and the value β depends on the calculation result R and the hash value h. The signature generating apparatus calculates an exclusive OR w of the recovery message mrec and the hash value u according to w=mrec(+)u ∈{0, 1}M, where (+) represents an exclusive-OR operator, and outputs the exclusive-OR value w. The signature generating apparatus calculates a value r which depends on an L+M-bit bit connection value h|w∈{0, 1}L+M in which the hash value h∈{0, 1}L is placed at a first bit position and the exclusive-OR value w∈{0, 1}M is placed at a second bit position, and from which the hash value h and the exclusive-OR value w can be recovered. Note that the first bit position does not always have to comprise L consecutive bit positions but can comprise L discrete bit positions. Similarly, the second bit position does not always have to comprise M consecutive bit positions but can comprise M discrete bit positions. However, the signature generating apparatus and the signature verifying apparatus have to use a common first bit position and a common second bit position. The signature generating apparatus applies a hash function H3: {0, 1}*→Z to a value γ which depends on the value r, to generate a hash value t=H3(γ)∈Z. Note that Z is an integer, and the hash function H3 outputs an integer in response to an input value. Then, the signature generating apparatus calculates s=k−t·x∈Z and outputs a signature σ=(r, s).
A signature verifying apparatus according to the present invention performs signature verification as described below. Note that a signature received by the signature verifying apparatus will be denoted by σ′=(r′, s′). A public key y=gx∈G of a signature generating apparatus is stored in a storage of the signature verifying apparatus.
The signature σ′=(r′, s′) is input to the signature verifying apparatus. A bit length M′ of a recovery message mrec′ associated with the signature σ′ is stored in a storage. A method by which the signature verifying apparatus acquires the value of the bit length M′ will be described later. The signature verifying apparatus applies a hash function H3: {0, 1}*→Z to a value γ′ which depends on r′ of the signature σ′, to generate a hash value t′=H3(γ′)∈Z . Note that Z is an integer, and the hash function H3 outputs an integer in response to an input value. The signature verifying apparatus calculates R′=gs′·yt′∈G to generate the calculation result R′. Note that “gs′·yt′∈G” means a calculation result obtained by calculation defined on the cyclic group G between a calculation result obtained by s′ times calculations defined on the cyclic group G to the generator g, and a calculation result obtained by t′ times calculations defined on the cyclic group G to the public key y (as described in detail later). The signature verifying apparatus applies a hash function H2: {0, 1}*→{0, 1}M′ to a value β′, to generate an M′-bit hash value u′=H2(β′)∈{0, 1}M′. Note that an output bit length M′ of the hash function H2 is determined according to the bit length M′ of the recovery message mrec′, and the value β′ depends on the calculation result R′ and an L-bit value h′∈{0, 1}L at a first bit position of r′. The signature verifying apparatus calculates an exclusive OR w′(+)u′ of a value w′∈{0, 1}M′ and the hash value u′, to generate a calculation result of the exclusive OR w′(+)u′ as the recovery message mrec′∈{0, 1}M′. Note that the value w′ depends on an M′-bit value at a second bit position of the value r′. The signature verifying apparatus applies a hash function H1: {0, 1}*→{0, 1}L to a value α′, to generate an L-bit hash value H1(α)∈{0, 1}L as calculation result. Note that the hash function H1 outputs an L-bit value in response to an input value, the value α′ depends on the calculation result R′ and the calculated recovery message mrec′. The signature verifying apparatus compares the L-bit value h′ and the hash value H1(α), and outputs information on the condition that h′=H1(α′) as that verification has succeeded. Note that the expression “a value depends on ε and δ” means not only that the value depends only on ε and δ but also that the value depends on ε, δ and other information. The expression “a value depends on ε” means not only that the value depends only on ε but also that the value depends on ε and other information. However, the configurations of the values α, β and γ used in the signature generating apparatus have to be the same as the configurations of the values α′, β′ and γ′ used in the signature verifying apparatus, respectively (as described in detail later).
According to the present invention, a hash function that has an output bit length that varies with the bit length of the recovery message and an innovative processing method are used, so that two operands of the exclusive-OR calculations can always have a common bit length even when the bit length of the recovery message varies. As a result, when the bit length of the recovery message is short, the number of bits involved in each calculation step and the number of bits of the signature σ can be reduced accordingly. In addition, even if the bit length of the recovery message is long, a message-recoverable signature intended for all the bits of the recovery message mrec can be generated.
Furthermore, according to the present invention, signature verification succeeds only when the hash values h and u calculated in the signature generating apparatus and the hash values h′ and u′ calculated in the signature verifying apparatus matches with each other. Therefore, the security is improved compared with the case where signature verification relies only on the match between the hash values h and h′.
Unlike the prior art, according to the present invention, all the bits of the message can be the target of the message-recoverable signature (m=mrec).
Alternatively, the target of the message-recoverable signature may not be all the bits of the message m. When the target of the message-recoverable signature is not all the bits of the message m, an M-bit recovery message mrec is the target of the message-recoverable signature, and an N-bit clear message mclr is the target of a normal signature that differs from the message-recoverable signature. In this case, preferably, the signature generating apparatus stores the N-bit clear message mclr ∈{0, 1}N in a storage, applies the hash function H3: {0, 1}*→Z to a value γ which depends on the value r and the clear message mclr, to calculate t=H3(γ)∈Z, calculates s=k−t·x∈Z, and outputs the signature σ=(r, s) and the clear message mclr. The signature σ′ and the clear message mclr′ are input to the signature verifying apparatus. The signature verifying apparatus applies the hash function H3: {0, 1}*→Z to a value γ′ which depends on r′ of the signature σ′ and the clear message mclr′, and outputs a hash value t′=H3(γ′)∈Z as calculation result.
Thus, the number of bits involved in each calculation step can be prevented from unnecessarily increasing because of all the bits of the message being designated as the target of the message-recoverable signature when all the bits of the message does not have to be the target of the message-recoverable signature. That is, a message-recoverable signature that can be flexibly applied to various messages of various bit lengths and can be used in various applications can be generated.
The present invention provides a message-recoverable signature that can be flexibly applied to various messages of various bit lengths.
In the following, best modes for carrying out the present invention will be described with reference to the drawings.
First, a first embodiment of the present invention will be described.
<General Configuration>
As shown in
<Configuration of Signature Generating Apparatus 10>
Next, a configuration of the signature generating apparatus 10 will be described.
[Hardware Configuration]
As illustrated in
[Cooperation Between Hardware and Program]
The CPU 11 (
As shown in
The storage 10a and the temporary memory 10t correspond to the register 11c, the auxiliary storage device 14 or the RAM 16 shown in
The programs described above may be of a type capable of serving the function alone or of a type that serves the function by reading another program from a library (not shown). At least part of the programs corresponds to the program that makes a computer perform the function of the signature generating apparatus 10.
<Configuration of Signature Verifying Apparatus 20>
Next, a configuration of the signature verifying apparatus 20 will be described.
