BIOMETRIC BASED IDENTITY BASED ENCRYPTION METHOD AND APPARATUS
Field of the Invention The present invention relates to security methods and apparatuses using biometric data; in particular, the present invention relates to such methods and apparatuses that utilise identifier-based encryption/decryption and analogous techniques.
As used herein, the term "biometric data" means any digital data, however measured or recorded, that represents characteristics of a biological individual intended to be unique to that individual. Thus, both digital image data of a human face and digital fingerprint data are examples of biometric data.
Background of the Invention The use of biometric data for authenticating individuals is well known. It is also known to use biometric authentication techniques in relation to memory-based identity cards - for example, such a card can carry fingerprint data concerning the card owner, this data being used to check whether a person presenting the card is the card owner by comparing the data from the card with that generated by a local fingerprint reader. Of course, the biometric data on such a card has to be trustable; more particularly, the card should have the properties of trustworthiness and unforgeability. Trustworthiness means that any information stored in the card must be issued by a trusted authority (that is, an authority trusted by the party relying on the authenticity of the stored biometric data). Unforgeability means that any information stored in the card cannot be modified by an unauthorized entity without being detected (a typical, but not the only, example of a suitable form of card would be one using a write-once memory chip).
It is also known to provide memory-based cards with cryptographic functions. For example, data may be stored in encrypted form on the card and only accessible after the card owner has entered a decryption key which the card uses to decrypt the stored data before it is output from the card.
It is an object of the present invention to provide improved security methods based on biometric data, such methods being usable, without limitation, in relation to memory-based security cards.
The present invention is in part based on the appreciation that Identifier-Based Encryption (IBE) has certain properties than can be adapted for use in memory-card based security systems and other applications.
Identifier-Based Encryption (IBE) is an emerging cryptographic schema. In this schema (see Figure 1 of the accompanying drawings), a data provider 10 encrypts payload data 13 using both an encryption key string 14, and public data 15 provided by a trusted authorityl2. This public data 15 is derived by the trusted authority 12 using private data 17 and a one-way function 18. The data provider 10 then provides the encrypted payload data <13> to a recipient 11 who decrypts it, or has it decrypted, using a decryption key computed by the trusted authority 12 in dependence on the encryption key string and its own private data.
A feature of identifier-based encryption is that because the decryption key is generated from the encryption key string, its generation can be postponed until needed for decryption.
Another feature of identifier-based encryption is that the encryption key string is crypto graphically unconstrained and canbe any kind of string, that is, any ordered series of bits whether derived from a character string, a serialized image bit map, a digitized sound signal, or any other data source. The string may be made up of more than one component and may be formed by data already subject to upstream processing. In order to avoid cryptographic attacks based on judicious selection of a key string to reveal information about the encryption process, as part of the encryption process the encryption key string is passed through a one-way function (typically some sort of hash function) thereby making it impossible to choose a cryptographically-prejudicial encryption key string. In applications where defence against such attacks is not important, it would be possible to omit this processing of the string.
Frequently, the encryption key string serves to "identify" the intended message recipient and the trusted authority is arranged to provide the decryption key only to this identified intended recipient. This has given rise to the use of the label "identifier-based" or "identity- based" generally for cryptographic methods of the type under discussion. However, depending on the application to which such a cryptographic method is put, the string may serve a different purpose to that of identifying the intended recipient and may be used to convey other information to the trusted authority or, indeed, may be an arbitrary string having no other purpose than to form the basis of the cryptographic processes. Accordingly, the use of the term "identifier-based" or "IBE" herein in relation to cryptographic methods and systems is to be understood simply as implying that the methods and systems are based on the use of a cryptographically unconstrained string whether or not the string serves to identify the intended recipient. Generally, in the present specification, the term "encryption key string" or "EKS" is used rather than "identity string" or "identifier string" ; the term "encryption key string" is also used in the shortened form "encryption key" for reasons of brevity.
