Patent Publication Number: US-7590854-B2

Title: Hierarchical identity-based encryption and signature schemes

Description:
RELATED APPLICATIONS 
   The present application is a continuation of U.S. patent application Ser. No. 11/552,076, filed Oct. 23, 2006, which is now U.S. Pat. No. 7,337,322, incorporated herein by reference, which is a division of U.S. patent application Ser. No. 10/384,328, filed Mar. 7, 2003, which is now U.S. Pat. No. 7,349,538, incorporated herein by reference, which claims priority under 35 U.S.C. § 119(e) to provisional U.S. patent applications Ser. No. 60/366,292, filed on Mar. 21, 2002, and Ser. No. 60/366,196, filed on Mar. 21, 2002, both of which are incorporated herein by reference. 

   BACKGROUND OF THE INVENTION 
   The present invention relates in general to cryptography and secure communication via computer networks or via other types of systems and devices, and more particularly to hierarchical, identity-based schemes for encrypting and decrypting communications. 
   Roughly speaking, identity-based cryptosystems are public key cryptosystems in which the public key of an entity is derived from information associated with the entity&#39;s identity. For instance, the identity information may be personal information (i.e., name, address, email address, etc.), or computer information (i.e., IP address, etc.). However, identity information may include not only information that is strictly related to an entity&#39;s identity, but also widely available information such as the time or date. That is, the importance of the concept of identity information is not its strict relation to the entity&#39;s identity, but that the information is readily available to anyone who wishes to encrypt a message to the entity. 
   An entity&#39;s private key is generated and distributed by a trusted party or logical process, typically known as a private key generator (“PKG”). The PKG uses a master secret to generate private keys. Because an entity&#39;s public key may be derived from its identity, when Alice wants to send a message to Bob, she does not need to retrieve Bob&#39;s public key from a database. Instead, Alice merely derives the key directly from Bob&#39;s identifying information. Databases of public keys are unnecessary. Certificate authorities (“CAs”) also are unnecessary. There is no need to “bind” Bob&#39;s identity to his public key because his identity is his public key. 
   The concept of identity-based cryptosystems is not new. It was proposed in A. Shamir,  Identity - Based Cryptosystems and Signatures Schemes , A DVANCES IN  C RYPTOGRAPHY —C RYPTO  &#39;84, Lecture Notes in Computer Science 196 (1984), Springer, 47-53. However, practical identity-based encryption schemes have not been found until recently. For instance, identity-based schemes were proposed in C. Cocks,  An Identity - Based Encryption Scheme Based on Quadratic Residues , available at http://www.cesg.gov.uk/technology/id-pkc/media/ciren.pdf; D. Boneh, M. Franklin,  Identity Based Encryption from the Weil Pairing , A DVANCES IN  C RYPTOLOGY —C RYPTO  2001, Lecture Notes in Computer Science 2139 (2001), Springer, 213-229; and D. Boneh, M. Franklin,  Identity Based Encryption from the Weil Pairing  (extended version), available at http://www.cs.stanford.edu/˜dabo/papers/ibe.pdf. Cocks&#39;s scheme is based on the “Quadratic Residuosity Problem,” and although encryption and decryption are reasonably fast (about the speed of RSA), there is significant message expansion (i.e., the bit-length of the ciphertext is many times the bit-length of the plaintext). The Boneh-Franklin scheme bases its security on the “Bilinear Diffie-Hellman Problem,” and it is quite fast and efficient when using Weil or Tate pairings on supersingular elliptic curves or abelian varieties. 
   However, the known identity-based encryption schemes have a significant shortcoming—they are not hierarchical. In non-identity-based public key cryptography, it has been possible to have a hierarchy of CAs in which the root CA can issue certificates for other CAs, who in turn can issue certificates for users in particular domains. This is desirable because it reduces the workload on the root CA. A practical hierarchical scheme for identity-based cryptography has not been developed. 
   Ideally, a hierarchical identity-based encryption scheme would involve a hierarchy of logical or actual PKGs. For instance, a root PKG may issue private keys to other PKGs, who in turn would issue private keys to users in particular domains. It also would be possible to send an encrypted communication without an online lookup of the recipient&#39;s public key or lower-level public parameters, even if the sender is not in the system at all, as long as the sender obtained the public parameters of the root PKG. Another advantage of a hierarchical identity-based encryption scheme would be damage control. For instance, disclosure of a domain PKG&#39;s secret would not compromise the secrets of higher-level PKGs, or of any other PKGs that are not direct descendents of the compromised domain PKG. The schemes taught by Cocks and Boneh-Franklin do not have these properties. 
   A secure and practical hierarchical identity-based encryption scheme has not been developed. A hierarchical identity-based key sharing scheme with partial collusion-resistance is given in G. Hanaoka, T. Nishioka, Y. Zheng, H. Imai,  An Efficient Hierarchical Identity - Based Key - Sharing Method Resistant Against Collusion Attacks , A DVANCES IN  C RYPTOGRAPHY —A SIACRYPT  1999, Lecture Notes in Computer Science 1716 (1999), Springer 348-362; and G. Hanaoka, T. Nishioka, Y. Zheng, H. Imai,  A Hierarchical Non - Interactive Key - Sharing Scheme With Low Memory Size and High Resistance Against Collusion Attacks , to appear in T HE  C OMPUTER  J OURNAL . In addition, an introduction to hierarchical identity-based encryption was provided in J. Horwitz, B. Lynn,  Toward Hierarchical Identity - Based Encryption , to appear in A DVANCES IN  C RYPTOGRAPHY —E UROCRYPT  2002, Lecture Notes in Computer Science. Springer. Horwitz and Lynn proposed a two-level hierarchical scheme with total collusion-resistance at the first level and partial collusion-resistance at the second level (i.e., users can collude to obtain the secret of their domain PKG and thereafter masquerade as that domain PKG). However, the complexity of the Horwitz-Lynn system increases with the collusion-resistance at the second level, and therefore that scheme cannot be both practical and secure. 
   Accordingly, there has been a need for a secure and practical hierarchical identity-based encryption scheme. It is therefore an object of the present invention to provide a secure and practical hierarchical identity-based encryption scheme. It is another object of the present invention to provide a secure and practical hierarchical identity-based signature scheme. It is a further object of the present invention that the encryption and signature schemes be fully scalable. It is a still further object of the present invention that the encryption and signature schemes have total collusion resistance on an arbitrary number of levels, and that they have chosen-ciphertext security in the random oracle model. 
   BRIEF SUMMARY OF THE PREFERRED EMBODIMENTS 
   In accordance with the present invention, methods are provided for implementing secure and practical hierarchical identity-based encryption and signature schemes. 
   According to one aspect of the present invention, a method is provided for encoding and decoding a digital message between a sender and a recipient in a system including a plurality of private key generators (“PKGs”). The PKGs include at least a root PKG and n lower-level PKG in the hierarchy between the root PKG and the recipient, wherein n≧1. A root key generation secret is selected and is known only to the root PKG. A root key generation parameter is generated based on the root key generation secret. A lower-level key generation secret is selected for each of the n lower-level PKGs, wherein each lower-level key generation secret is known only to its associated lower-level PKG. A lower-level key generation parameter also is generated for each of the n lower-level PKGs using at least the lower-level key generation secret for its associated lower-level private key generator. The message is encoded to form a ciphertext using at least the root key generation parameter and recipient identity information. A recipient private key is generated such that the recipient private key is related to at least the root key generation secret, one or more of the n lower-level key generation secrets associated with the n lower-level PKGs in the hierarchy between the root PKG and the recipient, and the recipient identity information. The ciphertext is decoded to recover the message using at least the recipient private key. 
   According to another aspect of the present invention, a method is provided for encoding and decoding a digital message between a sender and a recipient in a system including a plurality of private key generators (“PKGs”). The PKGs include at least a root PKG, m lower-level PKGs in the hierarchy between the root PKG and the sender, wherein m≧1, n lower-level PKG in the hierarchy between the root PKG and the recipient, wherein n≧1, and PKG l , which is a common ancestor PKG to both the sender and the recipient. In the hierarchy, l of the m private key generators are common ancestors to both the sender and the recipient, wherein l≧1. 
   According to this aspect of the invention, a lower-level key generation secret is selected for each of the m lower-level PKGs in the hierarchy between the root PKG and the sender. A sender private key is generated such that the sender private key is related to at least the root key generation secret, one or more of the m lower-level key generation secrets associated with the m lower-level PKGs in the hierarchy between the root PKG and the sender, and sender identity information. A recipient private key is generated such that the recipient private key is related to at least the root key generation secret, one or more of the n lower-level key generation secrets associated with the n lower-level PKGs in the hierarchy between the root PKG and the recipient, and recipient identity information. The message is encoded using at least the recipient identity information, the sender private key, and zero or more of the lower-level key generation parameters associated with the (m−l+1) private key generators at or below the level of the common ancestor PKG l , but not using any of the lower-level key generation parameters that are associated with the (l−1) PKGs above the common ancestor PKG l . The message is decoded using at least the sender identity information, the recipient private key, and zero or more of the lower-level key generation parameters associated with the (n−l+1) private key generators at or below the level of the common ancestor PKG l , but not using any of the lower-level key generation parameters that are associated with the (l−1) PKGs above the common ancestor PKG l . 