[Hardware Configuration]
The signature verifying apparatus 20 has the same hardware configuration as the signature generating apparatus 10 shown in
[Cooperation between Hardware and Program]
The signature verifying apparatus 20 is also implemented by a predetermined program loaded into a computer such as that shown in
As shown in
The storage 20a and the temporary memory 20p correspond to the register, the auxiliary storage device or the RAM of the computer or a storage region formed by a combination of these components, for example. The bit length extracting unit 20c, the hash calculators 20d, 20f, 20i and 20k, the group calculator 20e, the exclusive-OR calculator 20g, the bit extracting unit 20h, the exclusive-OR calculator 20j, the comparator 20l and the controller 20n are implemented by programs for realizing the respective processings loaded into the CPU. The output unit 20m and the communication unit 20b are driven under the control of the CPU loaded with a predetermined program. The signature verifying apparatus 20 performs each processing under the control of the controller 20n. Unless otherwise specified, every piece of data involved with the calculation process is read from or written to the temporary memory 20p.
The programs described above may be of a type capable of serving the function alone or of a type that serves the function by reading another program from a library (not shown). At least part of the programs corresponds to the program that makes a computer perform the function of the signature verifying apparatus 20.
<Processing>
Next, a processing performed in this embodiment will be described.
[Preprocessing]
A cyclic group G of order q used in the signature system 1, where the discrete logarithm problem in the cyclic group G is difficult to solve, and a generator g∈G thereof are determined. The cyclic group G may be a group of rational points on an elliptic curve or a multiplicative group of a finite field. In the case where a group of rational points on an elliptic curve is used, the generator g is a point g=(g1, g2) on the elliptic curve. In the case where a multiplicative group of a finite field is used, the generator g is an integer equal to or greater than 2. There are various methods of implementing a group of rational points on an elliptic curve by a computer (for example, “N. Koblitz, Elliptic Curve Cryptosystems, Math. Comp., Vol. 48, No. 17, pp. 203-209, 1987” or “Victor S. Miller, Use of Elliptic Curves in Cryptography, In Advances in Cryptology—CRYPTO '85, Vol. 218 of Lecture Notes in Computer Science, pp. 417-426, Springer, 1986”). In practice, there are various cryptographic methods based on a group of rational points on an elliptic curve that can be implemented by a computer. From the viewpoint of security, the order q is preferably a prime number. However, the order q may not be a prime number if the order q is hard to factorize into prime numbers. In addition, a bit length parameter L∈Z>0 (an integer greater than 0) used in the signature system 1 is determined.
In addition, a hash function H0: {0, 1}*→{0, 1}L+M that provides a variable-length output having an output bit length of L+M bits determined according to the bit length M of a recovery message mrec described later, and a hash function H2: {0, 1}*→{0, 1}M that provides a variable-length output having an output bit length of M bits determined according to the bit length M of the recovery message mrec. Processings of these hash functions will be described later.
In addition, a hash function H1: {0, 1}*→{0, 1}L that outputs an L-bit hash value in response to an input value and a hash function H3: {0, 1}*→Zq that outputs an element of Zq (a complete system of residues modulo q) in response to an input value are determined. The hash function H1 can be constructed in the same manner as the hash functions H0 and H2, and the hash function H3 can be constructed by performing a residue calculation modulo q for a hash value, such as SHA-1.
Information that identifies the cyclic group G and the hash functions H0 to H3 determined as described above is written in the programs implementing the signature generating apparatus 10 and the signature verifying apparatus 20, and this enables the signature generating apparatus 10 and the signature verifying apparatus 20 to perform calculations determined on the cyclic group G or calculations of the hash functions H0 to H3. The bit length parameter L∈Z>0, the order q and the generator g∈G are stored in the storage 10a of the signature generating apparatus 10 and the storage 20a of the signature verifying apparatus 20.
[Key Generation Processing]
Next, a key generation processing performed by the signature generating apparatus 10 will be described.
The secret key generator 10b of the signature generating apparatus 10 generates an arbitrary secret key x∈Zq. Generation of the secret key x may be performed by mapping of a pseudo random number to Zq or based on a value arbitrarily determined by the person who generates the signature. The generated secret key x is safely stored in the storage 10a of the signature generating apparatus 10. That is, an apparatus external to the signature generating apparatus 10 cannot acquire the secret key x from the storage 10a.
Then, the public key generator 10c of the signature generating apparatus 10 reads the secret key x and the generator g∈G of the cyclic group G from the storage 10a, performs a calculation defined on the cyclic group G according to
y=gx∈G (3)
to generate a public key y∈G associated with the secret key x, and stores the public key in the storage 10a. In the case where the cyclic group G is a group of rational points on an elliptic curve E, for example, the right side of the expression (3) means a multiplication of the generator g=(g1, g2) which is a point on the elliptic curve E, by x on the elliptic curve E (that is, x·g∈E), and the public key y is a point on the elliptic curve E. An exemplary specific method of scalar multiplication on an elliptic curve implemented on a computer involves using the dyadic expansion, the sliding window or the like by representing points on the elliptic curve by affine coordinates or projective coordinates (for example, see the reference document 1: Ian F. Blake, Gadiel Serrousi and Nigel P. Smart, “Elliptic Curve Cryptography”, published by Pearson Education, ISBN4-89471-431-0). In the case where the cyclic group G is a multiplicative group of a finite field, for example, the right side of the expression (3) means a calculation gx mod p (where g represents an integer equal to or greater than 2, and p=2q+1), and the public key y is a scalar value. The generated public key y is transmitted from the communication unit 10r to the public key server apparatus 30 via the network 40, and the public key server apparatus 30 presents the received public key y along with a public key certificate, for example. Presentation of the public key y and the like means that the public key y and the like are stored in a storage of the public key server apparatus 30 in such a manner that any apparatus capable of connecting to the network 40 can acquire the public key y and the like stored in the storage of the public key server apparatus 30. The signature verifying apparatus 20 receives the public key y from the public key server apparatus 30 at the communication unit 20b and stores the public key y in the storage 20a.
[Signature Generation Processing]
Next, a signature generation processing according to the first embodiment will be described.
First, a message m∈{0, 1}N+M and the bit length M(≧1) of the recovery message are input to the input unit 10d of the signature generating apparatus 10 (
Then, the message dividing unit 10e reads the message m∈{0, 1}N+M and the bit length M(≧1) of the recovery message from the storage 10a. Using the information, the message dividing unit 10e divides the message m∈{0, 1}N+M into the recovery message mrec ∈{0, 1}M having a bit length of M and a clear message mclr ∈{0, 1}N having a bit length of N (N≧0) (step S12). For example, the higher-order M bits of the message m∈{0, 1}N+M constitute the recovery message mrec ∈{0, 1}M, and the lower-order N bits constitute the clear message mclr ∈{0, 1}N. The message dividing method is not limited to that described above, and the bits of the message m∈{0, 1}N+M that constitute the recovery message mrec and the bits that constitute the clear message mclr can be arbitrarily chosen. The recovery message mrec ∈{0, 1}M having the bit length of M and the clear message mclr ∈{0, 1}N having the bit length of N generated as a result of the division are stored in the storage 10a.
Then, the arbitrary value generator 10f generates an arbitrary value k∈Zq and stores the generated arbitrary value k in the storage 10a (step S13). For example, generation of the arbitrary value k is performed by mapping of a pseudo random number to Zq.
Then, the group calculator 10g reads the generator g∈G and the arbitrary value k∈Zq from the storage 10a, calculates
R=gk∈G (4)
and outputs the calculation result R∈G to the storage 10a to store the calculation result in the storage 10a (step S14). In the case where the cyclic group G is a group of rational points on an elliptic curve E, for example, the right side of the expression (4) means a multiplication of the generator g=(g1, g2) which is a point on the elliptic curve E, by k on the elliptic curve E (that is, k·g∈E), and the calculation result R is a point on the elliptic curve E. An exemplary specific method of scalar multiplication on an elliptic curve implemented on a computer involves using the dyadic expansion, the sliding window or the like by representing points on the elliptic curve by affine coordinates or projective coordinates. In the case where the cyclic group G is a multiplicative group of a finite field, for example, the right side of the expression (4) means a calculation gk mod p, and the calculation result is a scalar value.
