CN116132070A - Heterogeneous aggregation signature method and equipment - Google Patents
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- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/32—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials
- H04L9/3247—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials involving digital signatures
- H04L9/3255—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials involving digital signatures using group based signatures, e.g. ring or threshold signatures
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- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/32—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols including means for verifying the identity or authority of a user of the system or for message authentication, e.g. authorization, entity authentication, data integrity or data verification, non-repudiation, key authentication or verification of credentials
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Abstract
The invention provides a heterogeneous aggregation signature method and equipment, wherein the method comprises the following steps: the signature terminal calculates a feature code S according to the public key system to which the signature terminal belongs i From the slave
R is selected randomly i Calculating R i =r i P, signature sequence (ID i ,m i ,R i ,S i ,T i ) Sending the verification result to a verification center; wherein ID i The identity information is signature terminal identity information; m is m i Is a message to be encrypted; t (T) i Is a time stamp; verification center verifies equation
If so, the signature verification of n user terminals is passed; otherwise, the verification is not passed. The invention supports the signing and authentication service of a plurality of signers from different cryptosystems, and the verifier only needs to initialize system parameters once and verify the aggregate signature once, thereby reducing the verificationThe system initialization overhead and signature verification overhead of the signer can simultaneously provide authentication services for a plurality of signers of different cryptographic systems.
Description
Technical Field
The invention relates to the technical field of information security data processing, in particular to a heterogeneous aggregation signature method and equipment.
Background
Aggregate signatures are an important way of signing cryptography. An aggregate signature protocol allows any number of users to send their respective digital signatures to the same verifier over a public, unsecured channel. The verifier can aggregate the signatures of any plurality of users, and can verify whether the signatures of all the users are legal or not only by carrying out a signature verification algorithm once, and the aggregated signature can greatly improve the signature verification efficiency.
The existing public key cryptosystem is divided into three types: 1. the public key system based on the certificate adopts a certificate mechanism to realize the secure correspondence between the identity of the user and the key of the user, and generally adopts public key infrastructure (Public Key Infrastructure: PKI) technology. 2. An Identity-based public key system (Identity-Based Cryptography: IBC) in which the private key of the user is computationally generated by a trusted third party (key generation center: KGC) in the system using an Identity private key generation algorithm; 3. based on a public key hierarchy (Certificateless Public Key Cryptography, CLC) without certificates, the user private key in this mechanism is determined by two secret factors: one is a key related to the identity of the user extracted from the key generation center, and the other is a key generated by the user himself. From one secret element the other cannot be calculated, i.e. the key generation center cannot calculate the partial key of the user nor the partial key generated by the key generation center. Thus, the certificateless cryptographic system has no key escrow (key escrow) function.
If all participants of the cryptographic protocol belong to the same cryptographic system, the protocol belongs to an isomorphic cryptographic protocol. Otherwise, the protocol belongs to a heterogeneous cryptographic protocol.
Currently, an aggregate signature protocol based on an isomorphic cryptographic protocol is mature, and mainly comprises the following steps:
in 2008, wen and Ma [ y.wen, j.ma, an aggregate signature scheme with constant pairing operations, 2008 International Conference on Computer Science and Software Engineering, 2008, pp. 830-833 ] proposed an aggregate signature protocol that required only a fixed number of bilinear pairs. This protocol allows any number of PKI users to sign, and the verifier can aggregate and verify all users' signatures at once. But this method is only applicable to PKI.
In 2019, yang et al [ X.yang, R.Liu, M.Wang and G.Chen, identity-based aggregate signature scheme in vehicle ad-hoc network, 2019 4th International Conference on Mechanical, control and Computer Engineering, 2019, pp., 1046-1049 ] proposed an aggregate signature protocol suitable for use in the Internet of vehicles. The protocol allows users of any multiple IBCs to sign, and a verifier can aggregate and verify the signatures of all users at once. But this method is only applicable to IBC.