   According to another aspect of the present invention, a method is provided for generating and verifying a digital signature of a message between a sender and a recipient in a system including a plurality of PKGs. The PKGs include at least a root PKG and n lower-level PKG in the hierarchy between the root PKG and the sender, wherein n≧1. A root key generation secret is selected and is known only to the root PKG. A root key generation parameter is generated based on the root key generation secret. A lower-level key generation secret is selected for each of the n lower-level PKGs, wherein each lower-level key generation secret is known only to its associated lower-level PKG. A lower-level key generation parameter also is generated for each of the n lower-level PKGs using at least the lower-level key generation secret for its associated lower-level private key generator. A private key is generated for the sender such that the private key is related to at least the root key generation secret and sender identity information. The message is signed to generate the digital signature using at least the sender private key. The digital message is verified using at least the root key generation parameter and the sender identity information. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The subsequent description of the preferred embodiments of the present invention refers to the attached drawings, wherein: 
       FIG. 1  shows a flow diagram illustrating a method of encoding and decoding a digital message according to one presently preferred embodiment of the invention; 
       FIG. 2  shows a flow diagram illustrating a method of encoding and decoding a digital message between a sender y and a recipient z according to another presently preferred embodiment of the invention; 
       FIG. 3  shows a block diagram illustrating a typical hierarchical structure in which this method of  FIG. 2  may be performed; 
       FIG. 4  shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention; 
       FIG. 5  shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention; 
       FIG. 6  shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention; 
       FIG. 7  shows a flow diagram illustrating a method of generating and verifying a digital signature according to another presently preferred embodiment of the invention; 
       FIG. 8  shows a flow diagram illustrating a method of generating and verifying a digital signature Sig of a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention; and 
       FIG. 9  shows a flow diagram illustrating a method of generating and verifying a digital signature Sig of a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   The presently preferred methods of the invention provide secure and practical hierarchical identity-based encryption (“HIDE”) and signature (“HIDS”) schemes. The hierarchical schemes are fully scalable, have total collusion resistance on an arbitrary number of levels, and have chosen-ciphertext security in the random oracle model. These objectives are achieved, in part, by introducing additional random information at each of the lower-level PKGs. One intuitively surprising aspect of these schemes is that, even though lower level PKGs generate additional random information, this does not necessitate adding public parameters below the root level of the hierarchy. In addition, the random information generated by a lower-level PKG does not adversely affect the ability of users not under the lower-level PKG to send encrypted communications to users under the lower-level PKG. 
   Each of the HIDE and HIDS schemes of the present invention requires a hierarchical structure of PKGs, including at least one root PKG and a plurality of lower-level PKGs. The hierarchy and the lower-level PKGs may be logical or actual. For instance, a single entity may generate both a root key generation secret and the lower-level key generation secrets from which lower-level users&#39; encryption or signature keys are generated. In this case, the lower-level PKGs are not separate entities, but are merely processes or information arranged in a logical hierarchy and used to generate keys for descendent PKGs and users in the hierarchy. Alternatively, each lower-level PKG may be a separate entity. Another alternative involves a hybrid of actual and logical lower-level PKGs. For purposes of this disclosure, the term “lower-level PKG” will be used generically to refer to any of these alternatives. 
   In the context of the hierarchical identity-based cryptosystems disclosed herein, identity-based public keys may be based on time periods. For instance, a particular recipient&#39;s identity may change with each succeeding time period. Alternatively, a recipient may arrange the time periods as children or descendents of itself in a hierarchy, and a sender would use the identity of the proper time period when encoding the message. Either way, each key may be valid for encrypting messages to Bob only during the associated time period. 
   The HIDE schemes of the present invention generally include five randomized algorithms: Root Setup, Lower-level Setup, Extraction, Encryption, and Decryption. Three of these algorithms rely upon the identities of the relevant entities in the hierarchy. Each user preferably has a position in the hierarchy that may be defined by its tuple of IDs: (ID 1 , . . . , ID t ). The user&#39;s ancestors in the hierarchy are the root PKG and the users, or PKGs, whose ID-tuples are {(ID 1 , . . . , ID i ): 1≦i≦(t−1)}. The ID-tuples preferably are represented as binary strings for purposes of computations. 
   In the Root Setup algorithm, the root PKG uses a security parameter k to generate public system parameters params and a root key generation secret. The system parameters include a description of the message space   and the ciphertext space  . The system parameters will be publicly available, while only the root PKG will know the root key generation secret. 
   In the Lower-level Setup algorithm, each lower-level PKG preferably generates its own lower-level key generation secret for purposes of extraction. Alternatively, a lower-level PKG may generate random one-time secrets for each extraction. 
   In the Extraction algorithm, a PKG (whether the root PKG or a lower-level PKG) generates a private key for any of its children. The private key is generated using the system parameters, the generating PKG&#39;s private key, and any other preferred secret information. 
   In the Encryption algorithm, a sender receives the system parameters from the root PKG, preferably via some secure means outside the present system. It is not necessary for the sender to receive any of the lower-level key generation parameters. The sender encodes a message M ε   to generate a ciphertext C ε   using params and the ID-tuple of the intended recipient. Conversely, in the Decryption algorithm, the recipient decodes the ciphertext C to recover the message M using params and the recipient&#39;s private key d. Encryption and decryption preferably satisfy the standard consistency constraint: 
   ∀M ε  : Decryption(params, d, C)=M 
   where C=Encryption(params, ID-tuple, M). 
   Like the HIDE schemes, the HIDS schemes of the present invention also generally include five randomized algorithms: Root Setup, Lower-level Setup, Extraction, Signing, and Verification. For Root Setup, the system parameters are supplemented to include a description of the signature space  . Lower-level Setup and Extraction preferably are the same as for HIDE, as described above. 
   In the Signing algorithm, the sender of a digital message signs the message M ε   to generate a signature S ε   using params and the sender&#39;s private key d. In the Verification algorithm, the recipient of the signed message verifies the signature S using params and the ID-tuple of the sender. The Verification algorithm preferably outputs “valid” or “invalid”. Signing and Verification also preferably satisfies a consistency constraint: 
   ∀M ε  : Verification(params, ID-tuple, S)=“valid” 
   where S=Signing(params, d, M). 
   Security of HIDE and HIDS Schemes 
   The security of the schemes embodying the present invention will now be discussed with respect to both HIDE and HIDS. It has been noted in the context of non-hierarchical identity-based cryptography that the standard definition of chosen-ciphertext security must be strengthened for identity-based systems. This is because it should be assumed, for purposes of a security analysis, that an adversary can obtain the private key associated with any identity of its choice (other than the particular identity being attacked). The same applies to hierarchical identity-based cryptography. Accordingly, to establish that the HIDE schemes of the present invention are chosen-ciphertext secure, a simulated attacker is allowed to make private key extraction queries. Also, the simulated adversary is allowed to choose the identity on which it wishes to be challenged. 
   It should also be noted that an adversary may choose the identity of its target adaptively or nonadaptively. An adversary that chooses its target adaptively will first make hash queries and extraction queries, and then choose its target based on the results of these queries. Such an adversary might not have a particular target in mind when it begins the attack. Rather, the adversary is successful when it is able to hack somebody. A nonadaptive adversary, on the other hand, chooses its target independently from results of hash queries and extraction queries. For example, such an adversary might target a personal enemy. The adversary may still make hash queries and extraction queries, but its target choice is based strictly on the target&#39;s identity, not on the query results. Obviously, security against an adaptively-chosen-target adversary is the stronger, and therefore preferable, notion of security. However, the security analysis of the HIDE schemes in the present invention address both types of security. 
   A HIDE scheme is said to be semantically secure against adaptive chosen ciphertext and adaptive chosen target attack if no polynomially bounded adversary   has a non-negligible advantage against the challenger in the following game. 
   SETUP: The challenger takes a security parameter k and runs the Root Setup algorithm. It gives the adversary the resulting system parameters params. It keeps the root key generation secret to itself. 
   PHASE 1: The adversary issues queries q 1 , . . . , q m , where q i  is one of:
         1. Public-key query (ID-tuple i ): The challenger runs a hash algorithm on ID-tuple i  to obtain the public key H (ID-tuple i ) corresponding to ID-tuple i .   2. Extraction query (ID-tuple i ): The challenger runs the Extraction algorithm to generate the private key d i  corresponding to ID-tuple i , and sends d i  to the adversary.   3. Decryption query (ID-tuple i , C i ): The challenger runs the Extraction algorithm to generate the private key d i  corresponding to ID-tuple i , runs the Decryption algorithm to decrypt C i  using d i , and sends the resulting plaintext to the adversary.
 