Then, the hash calculator 10h reads the calculation result R∈G, the bit length M of the recovery message and the bit length parameter L from the storage 10a. The hash calculator 10h applies the hash function H0: {0, 1}*→{0, 1}L±M having an output bit length of L+M bits determined according to the bit length M of the recovery message mrec to the calculation result R, and outputs the calculation result, that is, the L+M-bit hash value
Π=H0(R)∈{0,1}L+M (5)
to the storage 10a to store the calculation result in the storage 10a (step S15). In the case where the cyclic group G is a group of rational points on an elliptic curve E, for example, the right side of the expression (5) means a calculation that applies the hash function H0 to a value that can uniquely or restrictively determine the calculation result R which is a point on the elliptic curve E (for example, a combination of the x and y coordinates of the point R and the signs thereof, the x or y coordinate of the point R, or a bit connection value of the x and y coordinates of the point R). In the case where the cyclic group G is a group of rational points on an elliptic curve E, the expression “apply the hash function H0 to the calculation result R” means to apply the hash function H0 to a value that can uniquely or restrictively determine the calculation result R which is a point on the elliptic curve E. In the case where the cyclic group G is a multiplicative group of a finite field, for example, the right side of the expression (5) means a calculation that applies the hash function Ho to the calculation result R which is a scalar value.
[Example of Processing in Step S15]
First, the bit length M of the recovery message and the bit length parameter L are loaded into the number-of-hash-calculations calculator 10ha. The number-of-hash-calculations calculator 10ha calculates emax according to
emax=rounddown{(L+M)/length(H)} (5-1)
and stores emax in the temporary memory 10t (step S15a). In this expression, rounddown{*} means a calculation that truncates the fractional portion of *, length (*) means the bit length of *, and H means a well-known hash function. For example, the hash function may be SHA-1 (bit length of 160 bits) or MD5 (bit length of 128 bits). For example, if L+M=500, and the hash function H is SHA-1 [length(H)=160], emax=3.
Then, the controller 10s substitutes 0 in the variable e and stores the variable e in the temporary memory 10t (step S15b).
Then, the partial hash calculator 10h b reads the variable e from the temporary memory 10t, reads the calculation result R from the storage 10a, calculates a hash value
H(e,R) (5-2)
and stores the hash value in the temporary memory 10t (step S15c). In the case where the cyclic group G is a group of rational points on an elliptic curve E, for example, the expression (5-2) means a calculation that applies the hash function H to a bit connection value of the variable e and a value that can uniquely or restrictively determine the calculation result R which is a point on the elliptic curve E (for example, a combination of the x and y coordinates of the point R and the signs thereof, the x or y coordinate of the point R, or a bit connection value of the x and y coordinates of the point R). In the case where the cyclic group G is a multiplicative group of a finite field, for example, the expression (5-2) means a calculation that applies the hash function Ho to a bit connection value of the variable e and the calculation result R which is a scalar value.
Then, the controller 10s reads emax and the variable e from the temporary memory 10t and determines whether a relationship
e=emax (5-3)
holds or not (step S15d). If the relationship (5-3) does not hold, the controller 10s prepares e+1 as a new variable e, stores the new variable e in the temporary memory 10t (step S15e) and then returns the processing to step S15c. On the other hand, if the relationship (5-3) holds, the controller 10s issues an instruction to the bit connecting unit 10hc to read the hash values H(0, R), H(1, R), H(2, R), . . . , H(emax, R) from the temporary memory 10t, calculates the bit connection value thereof
HC(R)=H(0,R)| . . . |H(emax,R) (5-4)
and stores the bit connection value in the temporary memory 10t (step S15f).
Then, the bit deleting unit 10h d reads the bit connection value HC(R), the bit length M of the recovery message and the bit length parameter L from the temporary memory 10t, calculates
Π=H0(R)=delete{length(HC(R))−(L+M),HC(R)} (5-5)
and outputs the calculation result to the storage 10a (step S15g). In this expression, delete{δ, ε} means a processing of deleting the leading δ bits of ε. That is, the expression (5-5) means to delete some leading bits of HC(R) to generate Π=H0(R) having a total bit length of L+M.
The processing performed in step S15 is not limited to the processing described above. For example, instead of using the variable e, the bit length of the hash value may be expanded by hash chain. In this case, HC(R) in the expression (5-4) is as follows, for example.
HC(R)=H(R)|H(H(R))|H(H(H(R)))| . . . |H(H(H . . . (R) . . . ))
(This is the end of the description of “Example of Processing in step S15”).
Following step S15, the hash calculator 10i reads the hash value Π, the recovery message mrec and the bit length parameter L from the storage 10a. The hash calculator 10i applies the hash function H1: {0, 1}*→{0, 1}L that outputs an L-bit hash value in response to an input value to a value α which depends on the hash value Π and the recovery message mrec, and outputs the calculation result, that is, an L-bit hash value
h=H1(α)∈{0,1}L (6)
to the storage 10a to store the hash value in the storage 10a (step S16). In the first embodiment, α depends only on the hash value Π and the recovery message mrec(α=(Π, mrec)). Although the configuration of α is not limited to a particular one in this embodiment, it is assumed that α has the same configuration as α′ (described later) for the signature verifying apparatus 20 described later. For example, α can be configured as follows.
[α-1] a is an L+2M-bit value formed by connecting Π as the higher-order L+M bits and msec as the lower-order M bits to each other.
[α-2] α is an L+2M-bit value formed by connecting Π as the lower-order L+M bits and mrec as the higher-order M bits to each other.
[α-3] α is an L+2M-bit value formed by connecting mrec as the odd-numbered higher-order M bits and Π as the remaining L+M bits to each other.
Then, the hash calculator 10j reads the hash value Π, the hash value h and the bit length M of the recovery message from the storage 10a. The hash calculator 10j applies the hash function H2: {0, 1}*→{0, 1}M having an output bit length of M bits determined according to the bit length M of the recovery message mrec to a value β which depends on the hash value Π and the hash value h, and outputs the calculation result, that is, an M-bit hash value
u=H2(β)∈{0,1}M (7)
to the storage 10a to store the calculation result in the storage 10a (step S17). In the first embodiment, β depends only on the hash value Π and the hash value h(β=(Π, h)). Although the configuration of β is not limited to a particular one in this embodiment, it is assumed that β has the same configuration as β′ (described later) for the signature verifying apparatus 20 described later. For example, β can be configured as follows.
[β-1] β is a 2L+M-bit value formed by connecting Π as the higher-order L+M bits and h as the lower-order L bits to each other.
[β-2] β is a 2L+M-bit value formed by connecting Π as the lower-order L+M bits and h as the higher-order L bits to each other.
[β-3] β is a 2L+M-bit value formed by connecting h as the odd-numbered higher-order L bits and Π as the remaining L+M bits to each other.
[Example of Processing in Step S17]
First, the bit length M of the recovery message is loaded to the number-of-hash-calculations calculator 10ja. The number-of-hash-calculations calculator 10ja calculates emax according to
emax=rounddown{M/length(H)} (7-1)
and stores emax in the temporary memory 10t (step S17a).