2021, kar et al [ J.Kar, X.Liu and F.Li, CL-ASS: an efficient and low-cost certificateless aggregate signature scheme for wireless sensor networks, journal of Information Security and Applications, 2021, 61, p. 102905 ] proposed an aggregate signature protocol suitable for wireless sensor networks. The protocol allows users of any multiple CLCs to sign, and the verifier can aggregate and verify the signatures of all users at once. But this method is only applicable to CLCs.
Disclosure of Invention
In order to solve the technical problems, the invention aims to provide a method for enabling one verifier to verify the legitimacy of any plurality of user signatures from different password systems, thereby reducing the system initialization overhead and signature verification overhead of the verifier and providing signature and authentication services for users from different password systems.
In order to achieve the above object, the present invention provides a technical solution comprising:
the heterogeneous aggregation signature method is applied to a signature terminal and comprises the following steps:
the public system parameters spp= { p, G,G T , P, Q, 
, H 0 , H 1 , H 2 -a }; wherein p is a large prime number; g is the p-order addition cyclic group; g T Is a p factorial cyclic group;
is a bilinear map; h 0 ，H 1 And H 2 Respectively, are hash functions of collision resistance, and H 0 From {0,1} * Mapping to {0,1} n ，H 1 From {0,1} * Mapping to G, H 2 From {0,1} * Mapping to
；{0,1} * A binary sequence of arbitrary bit length;
a p-order integer domain obtained for removing zero elements; p and Q are G generator;
the signature terminal calculates a feature code S according to the public key system to which the signature terminal belongs i From the slave
R is selected randomly i Calculating R i = r i P, signature sequence (ID i , m i , R i , S i , T i ) Sending the verification result to a verification center; wherein ID i The identity information is signature terminal identity information; m is m i Is a message to be encrypted; t (T) i Is a time stamp;
for signature terminal A belonging to certificate-based public key system, its feature code S a And slave(s)
The first private key sk selected randomly in the list a Related to; for signature terminal B belonging to identity-based public key system, its signature S b With a master key s provided by a private key generating center 1 Related to; for signature terminal C belonging to public key system without certificate, its characteristic code S c With a second private key psk provided by a key generation center c And master key s 2 From
A randomly selected third private key usk c Related to; wherein a, b, c are the number of signature terminals A, B, C, respectively, and
。
in some preferred embodiments, the signature S a The acquisition method of (1) comprises the following steps: s is S a = (r a + h a sk a ) Q, where r a 
；h a = H 2 (ID a , m i , T i , R a )；sk a For signing terminal a slave
Is selected randomly and sk a = x a ；R a = r a P。
In some preferred embodiments, the signature S b The acquisition method of (1) comprises the following steps:
wherein r is b 
；h b = H 2 (ID b , m i , T i , R b )；
Is that
；P pub1 Master key s provided for a private key generation center 1 Corresponding public key and P pub1 = s 1 P；R b = r b P。
In some preferred embodiments, the signature S c The acquisition method of (1) comprises the following steps: s is S c = psk c + h c r c P pub2 + (r c + h c usk c ) Q, wherein psk c = s 2 H 1 (ID c )；h c = H 2 (ID c , m i , T i , R c )；r c 
；P pub2 For master key s provided with key generating center 2 Corresponding public key and P pub2 = s 2 P；R c = r c P is as follows; third private key usk c For signing terminal C slave
Is a randomly selected private key and usk c = x c 。
The heterogeneous aggregation signature method is applied to a verification center and comprises the following steps:
acquiring signature sequence (ID) transmitted by signature terminal i , m i , R i , S i , T i ) Calculation of
；
Based on signature codes of each type of signature terminalSeparately calculate
、
And
；
verification equation
If so, the signature verification of n user terminals is passed; otherwise, the verification is not passed.