These queries may be asked adaptively. In addition, the queried ID-tuple i  may correspond to a position at any level of the hierarchy.
       

   CHALLENGE: Once the adversary decides that Phase 1 is over, it outputs two equal-length plaintexts M 0 , M 1  ε   and an ID-tuple on which it wishes to be challenged. The only constraints are that neither this ID-tuple nor its ancestors appear in any private key extraction query in Phase 1. The challenger picks a random bit b ε {0,1} and sets C=Encryption(params, ID-tuple, M b ). It sends C as a challenge to the adversary. 
   PHASE 2: The adversary issues more queries q m+1 , . . . , q n  where q i  is one of:
         1. Public-key query (ID-tuple i ): The challenger responds as in Phase 1.   2. Extraction query (ID-tuple i ): The challenger responds as in Phase 1.   3. Decryption query (C, ID-tuple i ): The challenger responds as in Phase 1.
 
The queries in Phase 2 are subject to the constraint that the challenger cannot make an Extraction query on the ID-tuple associated with the challenge ciphertext C, or make a Decryption query using that ID-tuple and the ciphertext C. This same constraint also applies to all ancestors of the ID-tuple.
       

   GUESS: The adversary outputs a guess b′ ε {0,1}. The adversary wins the game if b=b′. The adversary&#39;s advantage in attacking the scheme is defined to be |Pr[b=b′]−½|. 
   A HIDE schemes is said to be a one-way encryption scheme if no polynomial time adversary has a non-negligible advantage in the game described below. In this game, the adversary   is given a random public key K pub  and a ciphertext C that is the encryption of a random message M using K pub , and outputs a guess for the plaintext. The adversary is said to have an advantage ε against the scheme if ε is the probability that   outputs M. The game is played as follows: 
   SETUP: The challenger takes a security parameter k and runs the Root Setup algorithm. It gives the adversary the resulting system parameters params. It keeps the root key generation secret to itself. 
   PHASE 1: The adversary makes public key and/or extraction queries as in Phase 1 of the chosen-ciphertext security analysis described above. 
   CHALLENGE: Once the adversary decides that Phase 1 is over, it outputs a new ID-tuple ID on which it wishes to be challenged. The challenger picks a random M ε   and sets C=Encryption(params, ID-tuple, M). It sends C as a challenge to the adversary. 
   PHASE 2: The adversary issues more public-key queries and more extraction queries on identities other than ID and its ancestors, and the challenger responds as in Phase 1. 
   GUESS: The adversary outputs a guess M′ ε  . The adversary wins the game if M=M′. The adversary&#39;s advantage in attacking the scheme is defined to be Pr[M=M′]. 
   The schemes of the present invention are secure against the challenges described above. In addition, the HIDS schemes of the present invention are secure against existential forgery on adaptively chosen messages. An adversary should be unable to forge its target&#39;s signature on other messages that the target has not signed previously, even after (adaptively) obtaining the target&#39;s signature on messages of the adversary&#39;s choosing. A HIDS adversary also will have the ability to make public key queries and private key extraction queries on entities other than the target and its ancestors, and the ability to choose its target. As with HIDE, the adversary&#39;s choice of target may be adaptive or nonadaptive. 
   Pairings 
   The presently preferred HIDE and HIDS schemes of the present invention are based on pairings, such as, for instance, the Weil or Tate pairings associated with elliptic curves or abelian varieties. The methods also are based on the Bilinear Diffie-Hellman problem. They use two cyclic groups    1  and    2 , preferably of the same large prime order q. The first group    1  preferably is a group of points on an elliptic curve or abelian variety, and the group law on    1  may be written additively. The second group    2  preferably is a multiplicative subgroup of a finite field, and the group law on    2  may be written multiplicatively. However, other types of groups may be used as    1  and    2  consistent with the present invention. 
   The methods also use a generator P 0  of the first group    1 . In addition, a pairing or function ê:    1 ×   1 →   2  is provided for mapping two elements of the first group    1  to one element of the second group    2 . The function ê preferably satisfies three conditions. First, the function ê preferably is bilinear, such that if Q and R are in    1  and a and b are integers, then ê(aQ, bR)=ê(Q, R) ab . Second, the function ê preferably is non-degenerate, such that the map does not send all pairs in    1 ×   1  to the identity in    2 . Third, the function ê preferably is efficiently computable. A function ê satisfying these three conditions is considered to be admissible. 
   The function ê also preferably is symmetric, such that ê(Q, R)=ê(R, Q) for all Q, R ε    1 . Symmetry, however, follows immediately from the bilinearity and the fact that    1  is a cyclic group. Weil and Tate pairings associated with supersingular elliptic curves or abelian varieties can be modified to create such bilinear maps according to methods known in the art. However, even though reference to elements of the first cyclic group    1  as “points” may suggest that the function ê is a modified Weil or Tate pairing, it should be noted that any admissible pairing ê will work. 
   The security of the HIDE and HIDS schemes of the present invention is based primarily on the difficulty of the Bilinear Diffie-Hellman problem. The Bilinear Diffie-Hellman problem is that of finding ê(P, P) abc  given a randomly chosen P ε    1 , as well as aP, bP, and cP (for unknown randomly chosen a, b, c ε  /q ). Solving the Diffie-Hellman problem in    1  solves the Bilinear Diffie-Hellman problem because ê(P, P) abc =ê(abP, cP). Similarly, solving the Diffie-Hellman problem in    2  solves the Bilinear Diffie-Hellman problem because, if g=ê(P, P), then g abc =(g ab ) c  where g ab =ê(aP, bP) and g c =ê(P, cP). For the Bilinear Diffie-Hellman problem to be hard,    1  and    2  should be chosen such that there is no known algorithm for efficiently solving the Diffie-Hellman problem in either    1  or    2 . If the Bilinear Diffie-Hellman problem is hard for a pairing ê, then it follows that ê is non-degenerate. 
   A randomized algorithm   is a Bilinear Diffie-Hellman generator if   takes a security parameter k&gt;0, runs in time polynomial in k, and outputs the description of two groups    1  and    2 , preferably of the same prime order q, and the description of an admissible pairing ê:    1 ×   1 →   2 . If   is a Bilinear Diffie-Hellman parameter generator, the advantage Adv ( ) that an algorithm   has in solving the Bilinear Diffie-Hellman problem is defined to be the probability that the algorithm   outputs ê(P, P) abc  when the inputs to the algorithm are    1 ,    2 , ê, P, aP, bP, and cP, where (   1 ,    2 , ê) is the output of   for a sufficiently large security parameter k, P is a random generator of    1 , and a, b, and c are random elements of  /q . The assumption underlying the Bilinear Diffie-Hellman problem is that Adv ( ) is negligible for all efficient algorithms  . 
   HIDE Schemes 
   Referring now to the accompanying drawings,  FIG. 1  shows a flow diagram illustrating a method of encoding and decoding a digital message according to one presently preferred embodiment of the invention. The method is performed in a HIDE system including a plurality of PKGs. The PKGs include at least a root PKG and n lower-level PKGs in the hierarchy between the root PKG and the recipient, wherein n≧1. 
   In block  102 , the root PKG selects a root key generation secret known only to the root PKG. The root key generation secret may be used to generate private keys for PKGs and/or users below the root PKG in the hierarchy. The root PKG then generates a root key generation parameter based on the root key generation secret in block  104 . The root key generation parameter is used to mask the root key generation secret. The root key generation parameter may be revealed to lower-level PKGs without compromising the root key generation secret. The lower-level PKGs select lower-level key generation secrets in block  106 . The lower-level key generation secret associated with a given lower-level PKG may be used to generate private keys for PKGs and/or users below the associated lower-level PKG in the hierarchy. Like the root key generation secret, each of the lower-level key generation secrets is known only to its associated lower-level PKG. 
   In block  108 , lower-level key generation parameters are generated for each of the n lower-level PKGs. Each of the lower-level key generation parameters is generated using at least the lower-level key generation secret for its associated lower-level PKG. Like the root key generation parameter, each of the lower-level key generation parameters masks its associated lower-level key generation secret. 
   Using at least the root key generation parameter and identity information associated with the recipient, the sender encodes the message in block  110  to form a ciphertext. For instance, the message may be encoded using only the root key generation parameter and the recipient&#39;s identity. Alternatively, one of the lower-level key generation parameters may be used, such as is described in more detail below with respect to dual-HIDE schemes. In block  112 , a lower-level PKG generates a private key for the recipient such that the private key is related to at least the root key generation secret, one or more of the n lower-level key generation secrets associated with the n lower-level PKGs in the hierarchy between the root PKG and the recipient, and the recipient&#39;s identity information. For instance, in addition to root key generation secret and the recipient&#39;s identity information, the recipient&#39;s private key preferably also is related at least to the lower-level key generation secret of the PKG that issued the private key to the recipient. Alternatively, the recipient&#39;s private key may be related to all n of its ancestral PKG&#39;s lower-level key generation secrets, as well as the root key generation secret. In block  114 , the recipient uses at least its private key to decode the ciphertext and recover the message. In addition to using its private key to decode, the recipient preferably also uses the n lower-level key generation parameters associated with the n lower-level PKGs in the hierarchy between the root PKG and the recipient. 