Then, the controller 10s substitutes 0 in the variable e and stores the variable e in the temporary memory 10t (step S17b).
Then, the partial hash calculator 10jb reads the variable e from the temporary memory 10t, reads the hash values Π and h from the storage 10a, calculates a hash value
H(e,β),β=(Π,h) (7-2)
and stores the hash value in the temporary memory 10t (step S17c).
Then, the controller 10s reads emax and the variable e from the temporary memory 10t and determines whether a relationship
e=emax (7-3)
holds or not (step S17d). If the relationship (7-3) does not hold, the controller 10s prepares e+1 as a new variable e, stores the new variable e in the temporary memory 10t (step S17e) and then returns the processing to step S17c. On the other hand, if the relationship (7-3) holds, the controller 10s issues an instruction to the bit connecting unit 10jc to read the hash values H(0, β), H(1, β), H(2, β), . . . , H(emax, β) from the temporary memory 10t, calculates the bit connection value thereof
HC(β)=H(0,β)| . . . |H(emax,β) (7-4)
and stores the bit connection value in the temporary memory 10t (step S17f).
Then, the bit deleting unit 10j d reads the bit connection value HC(β) and the bit length M of the recovery message from the temporary memory 10t, calculates
u=H2(β)=delete{length(HC(β))−M,HC(β)} (7-5)
and outputs the calculation result to the storage 10a (step S17g).
The processing performed in step S17 is not limited to the processing described above. For example, instead of using the variable e, the bit length of the hash value may be expanded by hash chain. (This is the end of the description of “Example of Processing in Step S17”).
Following step S17, the exclusive-OR calculator 10k reads the recovery message mrec and the hash value u form the storage 10a. The exclusive-OR calculator 10k calculates the exclusive OR of the recovery message mrec and the hash value u according to
w=mrec(+)u∈{0,1}M (8)
,where (+) represents an exclusive-OR operator, and outputs the exclusive OR value w to the storage 10a to store the value in the storage 10a (step S18).
Then, the bit connecting unit 10m reads the hash value h∈{0, 1}L and the exclusive-OR value w∈{0, 1}M from the storage 10a. The bit connecting unit 10m calculates an L+M-bit bit connection value
d=h|w∈{0,1}L+M (9)
in which the hash value h∈{0, 1}L is placed at a first bit position and the exclusive-OR value w∈{0, 1}M is placed at a second bit position, and outputs the bit connection value d to the storage 10a to store the value in the storage 10a (step S19). Although both the “first bit position” and the “second bit position” are not limited to a particular bit position, the signature generating apparatus 10 and the signature verifying apparatus 20 should determine the first and second bit positions based on the same criterion.
In the example shown in
Then, the exclusive-OR calculator 10n reads the hash value Π and the bit connection value d from the storage 10a. The exclusive-OR calculator 10n calculates the exclusive OR of the hash value Π and the bit connection value d according to
r=Π(+)d∈{0,1}L+M (10)
and outputs the exclusive-OR value r to the storage 10a to store the value in the storage 10a (step S20).
Then, the hash calculator 10p reads the exclusive-OR value r and the clear message mclr from the storage 10a. The hash calculator 10p applies the hash function H3: {0, 1}*→Zq that outputs an integer in response to an input value to a value γ which depends on the exclusive-OR value r and the clear message mclr, and outputs the calculation result, that is, a hash value
t=H3(γ)∈Zq (11)
to the storage 10a to store the hash value in the storage 10a (step S21). In the first embodiment, γ depends only on the exclusive-OR value r and the clear message mclr(γ=(r, mclr)). Although the configuration of γ is not limited to a particular one in this embodiment, it is assumed that γ has the same configuration as γ′ (described later) for the signature verifying apparatus 20 described later. For example, γ can be configured as follows.
[γ-1] γ is an L+M+N-bit value formed by connecting r as the higher-order L+M bits and mclr as the lower-order N bits to each other.
[γ-2] γ is an L+M+N-bit value formed by connecting r as the lower-order L+M bits and mclr as the higher-order N bits to each other.
[γ-3] γ is an L+M+N-bit value formed by connecting mclr as the odd-numbered higher-order N bits and r as the remaining L+M bits to each other.
Then, the integer calculator 10q reads the arbitrary value k, the hash value t and the secret keys x and q from the storage 10a. The integer calculator 10q calculates
s=k−t·x∈Zq (12)
and outputs the calculation result s to the storage 10a to store the result in the storage 10a (step S22).
Then, the exclusive-OR value r, the calculation result s and the clear message mclr are loaded into the communication unit 10r, and the communication unit l Or transmits the signature σ=(r, s) and the clear message mclr to the signature verifying apparatus 20 through the network 40 (step S23).
[Signature Verification Processing]
Next, a signature verification processing according to the first embodiment will be described.
First, the communication unit 20b of the signature verifying apparatus 20 (
Then, the bit length extracting unit 20c reads the bit length parameter L and r′ of the signature σ′=(r′, s′) from the storage 20a. The bit length extracting unit 20c calculates the bit length M′ of a recovery message mrec′ associated with the signature σ′ according to
M′=length(r′)−L (13)
and stores the bit length M′ in the storage 20a (step S42).
Then, the hash calculator 20d reads r′, the clear message mclr′ and q from the storage 20a. The hash calculator 20d applies the hash function H3: {0, 1}*→Zq, where the hash function H3 is the same as the hash function H3 used in the signature generating apparatus 10, to a value γ′ which depends on r′ and mclr′, and outputs the calculation result, that is,
t′=H3(γ′) (14)
to the storage 20a to store the calculation result in the storage 20a (step S43). γ′ has the same configuration as γ in the signature generating apparatus 10 described above (if r=r′, and mclr=mclr′).
Then, the group calculator 20e reads the generator g∈G, the public key y∈G of the signature generating apparatus 10, s′ of the signature σ′ and the hash value t′ from the storage 20a, calculates R′ according to
R′=gs′·yt′∈G (15)
and outputs the calculation result R′ to the storage 20a to store the calculation result R′ in the storage 20a (step S44). In the case where the cyclic group G is a group of rational points on an elliptic curve E, for example, the right side of the expression (15) means a calculation that multiplies the generator g=(g1, g2) by s′ on the elliptic curve E, multiplies the public key y=(y1, y2) by t′ on the elliptic curve E and sums the multiplication results on the elliptic curve E (s′·g+t′·y∈E), and the calculation result R′ is a point on the elliptic curve E. An exemplary specific method of scalar multiplication on an elliptic curve implemented on a CPU involves using the dyadic expansion, the sliding window or the like by representing points on the elliptic curve by affine coordinates or projective coordinates. In the case where the cyclic group G is a multiplicative group of a finite field, for example, the right side of the expression (15) means a calculation gs′·yt′ mod p, and the calculation result R′ is a scalar value.
Then, the hash calculator 20f reads the calculation result R′∈G, the bit length M′ of the recovery message mrec′ and the bit length parameter L from the storage 20a. The hash calculator 20f applies the hash function H0: {0, 1}*→{0, 1}L+M, where the hash function H0 is the same as the hash function H0 used in the signature generating apparatus 10, to the calculation result R′, and outputs the calculation result, that is, an L+M′-bit hash value
Π′=H0(R′)∈{0,1}L+M′ (16)
to the storage 20a to store the hash value in the storage 20a (step S45). Calculation of H0(R′) is the same as in the signature generating apparatus 10 (if R=R′).