In some preferred embodiments, the signature S of the signature terminal A belonging to a certificate-based public key hierarchy is obtained a ，S a = (r a + h a sk a ) Q is a group; wherein r is a 
；h a = H 2 (ID a , m i , T i , R a )；sk a For signing terminal a slave
Is selected randomly and sk a = x a ；R a = r a P；
Acquiring a signature code S of a signature terminal B belonging to an identity-based public key system b ，
The method comprises the steps of carrying out a first treatment on the surface of the Wherein r is b 
；h b = H 2 (ID b , m i , T i , R b )；
Is that
；P pub1 Master key s provided for a private key generation center 1 Corresponding public key and P pub1 = s 1 P；R b = r b P；
Acquiring a signature S belonging to a signature terminal C based on a public key system without certificates c ，S c = psk c + h c r c P pub2 + (r c + h c usk c ) Q is a group; wherein psk is c = s 2 H 1 (ID c )；h c = H 2 (ID c , m i , T i , R c )；r c 
；P pub2 For master key s provided with key generating center 2 Corresponding public key and P pub2 = s 2 P；R c = r c P is as follows; third private key usk c For signing terminal C slave
Is a randomly selected private key and usk c = x c 。
In some preferred embodiments, the signature S is a Feature code S b And feature code S c Substituting the obtained product into the verification equation, and obtaining after simplification:
wherein pk is a For the first private key sk a Corresponding public key and pk a = x a P；upk c For the third private key usk c Corresponding public key and upk c = x c P；
Will be
The verification code is set as the signature terminal A; will be
Verification code CA set as signature terminal B b The method comprises the steps of carrying out a first treatment on the surface of the Will respectively
And
verification code CA set as signature terminal C c2 And CA c1 ；
The above verification equation may be converted into:
a terminal device, comprising: a memory, a processor, a communication interface; wherein the memory has executable code stored thereon that, when executed by the processor, causes the processor to perform the aggregate signature method as described above.
An electronic device, comprising: a memory, a processor, a communication interface; wherein the memory has executable code stored thereon that, when executed by the processor, causes the processor to perform the heterogeneous aggregate signature method as described above.
Has the beneficial effects of.
The invention supports the signing and authentication service of a plurality of signers from different cipher systems, and the verifier only needs to initialize system parameters once and aggregate signature verification once, thereby reducing the system initialization cost and signature verification cost of the verifier and simultaneously providing authentication service for a plurality of signers of different cipher systems.
Drawings
FIG. 1 is a flow chart of the method for executing the signature terminal as the execution subject in the preferred embodiment of the invention;
FIG. 2 is a flow chart of the execution subject as the verification center in the preferred embodiment of the invention;
Detailed Description
The present invention will be further described with reference to the accompanying drawings, in order to make the objects, technical solutions and advantages of the present invention more apparent. In the description of the present invention, it should be understood that the terms "upper," "lower," "front," "rear," "left," "right," "top," "bottom," "inner," "outer," and the like indicate or are based on the orientation or positional relationship shown in the drawings, merely to facilitate description of the present invention and to simplify the description, and do not indicate or imply that the devices or elements referred to must have a specific orientation, be configured and operated in a specific orientation, and thus should not be construed as limiting the present invention.
The heterogeneous aggregation signature method provided by the embodiment of the invention can be executed by an electronic device, and the electronic device can be a terminal device such as a PC (personal computer), a notebook computer, a smart phone and the like. As shown in fig. 1, the present embodiment provides a heterogeneous aggregation signature method, an execution body of which may be a signature terminal, and it should be understood that the signature terminal may be implemented as software, or a combination of software and hardware, and specifically, the heterogeneous aggregation signature method may include:
101. the public system parameters spp= { p, G,G T , P, Q, 
, H 0 , H 1 , H 2 -a }; wherein p is a large prime number; g is the p-order addition cyclic group; g T Is a p factorial cyclic group;
is a bilinear map; h 0 ，H 1 And H 2 Respectively, are hash functions of collision resistance, and H 0 From {0,1} * Mapping to {0,1} n ，H 1 From {0,1} * Mapping to G, H 2 From {0,1} * Mapping to
；{0,1} * A binary sequence of arbitrary bit length;
a p-order integer domain obtained for removing zero elements; p and Q are G generator; it should be appreciated that the disclosure of the system parameter spp may be broadcast by any of the subscribing terminals to the participants within the network, or may be disclosed by a verification center or other trusted third party. The signature terminals respectively apply for registration to the public key system and initialize the system according to the system parameter spp.