A number of IBE algorithms are known and Figure 2 indicates, for three such algorithms, the following features, namely:
- the form of the encryption parameters 5 used, that is, the encryption key string and the public data of the trusted authority (TA);
- the conversion process 6 applied to the encryption key string to prevent attacks based on judicious selection of this string;
- the primary encryption computation 7 effected;
- the form of the encrypted output 8. The three prior art IBE algorithms to which Figure 2 relates are:
Quadratic Residuosity (QR) method as described in the paper: C. Cocks, "An identity based encryption scheme based on quadratic residues", Proceedings of the 8 IMA International Conference on Cryptography and Coding, LNCS 2260, pp 360-363, Springer- Verlag, 2001. A brief description of this form of IBE is given hereinafter. - Bilinear Mappings p using, for example, a modified Tate pairing t or modified Weil pairing e for which:
where
and G
2 denote two algebraic groups of prime order q and G
2 is a subgroup of a multiplicative group of a finite field. For the Tate pairing an asymmetric form is also possible: . G\ x Go G
2 where G is a further algebraic group the elements of which are not restricted to being of order q. Generally, the elements of the groups G
0 and G
\ are points on an elliptic curve though this is not necessarily the case. A description of this form of IBE method, using modified Weil pairings is given in the paper: D. Boneh, M. Franklin - "Identity- based Encryption from the Weil Pairing" in Advances in Cryptology - CRYPTO 2001 , LNCS 2139, pp. 213-229, Springer-Verlag, 2001.
- RS A-Based methods The RS A public key cryptographic method is well known and in its basic form is a two-party method in which a first party generates a public/private key pair and a second party uses the first party's public key to encrypt messages for sending to the first party, the latter then using its private key to decrypt the messages. A variant of the basic RS A method, known as "mediated RS A", requires the involvement of a security mediator in order for a message recipient to be able to decrypt an encrypted message. An IBE method based on mediated RSA is described in the paper "Identity based encryption using mediated RSA", D. Boneh, X. Ding and G. Tsudik, 3rd Workshop on Information Security Application, Jeju Island, Korea, Aug, 2002.
A more detailed description of the QR method is given below with reference to the entities depicted in Figure 1 and using the same notation as given for this method in Figure 2. In the QR method, the trust authority' s public data 15 comprises a value N that is a product of two random prime numbers p and q, where the values of p and q are the private data 17 of the trust authority 12. The values of p and q should ideally be in the range of 251 and 2512 and should both satisfy the equation: p,q ≡ 3 mod 4. However, p and q must not have the same value. Also provided is a hash function # which when applied to a string returns a value in the range 0 to N-l .
Each bit of the user's payload data 13 is then encrypted as follows:
The data provider 10 generates random numbers t+ (where t+ is an integer in the range [0, 2^]) until a value of t+ is found that satisfies the equationy'αcobtCt+,N)=m', where m' has a value of -1 or 1 depending on whether the corresponding bit of the user's data is 0 or 1 respectively. (As is well known, the jacobi function is such that where x2 ≡#modN the jacobi (#, N) = -1 if x does not exist, and = 1 if x does exist). The data provider 10 then computes the value: s+s (t++ iC/t+)modN where: s+ corresponds to the encrypted value of the bit m ' concerned, and
K = #(encryption key string)
Since K may be non-square, the data provider additionally generates additional random numbers t_ (integers in the range [0, 2^)) until one is found that satisfies the equation jacobi(t_,N)= m '. The data provider 10 then computes the value:
s_ s (t_ - χyt_)modN
as the encrypted value of the bit m concerned.
The encrypted values s+ and s. for each bit ' of the user's data are then made available to the intended recipient 11, for example via e-mail or by being placed in a electronic public area; the identity of the trust authority 12 and the encryption key string 14 will generally also be made available in the same way.
The encryption key string 14 is passed to the trust authority 12 by any suitable means; for example, the recipient 11 may pass it to the trust authority or some other route is used - indeed, the trust authority may have initially provided the encryption key string. The trust authority 12 determines the associated private key B by solving the equation : B2 ≡KmoάN ("positive" solution)
If a value of B does not exist, then there is a value of B that is satisfied by the equation:
B2 ≡ - KmodN ("negative" solution)
As Ν is a product of two prime numbers p, q it would be extremely difficult for any one to calculate the decryption key B with only knowledge of the encryption key string and N.
However, as the trust authority 12 has knowledge of p and q (i.e. two prime numbers) it is relatively straightforward for the trust authority 12 to calculate B.
Any change to the encryption key string 14 will result in a decryption key 16 that will not decrypt the payload data 13 correctly. Therefore, the intended recipient 11 cannot alter the encryption key string before supplying it to the trust authority 12.
The trust authority 12 sends the decryption key to the data recipient 11 along with an indication of whether this is the "positive" or "negative" solution for B.