Then, the exclusive-OR calculator 20g reads the hash value Π′ and r′ of the signature σ′ from the storage 20a, calculates the exclusive OR thereof
d′=Π(+)r′∈{0,1}L+M′ (17)
and outputs the exclusive-OR value d′ to the storage 20a to store the value in the storage 20a (step S46).
Then, the bit extracting unit 20h reads the exclusive-OR value d′ and the bit length M′ of the recovery message mrec′ from the storage 20a. The bit extracting unit 20h extracts an L-bit value h′∈{0, 1}L at the first bit position of the exclusive-OR value d′ and an M′-bit value w′∈{0, 1}M′ at the second bit position of the exclusive-OR value d′, and stores the values in the storage 20a (step S47). The first bit position and the second bit position are the same as the first bit position and the second bit position in the processing in the signature generating apparatus 10 (if d=d′).
Then, the hash calculator 20i reads the hash value Π′, a value h′ and the bit length M′ of the recovery message mrec′ from the storage 20a. The hash calculator 20i applies the hash function H2: {0, 1}*→{0, 1}M, which is the same as the hash function H2 used in the signature generating apparatus 10, to a value β′ which depends on the hash value Π′ and the value h′, and outputs the calculation result, that is, an M′-bit hash value
u′=H2(β′)∈{0,1}M′ (18)
to the storage 20a to store the hash value in the storage 20a (step S48). β′ has the same configuration as β in the signature generating apparatus 10 described above (if Π=Π′, and h=h′).
The exclusive-OR calculator 20j reads the value w′∈{0, 1}M′ and the hash value u′ from the storage 20a. The exclusive-OR calculator 20j calculates the exclusive OR of the value w′ and the hash value u′ according to
mrec′=w′(+)u′∈{0,1}M′ (19)
and outputs the calculation result, that is, the recovery message mrec′∈{0, 1}M′ to the storage 20a to store the recovery message in the storage 20a (step S49).
Then, the hash calculator 20k reads the hash value Π′ and the recovery message mrec′ from the storage 20a. The hash calculator 20k applies the hash function H1: {0, 1}*→{0, 1}L, which is the same as the hash function H1 used in the signature generating apparatus 10, to a value α′ which depends on the hash value Π′ and the recovery message mrec′, and outputs the calculation result, that is, an L-bit hash value
H1(α′)∈{0,1}L (20)
to the storage 20a to store the hash value in the storage 20a (step S50). α′ has the same configuration as α in the signature generating apparatus 10 described above (if Π=Π′, and mrec=mrec′).
Then, the comparator 20l reads the hash value H1(α′) and the value h′ from the storage 20a, and determines whether the relationship
h′=H1(α′) (21)
holds or not (step S51).
If the relationship (21) does not hold, the comparator 20l outputs a value 0 (indicating that verification failed) to the storage 20a to store the value in the storage 20a, and the output unit 20m outputs the value 0 (indicating that verification failed) received from the storage 20a (step S52). On the other hand, if the relationship (21) holds, the comparator 20l outputs a value 1 (indicating that verification succeeded) to store the value in the storage 20a, and the output unit 20m outputs the value 1 (indicating that verification succeeded) received form the storage 20a (step S53) and outputs the recovery message mrec′ (step S54).
Next, a second embodiment of the present invention will be described. The second embodiment differs from the first embodiment in that the clear message is not used. The following description will be mainly focused on differences from the first embodiment, and description of the things that are the same as in the first embodiment will be omitted.
<General Configuration>
In the signature system 1 according to the first embodiment, the signature generating apparatus 10 is replaced with a signature generating apparatus 110, and the signature verifying apparatus 20 is replaced with a signature verifying apparatus 120.
<Configuration of Signature Generating Apparatus 110>
Next, a configuration of the signature generating apparatus 110 will be described.
[Hardware Configuration]
The signature generating apparatus 110 has the same hardware configuration as the signature generating apparatus 10 according to the first embodiment.
[Cooperation Between Hardware and Program]
The signature generating apparatus 110 is also implemented by a predetermined program loaded into a computer.
As shown in
The bit length extracting unit 110e and the hash calculator 110p are implemented by programs for realizing the respective processings loaded into the CPU. The input unit 110d is driven under the control of the CPU loaded with a predetermined program, and the communication unit 110r is driven under the control of the CPU loaded with a predetermined program.
The programs described above may be of a type capable of serving the function alone or of a type that serves the function by reading another program from a library (not shown). At least part of the programs corresponds to the program that makes a computer perform the function of the signature generating apparatus 110.
<Configuration of Signature Verifying Apparatus 120>
Next, a configuration of the signature verifying apparatus 120 will be described.
[Hardware Configuration]
The signature verifying apparatus 120 has the same hardware configuration as the signature verifying apparatus 20 according to the first embodiment.
[Cooperation Between Hardware and Program]
The signature verifying apparatus 120 is also implemented by a predetermined program loaded into a computer.
As shown in
The hash calculator 120d is implemented by a program for realizing the processing loaded into the CPU. The communication unit 120b is driven under the control of the CPU loaded with a predetermined program. The programs described above may be of a type capable of serving the function alone or of a type that serves the function by reading another program from a library (not shown). At least part of the programs corresponds to the program that makes a computer perform the function of the signature verifying apparatus 120.
<Processing>
Next, a processing performed in this embodiment will be described.
[Preprocessing and Key Generation Processing]
The preprocessing and the key generation processing are the same as those in the first embodiment.
[Signature Generation Processing]
Next, a signature generation processing according to the second embodiment will be described.
First, a recovery message mrec ∈{0, 1}M is input to the input unit 110d of the signature generating apparatus 110 (
Then, the bit length extracting unit 110e reads the recovery message mrec ∈{0, 1}M from the storage 10a, extracts the bit length M of the recovery message and stores the bit length M in the storage 10a (step S112).
Then, the signature generating apparatus 110 performs steps S113 to S120, which are the same as steps S13 to S20 in the first embodiment, and then, the hash calculator 110p reads the exclusive-OR value r from the storage 10a. The hash calculator 110p applies the hash function H3: {0, 1}*→Zq, which is the same as the hash function H3 in the first embodiment, to a value γ which depends on the exclusive-OR value r, and outputs the calculation result, that is, a hash value
t=H3(γ)∈Zq (22)
to the storage 10a to store the hash value in the storage 10a (step S121). In the second embodiment, γ is a value that depends only on the exclusive-OR value r (γ=r). Although the configuration of γ is not limited to a particular one in this embodiment, it is assumed that γ has the same configuration as γ′ (described later) for the signature verifying apparatus 120 described later.
Then, the integer calculator 10q reads the arbitrary value k, the hash value t and the secret keys x and q from the storage 10a, calculates s according to the expression (12) described above, and outputs the calculation result s to the storage 10a to store the calculation result s in the storage 10a (step S122).
Then, the exclusive-OR value r and the calculation result s are loaded into the communication unit 110r, and the communication unit 110r transmits the signature σ=(r, s) to the signature verifying apparatus 120 through the network 40 (step S123).
[Signature Verification Processing]
Next, a signature verification processing according to the second embodiment will be described.
First, the communication unit 120b of the signature verifying apparatus 120 (
Then, the bit length extracting unit 20c reads the bit length parameter L and r′ of the signature σ′=(r′, s′) from the storage 20a, calculates the bit length M′ of the recovery message mrec′ associated with the signature σ′ according to the expression (13) described above and stores the bit length M′ in the storage 20a (step S142).