   Each lower-level PKG has a key generation secret, just like the root PKG. As described above, a lower-level PKG preferably uses this secret to generate a private key for each of its children, just as the root PKG does. As a result, the children&#39;s private keys are related to the lower-level PKG&#39;s key generation secret. This is true even if the lower-level PKG uses a modified version of its key generation secret to obscure that secret for purposes of restricting key escrow, as described more fully below. At the same time, the lower-level PKGs need not always use the same secret for each private key extraction. Rather, a new key generation secret could be generated randomly for each of the PKG&#39;s children, resulting in a different key generation parameter for each child. 
   Because a lower-level PKG is able to generate a private key for the recipient (block  112 ), the root PKG need not generate all of the private keys itself. In addition, because the lower-level PKGs use their own key generation secrets to generate private keys for their descendants, compromising a lower-level key generation secret causes only limited security damage to the hierarchy. Rather than compromising all of the private keys in the hierarchy, a breach of a lower-level PKG compromises only the private key of that PKG and those private keys that were generated using that PKG&#39;s key generation secret (i.e., the private keys of those users that are direct hierarchical descendants of the compromised PKG). 
   Another advantage of this embodiment is that the sender need not be in the hierarchy to send an encoded message to the recipient. The sender merely needs to know the identity information associated with the recipient and the system parameters generated by the root PKG. There are however, certain additional advantages of the HIDE schemes of the present invention that become available when the sender is positioned within the hierarchy. For instance, when both the sender and the recipient are in the hierarchy, the efficiency of the message encryption may be improved by using the identities of both parties. This type of HIDE scheme may be referred to as dual-HIDE because the identities of both the sender and the recipient are used as input for the encryption and decryption algorithms. A method of encoding and decoding a message using a dual-HIDE scheme will now be discussed with reference to  FIGS. 2 and 3 . 
   Dual-HIDE 
     FIG. 2  shows a flow diagram illustrating a method of encoding and decoding a digital message between a sender y and a recipient z according to another presently preferred embodiment of the invention.  FIG. 3  shows a block diagram illustrating a typical hierarchical structure in which this method may be performed. Like the previous embodiment, this method is performed in a HIDE system including at least a root PKG  302  and n lower-level PKGs  304   a,b,d  in the hierarchy between the root PKG  302  and the recipient z  308 , wherein n≧1. The sender y  306  in this embodiment also must be in the hierarchy, and the hierarchy also includes m lower-level PKGs  304   a,b,c  between the root PKG  302  and the sender y  306 , wherein m≧1. Of the m PKGs  304   a,b,c  between the root PKG  302  and the sender y  306 , and the n PKGs  304   a,b,d  between the root PKG  302  and the recipient z  308 , there are l PKGs  304   a,b  that are common ancestors to both the sender y  306  and the recipient z  308 , wherein 1≦l≦m, n. For instance, two of these l common ancestral PKGs (PKG y1 /PKG z1    304   a  and PKG yl /PKG zl    304   b ) are shown in  FIG. 3 . 
   The method of this embodiment begins in block  202 , when the root PKG  302  selects a root key generation secret known only to the root PKG  302 . The root PKG  302  then generates a root key generation parameter based on the root key generation secret in block  204 . The lower-level PKGs  304   a - d  select lower-level key generation secrets in block  206 . Like the root key generation secret, each of the lower-level key generation secrets is known only to its associated lower-level PKG  304   a - d . In block  208 , lower-level key generation parameters are generated for each of the n lower-level PKGs  304   a - d . Each of the lower-level key generation parameters is generated using at least the lower-level key generation secret for its associated lower-level PKG  304   a - d.    
   In block  210 , the sender&#39;s parent PKG ym    304   c  generates a private key for the sender y  306  such that the private key is related to at least the root key generation secret, one or more of the m lower-level key generation secrets associated with the m lower-level PKGs  304   a,b,c  between the root PKG  302  and the sender y  306 , and the sender&#39;s identity information. For instance, in addition to root key generation secret and the sender&#39;s identity information, the sender&#39;s private key preferably is related at least to the lower-level key generation secret of the sender&#39;s parent PKG ym    304   c . Alternatively, the sender&#39;s private key may be related to all m of its direct ancestral PKGs&#39; lower-level key generation secrets, as well as the root key generation secret. In block  212 , the recipient&#39;s parent PKG zn    304   d  generates a private key for the recipient z in a similar manner that the sender&#39;s parent PKG ym    304   c  used to generate the sender&#39;s private key. 
   In block  214 , the sender y encodes the message to form a ciphertext using at least the sender&#39;s private key and one or more of the lower-level key generation parameters associated with the (m−l+1) PKGs (i.e., PKG yl ,  304   b  and PKG ym    304   c ) between the root PKG  302  and the sender y  306  that are at or below the level of the lowest ancestor PKG (PKG yl /PKG zl    304   b ) that is common to both the sender y  306  and the recipient z  308 . In encoding the message, the sender y  306  preferably does not use any of the lower-level key generation parameters that are associated with the (l−1) PKGs (i.e., PKG y1    304   a ) that are above the lowest common ancestor PKG (PKG yl /PKG zl    304   b ). 
   The recipient z  308  then decodes the ciphertext to recover the message in block  216  using at least the recipient&#39;s private key and one or more of the lower-level key generation parameters associated with the (n−l+1) PKGs (i.e., PKG zl ,  304   b  and PKG zn    304   c ) between the root PKG  302  and the recipient z  308  that are at or below the level of the lowest ancestor PKG (PKG yl /PKG zl    304   b ) that is common to both the sender y  306  and the recipient z  308 . In decoding the message, the recipient z  306  preferably does not use any of the lower-level key generation parameters that are associated with the (l−1) PKGs (i.e., PKG z1    304   a ) that are above the lowest common ancestor PKG (PKG yl /PKG zl    304   b ). 
   This dual-HIDE embodiment of the invention provides a more efficient scheme for encoding and decoding the message because it requires the use of fewer key generation parameters. For instance, decoding in a regular HIDE scheme preferably requires all n of the key generation parameters, but decoding in a dual-HIDE scheme preferably requires only (n−l+1) of the key generation parameters. Dual-HIDE schemes require the sender y  306  to obtain its private key before sending an encoded message to the recipient z  308 , as opposed to merely obtaining the public system parameters of the root PKG. The dual-HIDE schemes also enable the sender y  306  and the recipient z  308  to restrict the scope of key escrow, as described more fully below. This shared secret is unknown to third parties other than their lowest common ancestor PKG yl /PKG zl    304   b.    
   BasicHIDE 
   In some embodiments, the scheme is as follows. Let Level i  be the set of entities at level i, where Level 0 ={Root PKG}. Let K be the security parameter given to the setup algorithm and let   be a BDH parameter generator. 
   Root Setup: The root PKG: 
   1. runs   on input K to generate groups    1 ,    2  of some prime order q and an admissible pairing ê:    1 ×   1 →   2 ; 
   2. chooses an arbitrary generator P 0 ε   1 ; 
   3. picks a random s 0 ε /q  and sets Q 0 =s 0 P 0 ; 
   4. chooses cryptographic hash functions H 1 :{0,1}*→   1  and H 2 :   2 →{0,1} n  for some n. The security analysis will treat H 1  and H 2  as random oracles. 
   The message space is  ={0,1} n . The ciphertext space  =   1   t ×{0,1} t  where t is the level of the recipient. The system parameters are params={   1 ,   2 ,ê,P 0 ,Q 0 ,H 1 ,H 2 }. The root PKG&#39;s secret is s 0 ε /q . 
   Lower-level Setup. Entity E t εLevel t  picks a random s t ε /q , which it keeps secret. 
   Extraction: Let E t  be an entity in Level t  with ID-tuple (ID 1 , . . . ,ID t ), where (ID 1 , . . . ,ID i ) for 1≦i≦t is the ID tuple of E t &#39;s ancestor at Level i . Set S 0  to be the identity element of    1 . The E t &#39;s parent: 
   1. computes P t =H 1 (ID 1 , . . . ,ID t )ε   1 ; 
   2. sets E t &#39;s secret point S t  to be 
   
     
       
         
           
             
               
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   3. also gives E t  the values of Q i =s i P 0  for 1≦i≦t−1. 
   Encryption: To encrypt M ε   with the ID-tuple (ID 1 , . . . ,ID t ), do the following: 
   1. Compute P i =H 1 (ID 1 , . . . ,ID i )ε   1  for 1≦i≦t. 
   2. Choose a random rε /q . 
   3. Set the ciphertext to be
 
 C=[rP   0   ,rP   2   , . . . ,rP   t   ,M⊕H   2 ( g   r )] where  g=ê ( Q   0   ,P   1 )ε    2 .
 