If the "positive" solution for the decryption key has been provided, the recipient 11 can now recover each bit m' of the payload data 13 using: m ' =jacobi(s++2B,N)
If the "negative" solution for the decryption key B has been provided, the recipient 11 recovers each bit m ' using: m ' —jacobi(sΛ2B,N)
Summary of the Invention
According to one aspect of the present invention, there is provided a security method, carried out by a trusted authority, comprising receiving biometric data of a specific individual, and using the biometric data both: as a biometric reference for comparison with biometric characteristics of a subject individual to determine whether the latter is said specific individual, and to generate a decryption key based on at least the biometric data and private data of the trusted authority.
The present invention also envisages apparatus and a computer program product conesponding to the foregoing security method of the invention.
According to another aspect of the present invention, there is provided a data access control method comprising:
(a) encrypting first data based on encryption parameters comprising public data of a trusted authority and an encryption key string formed using at least biometric data of a specific individual;
(b) providing the biometric data of said specific individual to the trusted authority which uses it both: as a biometric reference for comparison with biometric characteristics of a subject individual to determine whether the latter is said specific individual, and to generate a decryption key based on at least the biometric data and private data of the trusted authority, said public data of the trusted authority being related to its private data;
(c) using the decryption key to decrypt the encrypted first data.
hi one preferred embodiment, in step (a) the biometric data of said specific individual is read from a memory device of the specific individual and the encrypted first data is stored back to the memory device. When the specific individual wishes to retrieve the first data, that individual presents the memory device to the trusted authority which reads off the biometric data of said specific individual, and then if satisfied that the individual is the specific individual, the trusted authority decrypts the first data and makes it available to the specific individual. This embodiment provides a simple way for a person to store password data securely and later retrieve it.
In another prefened embodiment, step (a) is carried out by a data provider with the biometric data of said specific individual being image data derived from a photograph of the said specific individual, the biometric data of the specific individual being sent to a receiving party together with the encrypted first data for use by the trusted authority in step (b).
In a further preferred embodiment, step (a) is carried out by a data provider with said biometric data of the specific individual comprising data that is the same as biometric data stored on a memory device of said specific individual as a result of having been eitlier read from that card or provided from a common source. The aforesaid subject individual seeks to obtain the decryption key from the trusted authority by presenting a memory device to
the trusted authority to enable the latter to read off biometric data stored in the device and use it as said encryption key; the trusted authority only provides the decryption key to the subject individual if the latter is determined in step (b) to be said specific individual.
The present invention also envisages a system for implementing the foregoing data access control method of the invention.
Brief Description of the Drawings Embodiments of the invention will now be described, by way of non-limiting example, with reference to the accompanying diagrammatic drawings, in which:
. Figure 1 is a diagram illustrating the operation of a prior art encryption schema known as Identifier-Based Encryption;
. Figure 2 is a diagram illustrating how certain IBE operations are implemented by three different prior art IBE methods;
. Figure 3 is a diagram of a generalized system embodying the present invention;
. Figure 4A is a diagram of a data encryption stage of a first specific example of the Figure 3 system; and
. Figure 4B is a diagram of key generation and decryption stages of the first specific example of the Figure 3 system.
Best Mode of Carrying Out the Invention
Figure 3 illustrates a generalised system embodying the present invention, the system comprising: a data encryptor entity 20 for encrypting data D using an encryption key string KENC and public data of a trusted authority; a trusted authority entity 40 for generating a decryption key KDEC based on the encryption key string KENC and private data of the trusted authority, the public data being data generated by the entity 40 and being computationally related to the trusted authority's private data ; and a data decryptor entity 30 for using the decryption key KDEC and the public data to decrypt the encrypted data D. The entities 20, 30 and 40 are typically based around general-purpose processors executing stored programs but may include dedicated cryptographic hardware modules; furthermore, as will be
discussed below, certain functions of the trusted authority may be carried out by human operators. The computing entities 20, 30 and 40 inter-communicate as needed via, for example, the internet or other network, or by the transfer of data using portable storage devices; it is also possible that at least some of the entities actually reside on the same computing platform. Indeed, in certain embodiments the data decryptor entity 30 may be incorporated into the trusted authority entity 40 whilst in other embodiments the data encryptor entity 20 and the data decryptor entity 30 may be associated with the same individual and be provided by the same computing device.