102. The signature terminal calculates a feature code S according to the public key system to which the signature terminal belongs i From the slave
R is selected randomly i Calculating R i = r i P, signature sequence (ID i , m i , R i , S i , T i ) Sending the verification result to a verification center; wherein ID i The identity information is signature terminal identity information; m is m i Is a message to be encrypted; t (T) i Is a time stamp; wherein, (R) i , S i ) Is a signature. For heterogeneous cryptosystem, the signature code S of the belonged signature terminal i In a different manner, in particular, for signature terminals a belonging to the certificate-based Public Key Infrastructure (PKI), the signature code S a And slave(s)
The first private key sk selected randomly in the list a Related to; for signature terminals B belonging to an identity-based public key system (IBC), a signature code S b With a master key s provided by a private key generating center 1 Related to; for signature terminals C belonging to a certificate-free public key system (CLC), a signature code S thereof c With a second private key psk provided by a key generation center c And master key s 2 From
A randomly selected third private key usk c Related to; wherein the method comprises the steps ofa. b, c are the number of signature terminals A, B, C, respectively, and
. It should be appreciated that the signature S i The signature terminal is designed according to the subsequent verification requirement by integrating the characteristics of the public key system of the signature terminal, and the purpose is that the verification center does not need to initialize the authentication system parameters for different public key systems respectively when carrying out the aggregation signature authentication, but verifies the legality of the signature according to the identity information and the information of all signers, thereby reducing the system initialization overhead and the signature authentication overhead of the verification center and providing authentication services for users from different password systems. Obviously, when the signature terminal communicates with the public key system service provider, the method further comprises the step of verifying the respective identity and the validity of the secret key and/or the public key, and when the verification fails, the aggregation signature process is canceled, and as the content of the part is not the focus of the invention, the person skilled in the art can design according to the conventional method of the prior art, and the invention is not limited further.
The following presents a feature code S in some preferred embodiments i Specific acquisition method for specifically describing feature code S i Is not limited to the feature code S i The only method of acquisition.
102a, feature code S a The acquisition method of (1) comprises the following steps: s is S a = (r a + h a sk a ) Q, where r a 
；h a = H 2 (ID a , m i , T i , R a )；sk a For signing terminal a slave
Is selected randomly and sk a = x a ；R a = r a P。
102b, feature code S b The acquisition method of (1) comprises the following steps:
wherein r is b 
；h b = H 2 (ID b , m i , T i , R b )；
Is that
；P pub1 Master key s provided for a private key generation center 1 Corresponding public key and P pub1 = s 1 P；R b = r b P。
102c, feature code S c The acquisition method of (1) comprises the following steps: s is S c = psk c + h c r c P pub2 + (r c + h c usk c ) Q, wherein psk c = s 2 H 1 (ID c )；h c = H 2 (ID c , m i , T i , R c )；r c 
；P pub2 For master key s provided with key generating center 2 Corresponding public key and P pub2 = s 2 P；R c = r c P is as follows; third private key usk c For signing terminal C slave
Is a randomly selected private key and usk c = x c 。
The design thought of the feature code is as follows: under such design, the last two summands of the CLC feature code are identical to the form of the PKI feature code; the first two summand forms of the IBC feature code and the CLC feature code are the same. Three types of feature code aggregation can be performed in this way.
The heterogeneous aggregation signature method provided by the other embodiment of the present invention may be performed by an electronic device, which may be a terminal device such as a PC, a notebook, a smart phone, etc. As shown in fig. 2, the present embodiment provides a heterogeneous aggregation signature method, the execution subject of which may be a verification center, it being understood that the verification center may be implemented as software, or a combination of software and hardware, and specifically, the heterogeneous aggregation signature method may include:
201. acquiring signature sequence (ID) transmitted by signature terminal i , m i , R i , S i , T i ) Calculation of
；
Respectively calculating according to the feature codes of each type of signature terminal
、
And
；
verification equation
If so, the signature verification of n user terminals is passed; otherwise, the verification is not passed.