Then, the hash calculator 120d reads r′ and q from the storage 20a. The hash calculator 120d applies the hash function H3: {0, 1}*→Zq, where the hash function H3 is the same as the hash function H3 used in the signature generating apparatus 110, to the value y′ which depends on r′, and outputs the calculation result, that is, the hash value
t′=H3(γ′) (23)
to the storage 20a to store the hash value in the storage 20a (step S143). γ′ has the same configuration as γ in the signature generating apparatus 110 described above (if r=r′).
Then, the same steps as steps S44 to S54 in the first embodiment are performed to achieve signature verification (steps S144 to S154).
Next, a third embodiment of the present invention will be described. This embodiment is a modification of the first embodiment and differs from the first embodiment in that r of the signature σ=(r, s) is simplified. More specifically, although r=H0(R)(+)(H1(H0(R), mrec)|mrec(+)H2(H0(R), H1(H0(R), mrec))) in the first embodiment, r=H1(R, mrec)|mrec(+)H2(R, mrec)) in the third embodiment. As a result, the amount of calculation is reduced. The following description will be mainly focused on differences from the first embodiment, and description of the things that are the same as in the first embodiment will be omitted.
<General Configuration>
In the signature system 1 according to the first embodiment, the signature generating apparatus 10 is replaced with a signature generating apparatus 210, and the signature verifying apparatus 20 is replaced with a signature verifying apparatus 220.
<Configuration of Signature Generating Apparatus 210>
Next, a configuration of the signature generating apparatus 210 will be described.
[Hardware Configuration]
The signature generating apparatus 210 has the same hardware configuration as the signature generating apparatus 10 according to the first embodiment.
[Cooperation Between Hardware and Program]
The signature generating apparatus 210 is also implemented by a predetermined program loaded into a computer.
As shown in
The hash calculators 210i, 210j and 10p and the bit connecting unit 210m are implemented by programs for realizing the respective processings loaded into the CPU.
The programs described above may be of a type capable of serving the function alone or of a type that serves the function by reading another program from a library (not shown). At least part of the programs corresponds to the program that makes a computer perform the function of the signature generating apparatus 210.
<Configuration of Signature Verifying Apparatus 220>
Next, a configuration of the signature verifying apparatus 220 will be described.
[Hardware Configuration]
The signature verifying apparatus 220 has the same hardware configuration as the signature verifying apparatus 20 according to the first embodiment.
[Cooperation Between Hardware and Program]
The signature verifying apparatus 220 is also implemented by a predetermined program loaded into a computer.
As shown in
The hash calculators 220i and 220k and the comparator 20l are implemented by programs for realizing the respective processings loaded into the CPU. The programs described above may be of a type capable of serving the function alone or of a type that serves the function by reading another program from a library (not shown). At least part of the programs corresponds to the program that makes a computer perform the function of the signature verifying apparatus 220.
<Processing>
Next, a processing performed in this embodiment will be described.
[Preprocessing]
The preprocessing in this embodiment differs from that in the first embodiment in that the hash function H0 is not used.
[Key Generation Processing]
The key generation processing is the same as that in the first embodiment.
[Signature Generation Processing]
Next, a signature generation processing according to the third embodiment will be described.
First, the signature generating apparatus 210 performs the same steps as steps S11 to S14 in the first embodiment (steps S211 to S214). Then, the hash calculator 10i reads the calculation result R in step S214, the recovery message mrec and the bit length parameter L from the storage 10a. The hash calculator 10i applies the hash function H1: {0, 1}*→{0, 1}L, where the hash function H1 outputs an L-bit hash value in response to an input value, to a value α which depends on the calculation result R and the recovery message mrec (expression (6)), and outputs the calculation result, that is, an L-bit hash value h to the storage 10a to store the hash value in the storage 10a (step S215). In the third embodiment, α is a value that depends only on the calculation result R∈G and the recovery message mrec (α=(R, mrec)). In the case where the cyclic group G is a multiplicative group of a finite field, the configuration of a in this embodiment is the same as that in the first embodiment except that Π is replaced with R. In the case where the cyclic group G is a group of rational points on an elliptic curve E, the configuration of α in this embodiment is the same as that in the first embodiment except that Π is replaced with a value that can uniquely or restrictively determine the calculation result R, which is a point on the elliptic curve E (for example, a combination of the x and y coordinates of the point R and the signs thereof, the x or y coordinate of the point R, or a bit connection value of the x and y coordinates of the point R).
Then, the hash calculator 210j reads the calculation result R, the hash value h and the bit length M of the recovery message from the storage 10a. The hash calculator 210j applies the hash function H2: {0, 1}*→{0, 1}M having an output bit length of M bits determined according to the bit length M of the recovery message mrec to t a value β, where the value β depends on the calculation result and the hash value h (expression (7)), and R an outputs the calculation result, that is, an M-bit hash value u to the storage 10a to store the hash value in the storage 10a (step S216). In the third embodiment, β is a value that depends only on the calculation result R and the hash value h (β=(R, h)). In the case where the cyclic group G is a multiplicative group of a finite field, the configuration of β in this embodiment is the same as that in the first embodiment except that Π is replaced with R. In the case where the cyclic group G is a group of rational points on an elliptic curve E, the configuration of β in this embodiment is the same as that in the first embodiment except that Π is replaced with a value that can uniquely or restrictively determine the calculation result R, which is a point on the elliptic curve E (for example, the x or y coordinate of the point R, or a bit connection value of the x and y coordinates of the point R).
Then, the exclusive-OR calculator 10k reads the recovery message mrec and the hash value u from the storage 10a. The exclusive-OR calculator 10k calculates the exclusive-OR value w of the recovery message mrec and the hash value u (according to the expression (8)), and outputs the exclusive OR value w to the storage 10a to store the value in the storage 10a (step S217).
Then, the bit connecting unit 210m reads the hash value h∈{0, 1}L and the exclusive-OR value w∈{0, 1}M from the storage 10a. The bit connecting unit 210m calculates an L+M-bit bit connection value
r=h|w∈{0,1}L+M (24)
in which the hash value h∈{0, 1}L is placed at the first bit position and the exclusive-OR value w∈{0, 1}M is placed at the second bit position, and outputs the bit connection value r to the storage 10a to store the value in the storage 10a (step S218). The first bit position and the second bit position are the same as in the first embodiment.
Then, the same steps as steps S21 to S23 in the first embodiment are performed (steps S219 to S221).
[Signature Verification Processing]
Next, a signature verification processing according to the third embodiment will be described.
First, the signature verifying apparatus 220 performs the same steps as steps S41 to S44 in the first embodiment (steps S241 to S244).
Then, the bit extracting unit 220h reads r′ of the signature σ′=(r′, s′) and the bit length M′ of the recovery message mrec′ from the storage 20a. The bit extracting unit 220h extracts an L-bit value h′∈{0, 1}L at the first bit position of r′ and an M′-bit value w′∈{0, 1}M′ at the second bit position of r′, and stores the values in the storage 20a (step S245). The first bit position and the second bit position are the same as the first bit position and the second bit position in the processing in the signature generating apparatus 210 (if d=d′).