The system employs Identifier-Based Encryption with the entities 20, 30 and 40 having, in respect of IBE encryption/decryption processes, the roles of the data provider 10, data recipient 11 and trusted authority 12 of the Figure 1 IBE arrangement. The IBE algorithm used is, for example, the QR algorithm described above with respect to Figure 1 with the private data of the trusted authority being random prime numbers p,q and the conesponding public data being number N.
The encryption key string KENC s based on biometric data 50 of a specific individual 70. This biometric data is represented in Figure 3 by a face icon but can be any type of biometric data and is not limited to a facial image; possible types of biometric data include image data, fingerprint data, retina scan data etc. The biometric data can be compressed in form and can be obscured for privacy reasons, for example, by being subject to a known one-way function.
The biometric data 50 is provided in digital form to the encryptor entity 20 from a biometric data source 51 that may take a variety of fonns. For example, the biometric data source 51 maybe a capture device (such as a camera or fingerprint reader) for generating the biometric data directly from the individual 70 at the time it is required for use by the entity 20 - in other words, the individual 70 is present at the entity 20 at the time the data D is to be encrypted. Alternatively, the biometric data can be generated from an analogue storage source (such as a photographic print) or retrieved from a digital data storage medium; in particular, in one embodiment, the biometric data 50 is stored in digital form
on a memory card or other storage device 51 that belongs to the individual 70 and that preferably has the aforementioned properties of trustworthiness and unforgeability
The biometric data 50 is used by the entity 20 to form the encryption key string KENC3 the biometric data either being used directly as the key or after processing (see dashed operation oval 24) such as by concatenation with other data. The encryption key string KENC is then used to encrypt data D to form encrypted data E(KENC,N;D) where E() indicates that the elements appearing before the semi-colon inside the brackets are used to IBE encrypt the element appearing after the semi-colon. The encrypted data is then either stored to a storage medium for eventual transfer to the decryptor entity 30, or sent over a communications link directly or indirectly to the decryptor entity 30 (see arrow 61). Where the biometric data source is a storage device, the encrypted data may, in certain embodiments, be stored to this device as will be more fully described hereinafter. The biometric data 50 per se or as incorporated into the encryption key string KENC may be stored or transmitted along with the encrypted data.
When an individual 70A who may or may not be the same as the individual 70, wishes to access the encrypted data E(KENC>N;D) this individual presents themselves to the trusted authority entity 40 to which is also provided biometric data that may or may not be that used in the encryption key string KENC-
Considering first the situation where the biometric data is genuine - that is, it really is the biometric data used in the encryption key string KENC - this is represented in Figure 3 by the dashed arrows 63 and 64. The arrow 63 represents the case where the biometric data provided to the encryptor entity 20 and the trusted authority entity 40 is passed to each entity from the same source 51, for example, because the biometric data is provided from off the same memory card to both entities. The arrow 64 represents the case where the biometric data was output by the entity 20 along with the encrypted data (typically, but not necessarily, in its form incorporated into the encryption key string KENC) and has now been passed to the trusted authority. Where the biometric data only forms part of the encryption key string KENC but is provided in this form to the trusted authority, the latter is arranged to extract the biometric data from the key.
The trusted authority entity first uses the biometric data 50 as a biometric reference for comparison with biometric characteristics of the individual 70A to determine whether the latter is the individual 70 (see operation oval 44). As is well known to persons skilled in the art, this comparison and determination may be carried out automatically by comparing features represented in the reference biometric data 50 with features in measurement data produced by measurement of the subject individual 70A using biometric measurement equipment. Where the biometric data 50 is of obscured form (that is, the biometric measurements of individual 70 have been subject to a one-way function, for example, to produce the data 50), the un-obscured biometrics of individual 70 will have first been translated into biometric feature categories; accordingly, the same feature categorisation and obscuring functions must be applied to the biometric characteristics of the individual 70A to produce data for comparison with the biometric data 50. This categorisation is necessary because when the biometric data is in its obscured form, the comparison operation 44 can only be based on an exact match (a near match being meaningless).
Particularly where the biometric data comprises facial image data, an alternative to effecting an automatic biometric comparison, is to have a human operator presented with the biometric reference data (for example, as an image of a face where the biometric data is facial image data), this operator then judging whether the present individual 70A is the same as that represented by the biometric data.