   Decryption: Let C=[U 0 ,U 2 , . . . ,U t ,V]ε  be the ciphertext encrypted using the ID-tuple (ID 1 , . . . ,ID t ). To decrypt C, E t  computes: 
   
     
       
         
           
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   This concludes the description of one embodiment of our BasicHIDE scheme. 
   Remark 1. Each lower-level PKG—say, in Level t —has a secret s t ε /q , just like the root PKG. A lower-level PKG uses this secret to generate a secret point for each of its children, just as the root PKG does. An interesting fact, however, is that lower-level PKGs need not always use the same s t  for each private key extraction. Rather, s t  could be generated randomly for each of the PKG&#39;s children. 
   Remark 2. H 1  can be chosen to be an iterated hash function so that, for example, P i  may be computed as H 1 (P i−1 ,ID i ) rather than H 1 (ID 1 , . . . ,ID i ). 
     FIG. 4  shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. The recipient z  308  is n+1 levels below the root PKG in the hierarchy, as shown in  FIG. 3 , and is associated with the ID-tuple (ID z1 , . . . , ID z(n+1) ). The recipient&#39;s ID-tuple includes identity information ID z(n+1)  associated with the recipient, as well as identity information ID zi  associated with each of its n ancestral lower-level PKGs in the hierarchy. The method begins in block  402  by generating first and second cyclic groups    1  and    2  of elements. In block  404 , a function ê is selected, such that the function ê is capable of generating an element of the second cyclic group    2  from two elements of the first cyclic group    1 . The function ê preferably is an admissible pairing, as described above. A root generator P 0  of the first cyclic group    1  is selected in block  406 . In block  408 , a random root key generation secret s 0  associated with and known only to the root PKG  302  is selected. Preferably, s 0  is an element of the cyclic group  /q . A root key generation parameter Q 0 =s 0 P 0  is generated in block  410 . Preferably, Q 0  is an element of the first cyclic group    1 . In block  412 , a first function H 1  is selected such that H 1  is capable of generating an element of the first cyclic group    1  from a first string of binary digits. A second function H 2  is selected in block  414 , such that H 2  is capable of generating a second string of binary digits from an element of the second cyclic group    2 . The functions of blocks  402  through  414  are part of the HIDE Root Setup algorithm described above, and preferably are performed at about the same time. By way of example, the functions such as those disclosed in Boneh-Franklin may be used as H 1  and H 2 . 
   The next series of blocks (blocks  416  through  424 ) show the functions performed as part of Lower-level Setup algorithm. In block  416 , a public element P zi  is generated for each of the recipients&#39; n ancestral lower-level PKGs. Each of the public elements, P zi =H 1 (ID 1 , . . . , ID zi ) for 1≦i≦n, preferably is an element of the first cyclic group    1 . Although represented in a single block, generation of all the public elements P zi  may take place over time, rather than all at once. 
   A lower-level key generation secret s zi  is selected (block  418 ) for each of the recipients&#39; n ancestral lower-level PKGs  304   a,b,d . The lower-level key generation secrets s zi  preferably are elements of the cyclic group  /q  for 1≦i≦n, and each lower-level key generation secret s zi  preferably is known only to its associated lower-level PKG. Again, although represented in a single block, selection of all the lower-level key generation secrets s zi  may take place over time, rather than all at once. 
   A lower-level secret element S zi  is generated (block  420 ) for each of the sender&#39;s n ancestral lower-level PKGs. Each lower-level secret element, S zi =S z(i−1) +s z(i−1) P zi  for 1≦i≦n preferably is an element of the first cyclic group    1 . Although represented in a single block like the public elements P zi  and the secrets s zi , generation of all the secret elements S zi  may take place over time, rather than all at once. For purposes of these iterative key generation processes, S 0  may be defined to be the identity element of    1 . 
   A lower-level key generation parameter Q zi  also is generated (block  422 ) for each of the recipients&#39; n ancestral lower-level PKGs. Each of the key generation parameters, Q zi =s zi P 0  for 1≦i≦n, preferably is an element of the first cyclic group    1 . Again, although represented in a single block, generation of all the key generation parameters Q zi  may take place over time, rather than all at once. 
   The functions of the next two blocks (blocks  424  and  426 ) are performed as part of the Extraction algorithm described above. A recipient public element P z(n+1)  associated with the recipient z is generated in block  424 . The recipient public element, P z(n+1) =H 1 (ID z1 , . . . , ID z(n+1) ), preferably is an element of the first cyclic group    1 . A recipient secret element S z(n+1)  associated with the recipient z is then generated in block  426 . The recipient secret element 
               S     z   ⁡     (     n   +   1     )         =         S   zn     +       s   zn     ⁢     P     z   ⁡     (     n   +   1     )             =       ∑     i   =   1       n   +   1       ⁢       s     z   ⁡     (     i   -   1     )         ⁢     P   zi             ,         
also preferably is an element of the first cyclic group    1 .
 
   For convenience, the first function H 1  optionally may be chosen to be an iterated function so that, for example, the public points P i  may be computed as H 1 (P z(i−1) , ID zi ) rather than H 1  (ID 1 , . . . , ID zi ). 
   The last two blocks shown in  FIG. 4  (blocks  428  and  430 ) represent the Encryption and Decryption algorithms described above. In block  428 , the message M is encoded to generate a ciphertext C. The encoding preferably uses at least the root key generation parameter Q 0  and the ID-tuple (ID z1 , . . . , ID z(n+1) ). The ciphertext C is then decoded in block  430  to recover the message M The decoding preferably uses at least the lower-level key generation parameters Q zi  for 1&lt;i&lt;n, and the recipient secret element S z(n+1) . 
   The blocks shown in  FIG. 4  need not all occur in sequence. For instance, a sender who knows a recipient&#39;s identity may encrypt communications to the recipient before the recipient obtains its private key. 
   The specific use of the parameters and elements described above in the encoding and decoding of the message M and the ciphertext C will now be discussed with reference to  FIGS. 5 and 6 .  FIG. 5  shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. In this scheme, which may be referred to as BasicHIDE, the Root Setup, Lower-level Setup, and Extraction algorithms are the same as for the embodiment shown in blocks  402  through  426  of  FIG. 4 . The flow diagram of  FIG. 5  illustrates the Encryption and Decryption algorithms, beginning with the selection of a random encryption parameter r in block  528   a . Preferably, r is an integer of the cyclic group  /q . The ciphertext C is then generated in block  528   b  using the formula C=[U 0 , U 2 , . . . , U n+1 , V]. The ciphertext C includes elements U i =rP zi  for i=0 and for 2≦i≦n+1, which relate to the location of the recipient in the hierarchy. The other part of the ciphertext C is the actual message in encrypted form, V=M ⊕ H 2 (g r ), wherein g=ê(Q 0 , P z1 ). The element g preferably is a member of the second cyclic group    2 . After the message has been encoded, it may be decoded according to the BasicHIDE Decryption algorithm, in which the message M is recovered from the ciphertext C (block  530 ) using the formula 
           M   =     V   ⊕         H   2     (         e   ^     ⁡     (       U   0     ,     S     n   +   1         )           ∏     i   =   2       n   +   1       ⁢           ⁢       e   ^     ⁡     (       Q     i   -   1       ,     U   i       )           )     .             
FullHIDE
 
   It is known that Fujisaki-Okamato padding can be used to convert a basic IBE scheme to an IBE scheme that is chosen ciphertext secure in the random oracle model. In the same way, BasicHIDE can be converted to FullHIDE, a HIDE scheme that is chosen ciphertext secure in the random oracle model. Next we describe the scheme FullHIDE. One embodiment is as follows: 
   Setup: As in the BasicHIDE scheme, but in addition choose hash functions H 3 :{0,1} n ×{0,1} n → /q  and H 4 :{0,1} n →{0,1} n . 
   Extraction: As in the BasicHIDE scheme. 
   Encryption: To encrypt Mε  with the ID-tuple (ID 1 , . . . ,ID t ), do the following: 
   1. compute P i =H 1 (ID 1 , . . . ,ID i )ε   1  for 1≦i≦t, 
   2. choose a random σε{0,1} n , 
   3. set r=H 3 (σ,M), and 
   4. set the ciphertext to be
 
 C=[rP   0   ,rP   2   , . . . ,rP   t   ,σ ⊕ H   2 ( g   r ), M ⊕ H   4 (σ)]
 
   where g=ê(Q 0 ,P 1 )ε    2  as before. 
   Decryption: Let C=[U 0 ,U 2 , . . . ,U t ,V,W]ε  be the ciphertext encrypted using the ID-tuple (ID 1 , . . . ,ID t ). If (U 0 ,U 2 , . . . ,U t )∉    1   t , reject the ciphertext. To decrypt C, E t  does the following: 
   1. computes 
   
     
       
         
           
             
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   2. computes W ⊕ H 4 (σ)=M, 
   3. sets r=H 3 (σ,M) and tests that U 0 =rP 0  and U i =rP i  for i=2, . . . ,t. If not, it rejects the ciphertext. 
   4. outputs M as the decryption of C. 
   Note that M is encrypted as W=M ⊕ H 4 (σ). This can be replaced by 
           W   =       E       H   4     ⁡     (   σ   )         ⁡     (   M   )             
where E is a semantically secure symmetric encryption scheme.
 