The feature codes of the signature terminals are determined according to the characteristics of the public key system to which the signature terminals belong, and in some preferred embodiments, the feature codes correspond to the feature code acquisition method of the other embodiment, and this embodiment provides an example of acquiring the feature codes by the verification center:
obtaining a signature belonging to a certificate-based public key systemFeature code S of name terminal A a ，S a = (r a + h a sk a ) Q is a group; wherein r is a 
；h a = H 2 (ID a , m i , T i , R a )；sk a For signing terminal a slave
Is selected randomly and sk a = x a ；R a = r a P。
Acquiring a signature code S of a signature terminal B belonging to an identity-based public key system b ，
The method comprises the steps of carrying out a first treatment on the surface of the Wherein r is b 
；h b = H 2 (ID b , m i , T i , R b )；
Is that
；P pub1 Master key s provided for a private key generation center 1 Corresponding public key and P pub1 = s 1 P；R b = r b P；
Acquiring a signature S belonging to a signature terminal C based on a public key system without certificates c ，S c = psk c + h c r c P pub2 + (r c + h c usk c ) Q is a group; wherein psk is c = s 2 H 1 (ID c )；h c = H 2 (ID c , m i , T i , R c )；r c 
；P pub2 For master key s provided with key generating center 2 Corresponding public key and P pub2 = s 2 P；R c = r c P is as follows; third private key usk c For signing terminal C slave
Is a randomly selected private key and usk c = x c 。
It should be understood that when the feature value superposition value of each type of signature terminal is calculated one by one, the steps are more complicated, and the requirement on calculation force is higher, so in some preferred embodiments, the verification process can be more efficient by reasonably setting the feature formula in the formula to the verification code corresponding to each public key system. The method specifically comprises the following steps:
the feature code S obtained by the previous steps a Feature code S b And feature code S c Substituting into the verification equation and simplifying:
wherein pk is a For the first private key sk a Corresponding public key and pk a = x a P；upk c For the third private key usk c Corresponding public key and upk c = x c P；
Will be
The verification code is set as the signature terminal A; will be
Verification code CA set as signature terminal B b The method comprises the steps of carrying out a first treatment on the surface of the Will respectively
And
verification code CA set as signature terminal C c2 And CA c1 ；
The above verification equation may be converted into:
in addition, the embodiment of the invention also provides a terminal device, which is characterized by comprising: a memory, a processor, a communication interface; wherein the memory has executable code stored thereon which, when executed by the processor, causes the processor to perform the aggregate signature method as described above applied to a signature terminal.
The embodiment of the invention also provides a terminal device, which is characterized by comprising: a memory, a processor, a communication interface; wherein the memory has executable code stored thereon which, when executed by the processor, causes the processor to perform the aggregate signature method as described above for use in a verification center.
The foregoing has shown and described the basic principles, principal features and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, and that the above embodiments and descriptions are merely illustrative of the principles of the present invention, and various changes and modifications may be made without departing from the spirit and scope of the invention, which is defined in the appended claims. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (9)
1. The heterogeneous aggregation signature method is characterized by being applied to a signature terminal and comprising the following steps of:
obtaining public System parameters spp= { p, G T , P, Q, 
, H 0 , H 1 , H 2 -a }; wherein p is a large prime number; g is the p-order addition cyclic group; g T Is a p factorial cyclic group;
Is a bilinear map; h 0 ，H 1 And H 2 Respectively, are hash functions of collision resistance, and H 0 From {0,1} * Mapping to {0,1} n ，H 1 From {0,1} * Mapping to G, H 2 From {0,1} * Mapping to +.>
；{0,1} * A binary sequence of arbitrary bit length;
A p-order integer domain obtained for removing zero elements; p and Q are G generator;
the signature terminal calculates a feature code S according to the public key system to which the signature terminal belongs i From the slave
R is selected randomly i Calculating R i = r i P, signature sequence (ID i , m i , R i , S i , T i ) Sending the verification result to a verification center; wherein ID i The identity information is signature terminal identity information; m is m i Is a message to be encrypted; t (T) i Is a time stamp;