Then, the hash calculator 220i reads the calculation result R′ in step S244, the value h′ and the bit length M′ of the recovery message mrec′ from the storage 20a. The hash calculator 220i applies the hash function H2: {0, 1}*→{0, 1}M, which is the same as the hash function H2 used in the signature generating apparatus 210, to a value β′ which depends on the calculation result R′ and the value h′, (expression (18)), and outputs the calculation result, that is, an M′-bit hash value u′ to the storage 20a to store the hash value in the storage 20a (step S246). β′ has the same configuration as β in the signature generating apparatus 210 (if Π=Π′, and h=h′).
Then, the exclusive-OR calculator 20j reads the value w′∈{0, 1}M′ and the hash value u′ from the storage 20a. The exclusive-OR calculator 20j calculates the exclusive OR of the value w′ and the hash value u′ (according to the expression (10)), and outputs the calculation result, that is, the recovery message mrec′∈{0, 1}M′ to the storage 20a to store the recovery message in the storage 20a (step S247).
Then, the hash calculator 220k reads the calculation result R′ and the recovery message mrec′ from the storage 20a. The hash calculator 220k applies the hash function H1: {0, 1}*→{0, 1}L, which is the same as the hash function H1 used in the signature generating apparatus 210, to a value α′ which depends on the calculation result R′ and the recovery message mrec′, and outputs the calculation result, that is, an L-bit hash value (expression (20)) to the storage 20a to store the hash value in the storage 20a (step S248). α′ has the same configuration as a in the signature generating apparatus 210 (if Π=Π′, and mrec=mrec′).
Then, the same steps as steps S51 to S54 in the first embodiment are performed (steps S249 to S252).
Next, a fourth embodiment of the present invention will be described. This embodiment is a modification of the third embodiment. The fourth embodiment differs from the third embodiment in that the clear message is not used. The following description will be mainly focused on differences from the first to third embodiments, and description of the things that are the same as in the first to third embodiments will be omitted.
<General Configuration>
In the signature system 1 according to the first embodiment, the signature generating apparatus 10 is replaced with a signature generating apparatus 310, and the signature verifying apparatus 20 is replaced with a signature verifying apparatus 320.
<Configuration of Signature Generating Apparatus 310>
Next, a configuration of the signature generating apparatus 310 will be described.
[Hardware Configuration]
The signature generating apparatus 310 has the same hardware configuration as the signature generating apparatus 10 according to the first embodiment.
[Cooperation Between Hardware and Program]
The signature generating apparatus 310 is also implemented by a predetermined program loaded into a computer.
As shown in
<Configuration of Signature Verifying Apparatus 320>
Next, a configuration of the signature verifying apparatus 320 will be described.
[Hardware Configuration]
The signature verifying apparatus 320 has the same hardware configuration as the signature verifying apparatus 20 according to the first embodiment.
[Cooperation Between Hardware and Program]
The signature verifying apparatus 320 is also implemented by a predetermined program loaded into a computer.
As shown in
<Processing>
Next, a processing performed in this embodiment will be described.
[Preprocessing and Key Generation Processing]
The preprocessing and the key generation processing are the same as those in the first embodiment.
[Signature Generation Processing]
Next, a signature generation processing according to the fourth embodiment will be described.
The signature generating apparatus 310 first performs the same steps as steps S111 to S114 in the second embodiment (steps S311 to S314) and then performs the same steps as steps S215 to S218 in the third embodiment (steps S315 to S318). Then, the signature generating apparatus 310 performs the same steps as steps S121 to S123 in the second embodiment (steps S319 to S321).
[Signature Verification Processing]
Next, a signature verification processing according to the fourth embodiment will be described.
The signature verifying apparatus 320 first performs the same steps as steps S141 to S144 in the second embodiment (steps S341 to S344) and then performs the same steps as steps S245 to S252 in the third embodiment (steps S345 to S352).
[Basis for Adequacy of Signature Verification]
Next, the reason why the signature is appropriately verified by the processings by the signature verifying apparatuses 20, 120, 220 and 320 will be described.
Using the signature σ′=(r′, s′), the signature verifying apparatuses 20 and 120 calculate the hash value t′=H3(γ′) from the value γ′ that depends on r′ (according to the expressions (14) and (23)), calculate the value R′=gs′·yt′∈G (according to the expression (15)), and calculate the hash value Π′=H0(R′) (according to the expression (16)). If the signature σ′ is an authorized signature, r′=r, and s′=s (s=k−t·x∈Z), so that γ′=γ, t′=H3(γ′)=H3(γ)=t, and y=gx∈G, and therefore, R′=gs′·yt′=gs·yt=gk−t·x·gt·x=gk∈G. Therefore, Π′=H0(R′)=H0(gk)=Π.
In addition, the signature verifying apparatuses 20 and 120 determine the exclusive-OR value d′=Π′(+)r′ (according to the expression (17)). If the signature σ′ is an authorized signature, r′=r, r=Π(+)d, and Π′=Π, so that d′=d. Furthermore, the signature verifying apparatuses 20 and 120 determine the hash value u′=H2(β′) for the value β′ that depends on the hash value Π′ and the L-bit value h′∈{0, 1}L at the first bit position of the exclusive-OR value d′ (according to the expression (18)). If the signature σ′ is an authorized signature, d′=d, so that h′=h, and Π′=Π. Therefore, β′=β, and therefore, u′=u.
Furthermore, the signature verifying apparatuses 20 and 120 calculates the exclusive-OR value w′(+)u′ of the M′-bit value w′∈{0, 1}M′ at the second bit position of the exclusive-OR value d′ and the hash value u′ and regards the calculation result as the recovery message mrec′∈{0, 1}M′ (expression (19)). If the signature σ′ is an authorized signature, u′=u, M′=M, and d′=d. In this case, w′=w, and therefore, mrec′=w′(+)u′=w(+)u=mrec(+)u(+)u=mrec.
Then, the signature verifying apparatuses 20 and 120 determine the hash value H1(α′)∈{0, 1}L by applying the hash function H1 to the value α′ that depends on the hash value Π′ and the recovery message mrec′ (expression (20)). If the signature σ′ is an authorized signature, Π′=Π, mrec′=mrec, α′=α, and h′=h. In addition, h′=H1(α′), because h=H1(α) in the signature generating apparatus. That is, if the signature σ′ is an authorized signature, h′=H1(α′).
On the other hand, if it is difficult to solve the discrete logarithm problem in the cyclic group G, a third party who does not know the secret key x cannot determine the secret key x from the public key y=gx∈G and, therefore, cannot generate the signature σ′=(r′, s′) that passes the verification described above. Therefore, the signature σ′=(r′, s′) can be identified as an authorized signature generated by a person who knows the secret key x.
Using the signature σ′=(r′, s′), the signature verifying apparatuses 220 and 320 calculate the hash value t′=H3(γ′) from the value γ′ that depends on r′, and calculate the value R′=gs′·yt′∈G. If the signature σ′ is an authorized signature, r′=r, and s′=s (s=k−t·x∈Z), so that γ′=γ, t′=H3(γ′)=H3(γ)=t, and y=gx∈G, and therefore, R′=gs′·yt′=gs·yt=gk−t·x·gt·x=gk=R.
In addition, the signature verifying apparatuses 220 and 320 determine the hash value u′=H2(β′) for the value β′ that depends on the calculation result R′ and the L-bit value h′∈{0, 1}L at the first bit position of r′ of the signature σ′. If the signature σ′ is an authorized signature, r′=r, so that h′=h, and R′=R. Therefore, β′=β, and therefore, u′=u.