   Using known methods for making one-way encryption schemes secure against chosen-ciphertext attacks, a BasicHIDE scheme may be converted to a FullHIDE scheme that is chosen ciphertext secure in the random oracle model. A FullHIDE scheme that is chosen ciphertext secure will now be discussed with reference to  FIG. 6 . 
     FIG. 6  shows a flow diagram illustrating a method of encoding and decoding a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. The Root Setup, Lower-level Setup, and Extraction algorithms are the same for this embodiment of the invention as for the embodiment described with reference to  FIG. 4 , except that the Root Setup algorithm of this embodiment requires two additional functions. Accordingly, the flow diagram of  FIG. 6  begins with the selection of the additional functions (blocks  615   a  and  615   b ) and continues with the Encryption and Decryption algorithms (blocks  628   a  through  630   d ). 
   The Root Setup algorithm is completed by selecting a third function H 3  (block  615   a ) and a fourth function H 4  (block  615   b ). The third function H 3  preferably is capable of generating an integer of the cyclic group  /q  from two strings of binary digits. The fourth function H 4  preferably is capable of generating one binary string from another binary string. 
   The Encryption algorithm begins with block  628   a , which shows the selection of a random binary string σ. The random binary string σ is then used to generate a random integer r=H 3 (σ, M, W), as shown in block  628   b , wherein W is a symmetric encryption of the actual message M. The encryption preferably is generated using a symmetric encryption algorithm E, and using H 4 (σ) as the encryption key. Accordingly, W=E H     4     (σ) (M). In block  628   c , the ciphertext C=[U 0 , U 2 , . . . , U n+1 , V, W] is generated. The ciphertext C includes elements U i =rP zi  for i=0 and for 2≦i≦n+1, which relate to the location of the recipient in the hierarchy. The second part of the ciphertext C is the random binary string σ in encrypted form, V=σ ⊕ H 2 (g r ), wherein g=ê(Q 0 , P z1 ). The element g preferably is a member of the second cyclic group    2 . The third part of the ciphertext C is W, the actual message in symmetrically encrypted form, as described above. 
   The Decryption algorithm begins with block  630   a , which shows the recovery of the random binary string σ. The random binary string σ is recovered using the formula 
           σ   =     V   ⊕         H   2     (         e   ^     ⁡     (       U   0     ,     S     z   ⁡     (     n   +   1     )           )           ∏     i   =   2       n   +   1       ⁢       e   ^     ⁡     (       Q     i   -   1       ,     U   i       )           )     .             
The message M is then recovered from the ciphertext C (block  630   b ) using the formula M=E H     4     (σ)   −1 (W). The ciphertext optionally may be checked for internal consistency. For instance, an experimental random integer r′=H 3 (σ, M, W) may be generated, as shown in block  630   c . The experimental random integer r′ then may be used in block  630   d  to confirm that U 0 =r′P 0  and U i =r′P zi  for 2≦i≦n+1. If so, then the ciphertext C is considered to be authentic.
 
Dual-BasicHIDE and Dual-FullHIDE
 
   In 2000, Sakai, Ohgishi and Kasahara presented a “key sharing scheme” based on the Weil pairing. The idea was quite simple: suppose a PKG has a master secret s, and it issues private keys to users of the form sP y  where P y =H 1 (ID y ) and ID y  is the ID of user y (as in Boneh-Franklin). Then users y and z have a shared secret that only they (and the PKG) may compute, namely ê(sP y ,P z )=ê(P y ,sP z ). They may use this shared secret to encrypt their communications. Notice that this “key sharing scheme” does not require any interaction between the parties. We can view Sakai, Ohgishi and Kasahara&#39;s discovery as a type of “dual-identity-based encryption,” where the word “dual” indicates that the identities of both the sender and the recipient (rather than just the recipient) are required as input into the encryption and decryption algorithms. The only significant practical difference between this scheme and the Boneh-Franklin IBE scheme is that the sender must obtain its private key from the PKG before sending encrypted communications, as opposed to merely obtaining the public parameters of the PKG. 
   In the non-hierarchical context, Dual-IBE does not appear to have any substantial advantages over IBE. In the hierarchical context, however, Dual-HIDE may be more efficient than HIDE if the sender and recipient are close to each other in the hierarchy tree. Suppose two users, y and z, have the ID-tuples (ID y1 , . . . ,ID yl , . . . ,ID ym ) and (ID z1 , . . . ,ID zl , . . . ,ID zn ), where (ID y1 , . . . ,ID yl )=(ID z1 , . . . ,ID zl ). In other words, user y is in Level m , user z is in Level n , and they share a common ancestor in Level l . User y may use Dual-HIDE to encrypt a message to user z as follows: 
   Encryption: To encrypt Mε , user y: 
   1. Computes P zi =H 1 (ID z1 , . . . ,ID zi )ε   1  for l+1≦i≦n. 
   2. Chooses a random rε /q . 
   3. Set the ciphertext to be:
 
 C=[rP   0   ,rP   z(l+1)   , . . . ,rP   zn   ,M⊕H   2 ( g   yl   r )]
 
where
 
               g   yl     =           e   ^     ⁡     (       P   0     ,     S   y       )           ∏     i   =     l   +   1       m     ⁢           ⁢       e   ^     ⁡     (       Q     y   ⁡     (     i   -   1     )         ,     U   i       )           =       e   ^     ⁡     (       P   0     ,     S   yl       )           ,         
S y  is y&#39;s secret point, S yl  is the secret point of y&#39;s and z&#39;s common ancestor at level l, and Q yi =s yi P 0  where s yi  is the secret number chosen by y&#39;s ancestor at level i.
 
   Decryption: Let C=[U 0 ,U l+1 , . . . ,U n ,V] be the ciphertext. To decrypt C, user z computes: 
   
     
       
         
           
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   Note that if y and z have a common ancestor below the root PKG, then the ciphertext is shorter with Dual-HIDE than with non-dual HIDE. Further, using Dual-HIDE, the encrypter y computes m−l+1 pairings while the decryptor z computers n−l+1 pairings. (Note that m+n−2l is the “length” of the path between y and z in the hierarchy tree.) In the non-dual HIDE scheme, the encrypter computes one pairing (or receives it as a per-computer value) while the decrypter computers n pairings. Thus when m&lt;2l−1, the total work is less with Dual-HIDE than with non-dual HIDE. The relative computing power of the sender and recipient can also be taken into account. 
   The number of pairings that y and z must compute can be decreased to m+n−2l+1 if their common ancestor in Level l  always uses the same s l  rather than generating this number randomly with each private key extraction. Encryption and decryption proceed as follows: 
   Encryption: To encrypt Mε , user y: 
   1. Computes P zi =H 1 (ID z1 , . . . ,ID zi )ε   1  for l+1≦i≦n. 
   2. Chooses a random rε /q . 
   3. Set the ciphertext to be:
 
 C=[rP   0   ,r ( P   z(l+1)   −P   z(l+1) ), rP   z(l+2)   . . . ,rP   zn   ,M⊕H   2 ( g   y(l+1)   r )]
 
where
 
               g     y   ⁡     (     l   +   1     )         =           e   ^     ⁡     (       P   0     ,     S   y       )           ∏     i   =     l   +   2       m     ⁢           ⁢       e   ^     ⁡     (       Q     y   ⁡     (     i   -   1     )         ,     P   yi       )           =       e   ^     ⁡     (       P   0     ,     S     y   ⁡     (     l   +   1     )           )           ,         
where S y  is y&#39;s secret point and S y(l+1)  is the secret point of y&#39;s ancestor at level l+1.
 