for signature terminal A belonging to certificate-based public key system, its feature code S a And slave(s)
The first private key sk selected randomly in the list a Related to; for signature terminal B belonging to identity-based public key system, its signature S b With a master key s provided by a private key generating center 1 Related to; for signature terminal C belonging to public key system without certificate, its characteristic code S c With a second private key psk provided by a key generation center c And master key s 2 From->
A randomly selected third private key usk c Related to; wherein a, b, c are the number of signature terminals A, B, C, respectively, and +.>
。
2. The heterogeneous aggregated signature method of claim 1, wherein:
the feature code S a The acquisition method of (1) comprises the following steps: s is S a = (r a + h a sk a ) Q, where r a 
；h a = H 2 (ID a , m i , T i , R a )；sk a For signing terminal A from->
Is selected randomly and sk a = x a ；R a = r a P。
3. The heterogeneous aggregated signature method of claim 1, wherein:
the feature code S b The acquisition method of (1) comprises the following steps:
wherein r is b 
；h b = H 2 (ID b , m i , T i , R b )；
Is->
；P pub1 Master key s provided for a private key generation center 1 Corresponding public key and P pub1 = s 1 P；R b = r b P。
4. The heterogeneous aggregated signature method of claim 1, wherein:
the feature code S c The acquisition method of (1) comprises the following steps: s is S c = psk c + h c r c P pub2 + (r c + h c usk c ) Q, wherein psk c = s 2 H 1 (ID c )；h c = H 2 (ID c , m i , T i , R c )；r c 
；P pub2 For master key s provided with key generating center 2 Corresponding public key and P pub2 = s 2 P；R c = r c P is as follows; third private key usk c For signing terminal C from->
Is a randomly selected private key and usk c = x c 。
5. The heterogeneous aggregation signature method is characterized by being applied to a verification center and comprising the following steps of:
acquiring signature sequence (ID) transmitted by signature terminal i , m i , R i , S i , T i ) Calculation of
；
Respectively calculating according to the feature codes of each type of signature terminal
、
And->
；
Verification equation
If so, the signature verification of n user terminals is passed; otherwise, the verification is not passed. />
6. The heterogeneous aggregated signature method of claim 5, further comprising:
acquiring signature code S of signature terminal A belonging to certificate-based public key system a ，S a = (r a + h a sk a ) Q is a group; wherein r is a 
；h a = H 2 (ID a , m i , T i , R a )；sk a For signing terminal A from->
Is selected randomly and sk a = x a ；R a = r a P；
Acquiring a signature code S of a signature terminal B belonging to an identity-based public key system b ，
The method comprises the steps of carrying out a first treatment on the surface of the Wherein r is b 
；h b = H 2 (ID b , m i , T i , R b )；
Is->
；P pub1 Master key s provided for a private key generation center 1 Corresponding public key and P pub1 = s 1 P；R b = r b P；
Acquiring a signature S belonging to a signature terminal C based on a public key system without certificates c ，S c = psk c + h c r c P pub2 + (r c + h c usk c ) Q is a group; wherein psk is c = s 2 H 1 (ID c )；h c = H 2 (ID c , m i , T i , R c )；r c 
；P pub2 For master key s provided with key generating center 2 Corresponding public key and P pub2 = s 2 P；R c = r c P is as follows; third private key usk c For signing terminal C from->
Is a randomly selected private key and usk c = x c 。
7. The heterogeneous aggregated signature method of claim 6, further comprising:
will characteristic code S a Feature code S b And feature code S c Substituting the obtained product into the verification equation, and obtaining after simplification:
wherein pk is a For the first private key sk a Corresponding public key and pk a = x a P；upk c For the third private key usk c Corresponding public key and upk c = x c P；
Will be
Verification code CA set as signature terminal A a The method comprises the steps of carrying out a first treatment on the surface of the Will->
Verification code CA set as signature terminal B b The method comprises the steps of carrying out a first treatment on the surface of the Will be->
And->
Verification code CA set as signature terminal C c2 And CA c1 ；
The above verification equation may be converted into:
8. a terminal device, comprising: a memory, a processor, a communication interface; wherein the memory has stored thereon executable code which, when executed by the processor, causes the processor to perform the aggregate signature method of any of claims 1 to 4.
9. An electronic device, comprising: a memory, a processor, a communication interface; wherein the memory has stored thereon executable code which, when executed by the processor, causes the processor to perform the heterogeneous aggregate signature method of any of claims 5 to 7.
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