If no match is found between the individual 70A and that represented by the biometric data 50, the trusted authority 40 refuses to proceed with the generation of the decryption key KDEC needed to access the encrypted data. However, if a match is found in operation 44, the trusted authority proceeds. Where biometric data 50 does not constitute the encryption key string KENC n its entirety, the next operation is to re-form the encryption key string (see dashed operation oval 45) - this may involve the concatenation of the biometric data with other data known to both the entities 20 and 40. For example, this other data may simply be an item of non-confidential data or it may be a shared secret; this other data may vary between encryption operations of the entity 20. Of course, where the encryption key
string KE C itself was provided to the trusted authority, then this is used directly without needing to reform it.
Once the encryption key string has been obtained, the trusted authority uses it, along with its private data , q, to generate the decryption key KDEC (see operation oval 46). As can be seen, the same biometric data that was used as the biometric reference data in operation 44 is also used in the process 46 of generating the decryption key KDEC-
The decryption key KDEC s then transferred (see arrow 66) to the data decryptor entity 30 to which the encrypted data E(KENC,N;D) is also supplied (see arrow 67). The transfer of the decryption key to the entity 30 from the entity 40 may be effected over a communications link or via a data storage device; as already indicated, in certain embodiments, the decryption entity 30 is actually part of the trusted authority so no transfer is required. The decryption key KDEC is thereafter used to decrypt the encrypted data to recover the data D in clear (operation 35). Where the decryption is effected by the trusted party entity 40, the recovered data D is typically then provided to the individual 70A (now known to be the individual 70) either by displaying it or by the transfer of an electronic or paper copy to the individual; however, the trusted authority may decide not to disclose the data D.
It will be appreciated that the trusted authority can carry out the key generation operation 46 in parallel with, or even before, having determined that the individual 70A is the individual 70 - what is important is that the entity 40 does not provide the decryption key (or where it also effects the decryption operation 35, the recovered data D) to the individual 70A until the latter is determined to be the individual 70.
The foregoing description of the operation of the trusted authority entity 40 was for the situation of the biometric data provided to the entity being the genuine biometric data 50 used in the encryption key string KENC- If the biometric data presented to the trusted authority entity 40 is not that used for the encryption key string (represented by dashed arrow 65 in Figure 3) as maybe the case where the individual 70A is not the individual 70 and tries to fool the trusted authority by presenting their own biometric data, then even
though the trusted authority may be fooled into generating a decryption key, this key will not serve to decrypt the encrypted data E(KENC,N;D). This is because the trusted authority uses the same biometric data for both operations 44 and 46.
Figures 4A and 4B illustrates a first specific embodiment of the generalized Figure 3. system. In this embodiment, the biometric data of the individual 70 is stored on a memory card 52 that serves as a security card for an organisation such as a commercial enterprise. The card has, for example, a picture of the individual 70 on its front face and an embedded memory chip divided into a write-once first portion 53 holding the biometric data 50 and a re-writable second portion 55. For simplicity, in the present example it is assumed that the biometric data 50 directly constitutes the encryption key string KENC to be used by the encryptor entity 20 to encrypt data D.
Suppose the individual 70 wishes to safely store all the many passwords that he/she has for accessing various services. In this case, the individual supplies these passwords as the data D to an encryptor entity 20 that includes a card reader 26. The entity 20 reads the biometric data 50 from the card 52 and uses it as the encryption key string KENC to encrypt the passwords (operation 25); the entity 20 then writes the encrypted data to the rewritable portion 55 of the memory card 52. The individual 70 now has their passwords safely stored in their memory card 52. The entity 20 can be provided by a computer or other device under the control of the individual or can be provided by the trusted authority - in this example, the trusted authority may be the security office of the enterprise.
Should the individual 70 forget any of their passwords, he/she goes to the trusted authority (security office) and presents their memory card 52 (see Figure 4B). The biometric data 50 is read off this card by the trusted authority entity and used in operation 44 to check that the individual presenting the memory card 52 is the owner of the card as indicated by the biometric data on the card. Preferably, the biometric data 50 is a facial image of the individual enabling a security office member to readily check that the individual presenting the card is the card owner. Assuming that the check 44 is passed, the decryptions key KDEC is generated and used by the trusted authority entity to decrypt the password data D held on
the card; this password data is then displayed to the individual 70 on a display 48 in a manner such that the data D is not visible to members of the security office.
Rather than the password storage device being an enterprise security card and the trusted authority being a security office, the individual can store their passwords on any storage medium they deem appropriate and select any party as a trusted authority provided the latter can be trusted to keep their private data (p,q) confidential and not to retain copies of the decrypted passwords. Another possible trusted authority would be a trusted computing platform having functionality such as specified, for example, in "TCPA - Trusted Computing Platform Alliance Main Specification vl .1 " www.trustedcomputing.org.2001 and described in the book "trusted computing platforms - tcpa technology in context"; Pearson (editor); Prentice Hall; ISBN 0-13-009220-7".