Furthermore, the signature verifying apparatuses 220 and 320 calculates the exclusive-OR value w′(+)u′ of the M′-bit value w′∈{0, 1}M′ at the second bit position of r′ of the signature σ′ and the hash value u′, and regards the calculation result as the recovery message mrec ∈{0, 1}M′. If the signature σ′ is an authorized signature, u′=u, M′=M, and r′=r. In this case, w′=w, and therefore, mrec′=w′(+)u′=w(+)u=mrec(+)u(+)u=mrec.
Then, the signature verifying apparatuses 220 and 320 determine the hash value H1(α′)∈{0, 1}L by applying the hash function H1 to the value α′ that depends on the calculation result R′ and the recovery message mrec′. If the signature σ′ is an authorized signature, R′=R, mrec′=mrec, α′=α, and h′=h. In addition, h′=H1(α′), because h=H1(α) in the signature generating apparatus. That is, if the signature σ′ is an authorized signature, h′=H1(α′).
On the other hand, if it is difficult to solve the discrete logarithm problem in the cyclic group G, a third party who does not know the secret key x cannot determine the secret key x from the public key y=gx∈G and, therefore, cannot generate the signature σ′=(r′, s′) that passes the verification described above. Therefore, the signature σ′=(r′, s′) can be identified as an authorized signature generated by a person who knows the secret key x.
[Modifications]
The present invention is not limited to the embodiments described above. For example, although α is a value that depends only on Π and mrec, and α′ is a value that depends only on Π′ and mrec′ in the first and second embodiments, α may be a value that depends on Π, mrec and some third information, and α′ may be a value that depends on Π′, mrec′ and the third information. For example, the third information may include a parameter that identifies the clear message mclr, the public key y or the group G. The same holds true for β and β′ and γ and γ′. If the third information is used, the precision of the signature verification is improved. In particular, when the third information is a parameter that identifies the group G, an unauthorized signature generated by using an unauthorized group (a group for which the discrete logarithm problem can be easily solved and the result of calculation by the group calculator 20e is the same as the calculation result for the authorized cyclic group G, for example) can be prevented from passing the verification.
Similarly, although α is a value that depends only on R and mrec, and α′ is a value that depends only on R′ and mrec′ in the third and fourth embodiments, α may be a value that depends on R, mrec and some third information, and α′ may be a value that depends on R′, mrec′ and the third information. The same holds true for β and β′ and γ and γ′.
Furthermore, although the signature generating apparatuses 10, 110, 210 and 310 perform key generation in the embodiments described above, another apparatus may perform key generation. Furthermore, although the public key server apparatus 30 presents the public key y in the embodiments described above, the signature generating apparatuses 10, 110, 210 and 310 may transmit the public key y to the signature verifying apparatuses 20, 120, 220 and 320. Furthermore, Zq (a complete residue system modulo q) in the processings may be replaced with Z (integer).
Furthermore, although the signature verifying apparatuses 20, 120, 220 and 320 calculate the bit length of the recovery message from the bit length of r′ of the signature σ′ and the bit length parameter L in the embodiments described above, the signature generating apparatuses 10, 110, 210 and 310 may transmit the bit length of the recovery message to the signature verifying apparatuses 20, 120, 220 and 320.
Furthermore, at least the recovery message mrec is a signature target in the embodiments described above. That is, the bit lengths M and M′ of the recovery messages mrec and mrec′ are equal to or greater than 1. Alternatively, however, in the first and third embodiments, the recovery messages mrec and mrec′ may be null, and only the clear messages mclr and mclr′ may be signature targets. This means that the bit lengths M and M′ of the recovery messages mrec and mrec′ are 0. Alternatively, the bit lengths M and M′ may be configurable within a range M≧0. In this case, it is possible to switch between the message recovery signature and the normal signature depending on the settings of the bit lengths M and M′. The processings that become unnecessary as a result of setting the recovery messages mrec and mrec′ at null and setting the bit lengths M and M′ at 0 can be omitted. The operation of the parts responsible for the unnecessary processings can be stopped.
The “hash function” in the present invention refers to a function that calculates a representative value for certain data. According to the present invention, the hash function is not limited to SHA-1, MD5 or the like but can be a common key cryptography function, such as DES and Camellia, into which a common key is substituted.
Furthermore, the processings described above may be performed in time series in the order described above or may be performed in parallel or separately as required or depending on the processing capability of the apparatuses that perform the processings. Furthermore, of course, various other modifications can be appropriately made without departing from the spirit of the present invention.
When the configurations described above are implemented on a computer, the specific capabilities of the apparatuses are described as programs. The specific capabilities are implemented on the computer by executing the programs on the computer.
The programs describing the specific capabilities can be recorded in a computer-readable recording medium. The computer-readable recording medium may be any type of medium, such as a magnetic recorder, an optical disk, a magneto-optical recording medium and a semiconductor memory. To be specific, the magnetic recorder may be a hard disk drive, a flexible disk or a magnetic tape, for example. The optical disk may be a digital versatile disc (DVD), a digital versatile disc random access memory (DVD-RAM), a compact disc read only memory (CD-ROM), a compact disc recordable (CD-R) or a compact disc rewritable (CD-RW), for example. The magneto-optical recording medium may be a magneto-optical disc (MO), for example. The semiconductor memory may be an electronically erasable and programmable read only memory (EEP-ROM), for example.
The programs are distributed by sale, transfer, rental or the like of a portable recording medium, such as a DVD and a CD-ROM, on which the programs are recorded. Alternatively, the programs may be stored in a storage device of a server computer and distributed by the server computer transferring the programs to other computers over a network.
For example, the computer that executes such a program first stores the program recorded on a portable recording medium or transferred from the server computer in a storage device thereof. When the computer performs the processing, the computer reads the program from the storage device thereof and performs the processing according to the read program. Alternatively, the computer may read the program directly from the portable recording medium and perform the processing according to the program. As a further alternative, the computer may perform the processing according to the program each time the computer receives a program transferred from the server computer. As a further alternative, the processing may be performed by on an application service provider (ASP) basis, in which the server computer does not transmit the program to the computer, and the specific capabilities are implemented only through execution instruction and result acquisition. The programs according to the embodiments of the present invention include a quasi-program, which is information processed by a computer (data or the like that is not a direct instruction to a computer but has a property that defines the processing performed by the computer).
In the above description, the apparatuses according to the embodiments of the present invention are implemented by executing a predetermined program on a computer. However, at least part of the processings may be implemented in the form of hardware.
The present invention can be applied to various applications using the electronic signature.
| Number | Date | Country | Kind |
|---|---|---|---|
| 2007-280287 | Oct 2007 | JP | national |
| Filing Document | Filing Date | Country | Kind | 371c Date |
|---|---|---|---|---|
| PCT/JP2008/057962 | 4/24/2008 | WO | 00 | 6/22/2010 |
| Publishing Document | Publishing Date | Country | Kind |
|---|---|---|---|
| WO2009/057338 | 5/7/2009 | WO | A |
| Number | Name | Date | Kind |
|---|---|---|---|
| 6266771 | Bellare et al. | Jul 2001 | B1 |
| 20020046339 | Bellare et al. | Apr 2002 | A1 |
| 20050018850 | Venkatesan et al. | Jan 2005 | A1 |
| Number | Date | Country |
|---|---|---|
| 0 977 165 | Feb 2000 | EP |
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| Number | Date | Country | |
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| 20100268957 A1 | Oct 2010 | US |