   Decryption: Let C=[U 0 ,U l+1 , . . . ,U n ,V] be the ciphertext. To decrypt C, user z computes: 
   
     
       
         
           
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   The concept of dual-HIDE described with reference to  FIGS. 2 and 3  may be applied to BasicHIDE and FullHIDE schemes. When both the sender and recipient are within the hierarchical structure, as shown in  FIG. 3 , dual-HIDE allows them to increase the efficiency and security of their encrypted communications. The application of dual-HIDE to BasicHIDE and FullHIDE schemes requires the determination of additional information, most of which is determined via the Lower-level Setup algorithm described above. For instance, public elements P yi , lower-level key generation secrets s yi , lower-level secret elements S yi , and lower-level key generation parameters Q yi  must be determined for the sender&#39;s m ancestral lower-level PKGs. Note, however, that for the lower-level PKGs that are common ancestors to both the sender y and the recipient z, these parameters preferably will be the same for purposes of analyzing both the sender y and the recipient z (i.e., preferably for all i≦l:P yi =P zi , s yi =s zi , S yi =S zi , and Q yi =Q zi ). Dual-HIDE also requires determination of a sender public element P y(m+1)  and a sender secret element S y(m+1)  for the sender, using the same methods for which these parameters are determined for the recipient as described above. 
   Given these additional parameters, a message M may be encoded to generate a ciphertext C according the principles of dual-HIDE by using the lower-level key generation parameters Q yi  for i≧l and the sender secret element S y(m+1) , but not using the lower-level key generation parameters Q yi  for i&lt;l. Similarly, the ciphertext C may be decoded to recover the message M using the lower-level key generation parameters Q zi  for i≧l and the recipient secret element S z(n+1) , but not using the lower-level key generation parameters Q zi  for i&lt;l. 
   For instance, in a BasicHIDE scheme ( FIGS. 4 and 5 ), application of dual-HIDE changes the encoding of the message M to generate a ciphertext C=[U 0 , U l+1 , . . . , U n+1 , V], wherein U i =rP zi  for i=0 and for l+1≦i≦n+1, wherein V=M ⊕ H 2 (g yl   r ), and wherein 
             g   yl     =           e   ^     ⁡     (       P   0     ,     S     y   ⁡     (     m   +   1     )           )           ∏     i   =     l   +   1         m   +   1       ⁢           ⁢       e   ^     ⁡     (       Q     y   ⁡     (     i   -   1     )         ,     P   yi       )           .           
The U i  factors are calculated in the same way as before, but fewer of them are necessary. However, dual-BasicHIDE does require the sender y to use more key generation parameters Q yi  to generate g yl  than are necessary to generate g as describe above. This is because the sender&#39;s identity is being incorporated into the Encryption algorithm.
 
   The increase in efficiency of the Decryption algorithm is more dramatic. The message M is recovered using 
           M   =     V   ⊕         H   2     (         e   ^     ⁡     (       U   0     ,     S     z   ⁡     (     n   +   1     )           )           ∏     i   =     l   +   1         n   +   1       ⁢           ⁢       e   ^     ⁡     (       Q     z   ⁡     (     i   -   1     )         ,     U   zi       )           )     .             
Again, fewer U i  parameters are necessary. Similarly, the recipient requires fewer key generation parameters Q zi  for dual-HIDE than would otherwise be necessary.
 
   FullHIDE also may be modified to create a dual-FullHIDE scheme. Generation of the ciphertext C in the Encryption algorithm is modified such that C=[U 0 , U l+1 , . . . , U n+1 , V, W], wherein U i =rP zi  for i=0 and for l+1≦i≦n+1. The W and r parameters is still generated the same way, W=E H     4     (σ) (M), and the g yl  parameter in V=σ ⊕ H 2 (g yl   r ) is generated using 
   
     
       
         
           
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   The Decryption algorithm also is modified in a dual-FullHIDE scheme. The random binary string σ is recovered using 
           σ   =     V   ⊕         H   2     (         e   ^     ⁡     (       U   0     ,     S     z   ⁡     (     n   +   1     )           )           ∏     i   =     l   +   1         n   +   1       ⁢           ⁢       e   ^     ⁡     (       Q     z   ⁡     (     i   -   1     )         ,     U   zi       )           )     .             
Otherwise, recovery of the message M does not change.
 
   Although these dual-HIDE schemes have been described using PKG l    304   b  as the lowest ancestor PKG common to both the sender y and the recipient z, PKG l    304   b  may be any common ancestor PKG. The encryption and decryption algorithms are the same. For maximum efficiency however, it is preferable that PKG l    304   b  be the lowest common ancestor PKG. 
   In addition to the increase in efficiency, the dual-HIDE schemes of the present invention also offer increased security by restricting key escrow. In the BasicHIDE and FullHIDE schemes described above, all of the recipient&#39;s direct ancestor PKGs are able to decrypt messages to the recipient. However, because the dual-HIDE schemes incorporate the key generation secret of PKG l−1  (the immediate parent of PKG l ), which is unknown to the common ancestor PKGs above PKG l−1 , those common ancestor PKGs are not able to decrypt messages between the sender y and the recipient z. The immediate parent of PKG l    304   b  is still able to decrypt messages, however, because it knows its own key generation secret. 
   Key escrow may be further restricted such that even the immediate parent of PKG l  may not decrypt messages between the sender y and the recipient z. This may be accomplished by obscuring PKG l &#39;s private key in the process of generating private keys for the sender y and the recipient z (or private keys for children of PKG l  that are ancestors of the sender y and the recipient z). For instance, PKG l    304   b  may easily change its private key by setting S l ′:=S l +bP l , and Q l−1 ′:=Q l−1 +bP 0 , for some random b ε  /q . The new private key S l ′ is just as effective, but is unknown to PKG l &#39;s immediate parent. Accordingly, no PKGs above PKG l  are able to decode messages encrypted to the recipient z. More specifically, only ancestors of the recipient z that are within PKG l &#39;s domain are able to decrypt messages to the recipient z. 
   When PKG l    304   b  changes its private key by setting S l ′:=S l +bP l , and Q l−1 ′:=Q l−1 +bP 0 , the new private key is still related to PKG l−1 &#39;s key generation secret s l−1 , because the new private key is derived from a private key generated by PKG l−1  using s l−1 . In general, in all of the schemes discussed herein, a user or PKG may change its own secret element S z(n+1)  and key generation parameters Q zi  for 1≦i≦n by choosing values for b i  for 1≦i≦n and setting 
               S     z   ⁡     (     n   +   1     )       ′     ⁢     :     =     S     z   ⁡     (     n   +   1     )           +       ∑     i   =   1     n     ⁢       b   i     ⁢     P     z   ⁡     (     i   +   1     )                   
and Q zi ′:=Q zi +b i P 0  for 1≦i≦n. For purposes of the present invention, however, this new private key is still considered to be related to the original private key, and is thus related to the original values of the key generation secrets s zi .
 
Dual-HIDE Scheme With More Efficient Encryption or Decryption
 
   In the dual-HIDE schemes described above, it is possible to decrease by one the number of values of the pairing that the encrypter must compute without increasing the number of values of the pairing that the decrypter must compute. For instance, the dual-BasicHIDE Encryption algorithm described above may be modified such that the ciphertext C=[rP 0 ,r(P y(l+1) −P z(l+1) ),rP z(l+2) , . . . ,rP z(n+1) ,M ⊕ H 2 (g y(l+1)   r )], where 
             g     y   ⁡     (     l   +   1     )         =           e   ^     ⁡     (       P   0     ,     S     y   ⁡     (     n   +   1     )           )           ∏     i   =     l   +   2       m     ⁢           ⁢       e   ^     ⁡     (       Q     y   ⁡     (     i   -   1     )         ,     P   yi       )           =         e   ^     ⁡     (       P   0     ,     S     y   ⁡     (     l   +   1     )           )       .             
If the ciphertext is represented as C=[U 0 ,U l+1 , . . . ,U n+1 ,V], then it may be decrypted using
 
   
     
       
         
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   Likewise, it is possible to decrease by one the number of values of the pairing that the decrypter must compute without increasing the number of values that the encrypter must compute. For instance, the dual-BasicHIDE Encryption algorithm may be modified such that the ciphertext C=[rP 0 ,rP y(l+2) , . . . ,rP y(n) ,M ⊕ H 2 (g z(l+1)   r )], where 
             g     z   ⁡     (     l   +   1     )         =             e   ^     ⁡     (       P   0     ,     S     y   ⁡     (     m   +   1     )           )       ⁢       e   ^     ⁡     (       Q   yl     ,     (       P     z   ⁡     (     l   +   1     )         -     P     y   ⁡     (     l   +   1     )           )       )             ∏     i   =     l   +   2       m     ⁢           ⁢       e   ^     ⁡     (       Q     y   ⁡     (     i   -   1     )         ,     P   yi       )           =         e   ^     ⁡     (       P   0     ,     S     z   ⁡     (     l   +   1     )           )       .             
If the ciphertext is represented as C=[U 0 ,U l+2 , . . . ,U n ,V], then it may be decrypted using
 
           M   =     V   ⊕         H   2     (         e   ^     ⁡     (       U   0     ,     S     z   ⁡     (     n   +   1     )           )           ∏     i   =     l   +   2       n     ⁢           ⁢       e   ^     ⁡     (       Q     z   ⁡     (     i   -   1     )         ,     U   i       )           )     .             
Authenticated Lower-Level Root PKGs
 