According to a second specific embodiment of the generalized Figure 3. system, the encryptor entity 20 is operated by a data provider that wishes to send data D to the individual 70 in a secure manner. The data provider has a reasonably current photograph of the individual 70 and so the data provider scans in the photograph to produce digital image data which is then used as the biometric data 50 from which an encryption key string KE C is formed. The data D is then encrypted using this key and the public data of a trusted authority such as a post office local to the individual. The data provider sends the encrypted data and the encryption key string KENC to the individual 70 who extracts the encryption key string, puts it on a floppy disc (or other storage device) and takes it to their local post office acting as a trusted authority. The post office reads the biometric data and brings up an image which a counter clerk then uses to determine if the individual presenting the biometric data is that represented by the data; if so, the clerk causes the biometric data to be used to generate a decryption key which is then stored to the storage device of the individual. The individual can now take away the decryption key and use it to recover the data D in clear.
According to a third specific embodiment of the generalized Figure 3 system, an individual has a memory card holding their biometric data. This individual wishes to store sensitive data D (such as their medical records) from a data provider and accordingly presents the
memory card to the data provider. The data provider reads off the biometric data and first confirms that this data corresponds to the individual present. Assuming this is the case, the data provider encrypts the data D and stores it back to the card. If at any time in the future, access is required to the sensitive data, a trusted authority reads off the biometric data and confirms that the card belongs to the individual concerned before generating the decryption key and decrypting the data D. In this embodiment, one would normally require some consensual act by the card owner (such as presentation of the card to the trusted authority); however, this embodiment also allows the trusted authority to access the data D in an emergency situation - as might be needed where the individual has had a road traffic accident and the attending medical staff need urgently to access the medical record data D recorded on the card (in this case, the trusted authority would be the emergency services).
In the foregoing example, the data provider could in fact have initially obtained the biometric data not from the card but from a common source - for example, the card and the encrypted data may be created simultaneously using the same biometric data.
As already noted, the encryption key string KENC may comprise data additional to the biometric data 50. This additional data may, for example, be conditions placed by the data provider on the release of the data D, these conditions being checked by the trusted authority before generation of the decryption key and/or release of the decryption key / the decrypted data.
It will be appreciated that instead of the QR IBE method, the above-described embodiments can be implemented using any other suitable IBE algorithm, such as those mentioned above that use of Weil or Tate pairings, or are RSA based; analogous cryptographic algorithms can also be used.
Whilst in the foregoing example the biometric data has concerned human individuals, the biometric data can alternatively be that of another type of biological organism such as a dog or horse.
Furthermore, although in the described examples the individuals have presented themselves to the trusted authority, the trusted authority or a party associated with it may be more pro-active and approach or otherwise select an individual (for example, a customs officer may select a traveller at an airport and ask to see their identity card which is a memory card with biometric data).
The trusted authority may be distributed in nature having, for example, a remote station at which an individual presents themselves for biometric measurement, and a central station where biometric data is compared and decryption key generation is carried out.
It is possible to require the involvement of multiple trust-authority entities effectively forming a compound trust authority. This maybe desirable where a single authority is not trusted to be entirely reliable. One way of achieving this would be for the data encryptor to recursively encrypt the data D, with each iteration being done using the same encryption key string but the public data of a different trusted authority - the individual must then go to several trust authorities in turn to successively roll back each encryption iteration. An alternative approach is for the data provider to encrypt the data D using a public base key associated with each of the trusted authorities, decryption of the encrypted item only being possible by obtaining a decryption sub-key from the trusted delegate entity acting for each trusted authority in turn. This can be expressed as: Encryption: ciphertext = E(K_all, data) Decryption: data = D(K_all, ciphertext) where K_all is encryption key string related to all trusted authorities, K'_all is the conesponding decryption key; K'_all is retrieved from all decryption sub-keys. Further information about how multiple trusted authorities can be used is given in:
Chen L., K. Harrison, A. Moss, N.P. Smart and D. Soldera. "Certification of public keys within an identity based system" Proceedings of Information Security Conference 2002, ed. A. H. Chan and V. Gligor, LNCS 2433, pages 322-333, Springer-Verlag, 2002.