   The efficiencies of the dual-HIDE schemes described above may be extended to message senders who are outside the hierarchy by creating an authenticated lower-level root PKG. To “authenticate” the lower-level PKG, the root PKG may issue an additional parameter, such as a random message M′. The lower-level PKG then “signs” M′, generating the signature Sig=S zl +s zl P M′ , where S l  is the lower-level PKG&#39;s private key, and s t  is its lower-level key generation secret. The lower-level PKG also publishes Q i  for 1≦i≦t. 
   Taking advantage of the authenticated lower-level root PKG, a sender outside the hierarchy may send an encrypted message to the recipient z without computing public elements P zi  for all n of the recipient&#39;s ancestor PKGs. Rather, the sender may use the parameters for the lower-level authenticated root PKG to encrypt the message more efficiently. In particular, the sender computes P zi =H 1 (ID 1 , . . . , ID zi ) ε    1  for l+1≦i≦n+1. The sender then chooses a random r ε  /q , and generates the ciphertext C=[rP 0 ,rP z(l+1) , . . . ,rP z(n+1) ,M ⊕ H 2 (g zl   r )], where 
             g   zl     =           e   ^     ⁡     (       P   0     ,   Sig     )           e   ^     ⁡     (         s   zl     ⁢     P   0       ,     P     M   ′         )         =         e   ^     ⁡     (       P   0     ,     S   zl       )       .             
Letting the received ciphertext C=[U 0 , U l+1 , . . . , U n+1 , V], the recipient may then decrypt the ciphertext to recover the message
 
             M   =     V   ⊕       H   2     [         e   ^     ⁡     (       U   0     ,     S     z   ⁡     (     n   +   1     )           )           ∏     i   =     l   +   1         n   +   1       ⁢           ⁢       e   ^     ⁡     (       Q     z   ⁡     (     i   -   1     )         ,     U   i       )           ]         ,         
where S z(n+1)  is the recipient&#39;s private key.
 
Distributed PKGs
 
   To further protect the key generation secrets of the HIDE schemes described above, and to make the schemes robust against dishonest PKGs, the key generation secrets and private keys may be distributed using known techniques of threshold cryptography. 
   More Efficient Encryption 
   The efficiency of encryption for the HIDE schemes described above may be increased by merging the highest two levels of the hierarchy into a single root PKG. In that case, g=ê(Q 0 ,P 1 ) is included in the system parameters. This saves encrypters the task of computing the value of this pairing. However, the decrypters must compute one extra pairing (as a result of being one level lower down the tree). 
   HIDS Schemes 
   It has been noted by Moni Naor that an IBE scheme can be immediately converted into a public key signature scheme as follows: the signer&#39;s private key is the master key in the IBE scheme. The signer&#39;s signature on M is the IBE decryption key d corresponding to the “public key” H 1 (ID)=H 1 (M). The verifier checks the signature by choosing a random message M′, encrypting M′ with H 1 (M), and trying to decrypt the resulting ciphertext with d. If the ciphertext decrypts correctly, the signature is considered valid. 
   This observation can be extended to a hierarchical context: a HIDE scheme can be immediately converted to a hierarchical ID=based signature (HIDS) scheme. Suppose the signer has ID-tuple (ID 1 , . . . ,ID t ). To sign M, the signer computes the private key d for the ID-tuple (ID 1 , . . . ,ID t ,M), and sends d to the verifier. As before, the verifier checks the signature by choosing a random message M′, encrypting M′ with the “public key” (ID 1 , . . . ,ID t ,M), and trying to decrypt the resulting ciphertext with d. The security of this HIDS scheme follows immediately from the security of our HIDE scheme, since forging a signer&#39;s signature is equivalent to recovering the private key of one of the signer&#39;s children. In fact, the security of the HIDS scheme is based on the difficulty of solving the Diffie-Hellman problem in the group    1 . 
   A pitfall in the HIDS scheme just described is that an attacker might try to get the signer to sign M=ID t+1  where ID t+1  represents an actual identity. In this case, the signer&#39;s signature will actually be a private key, which thereafter may be used to decrypt messages and forge signatures. One solution to this problem is to use some expedient—such as a bit prefix—that distinguishes between signing and private key extraction. Here is one embodiment. 
   Let Level i  be the set of entities at level i, where Level 0 ={Root PKG}. Let K be the security parameter given to the setup algorithm and let   be a BDH parameter generator. 
   Root Setup: The root PKG: 
   1. runs   on input K to generate groups    1 ,    2  of some prime order q and an admissible pairing ê:    1 ×   1 →   2 ; 
   2. chooses an arbitrary generator P 0 ε   1 ; 
   3. picks a random s 0 ε /q  and sets Q 0 =s 0 P 0 ; 
   4. chooses cryptographic hash functions H 1 :{0,1}*→   1  and H 3 :{0,1}*→   1 . The security analysis will treat H 1  and H 3  as random oracles. 
   The signature space is S=   1   t+1  where t is the level of the recipient. The system parameters are params={   1 ,   2 ,ê,P 0 ,Q 0 ,H 1 ,H 3 }. The root PKG&#39;s secret is s 0 ε /q . 
   Lower-level Setup. As in BasicHIDE. 
   Extraction: As in BasicHIDE. 
   Signing: To encrypt M with the ID-tuple (ID 1 , . . . ,ID t ) (using the secret point 
             S   t     =       ∑     i   =   1     t     ⁢       s     i   -   1       ⁢     P   i               
and the points Q i =s i P 0  for 1≦i≦t), do the following:
 
   1. Compute P i =H 1 (ID 1 , . . . ,ID i )ε   1  for 1≦i≦t. (These can be precomputed.) 
   2. Compute P M =H 3 (ID 1 , . . . ,ID t ,M)ε   1 . (As suggested above, we might use a bit-prefix or some other method, instead of using a totally different hash function.) 
   3. Compute Sig(ID-tuple,M)=S t +s t P M . 
   4. Send Sig(ID-tuple,M) and Q i =s i P 0  for 1≦i≦t 
   Verification: Let [Sig,Q 1 , . . . ,Q t ]ε  be the signature for (ID-tuple,M). The verifier confirms that: 
   
     
       
         
           
             
               
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   By modifying the above HIDS scheme so that an entity&#39;s point-tuple (P 1 , . . . ,P t ) is computed as a function not only of its ID-tuple (ID 1 , . . . ,ID t ), but also as a function of the points Q i =s i P 0 , i.e. P i =H 1 (ID 1 , . . . ,ID i ,Q 1 , . . . ,Q i ) for 1≦i≦t, we are able to obtain a stronger security proof. See the aforementioned U.S. patent application No. 60/366,196. 
   A dual HIDS scheme can be obtained as follows. If users y and z have a common ancestor in Level l , then y only needs to send Q yi  for l+1≦i≦m. This makes the length of the signature proportional to m−l rather than m. 
   Turning now to the signature, or HIDS, schemes of the present invention,  FIG. 7  shows a flow diagram illustrating a method of generating and verifying a digital signature according to another presently preferred embodiment of the invention. The method is performed in a HIDS system including a plurality of PKGs. The PKGs include at least a root PKG and n lower-level PKGs in the hierarchy between the root PKG and the sender, or signer, wherein n≧1. In block  702 , the root PKG selects a root key generation secret known only to the root PKG. The root key generation secret may be used to generate private keys for PKGs or users below the root PKG in the hierarchy. The root PKG then generates a root key generation parameter based on the root key generation secret in block  704 . The lower-level PKGs select lower-level key generation secrets in block  706 . The lower-level key generation associated with a given lower-level PKG may be used to generate private keys for PKGs or users below the associated lower-level PKG in the hierarchy. Like the root key generation secret, each of the lower-level key generation secrets is known only to its associated lower-level PKG. In block  708 , lower-level key generation parameters are generated for each of the n lower-level PKGs. Each of the lower-level key generation parameters is generated using at least the lower-level key generation secret for its associated lower-level PKG. 
   In block  710 , a lower-level PKG generates a private key for the recipient such that the private key is related to at least one of the n lower-level key generation secrets. For instance, the sender&#39;s private key may be related at least to the lower-level key generation secret of the PKG that issued the private key to the recipient. Preferably, however, the recipient&#39;s private key may be related to all n of its ancestral PKG&#39;s lower-level key generation secrets, as well as the root key generation secret. In block  712 , the sender uses at least its private key to sign the message and generate the digital signature. The recipient, or verifier, then verifies the digital signature in block  714  using at least one of the lower-level key generation parameters. For instance, the signature may be verified using only the root key generation parameter. Alternatively, one or more of the lower-level key generation parameters also may be used. 
     FIG. 8  shows a flow diagram illustrating a method of generating and verifying a digital signature Sig of a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. The sender y  306  is m+1 levels below the root PKG in the hierarchy, as shown in  FIG. 3 , and is associated with the ID-tuple (ID y1 , . . . , ID y(m+1) ). The sender&#39;s ID-tuple includes identity information ID y(m+1)  associated with the sender, as well as identity information ID yi  associated with each of its m ancestral lower-level PKGs in the hierarchy. The method begins in block  802  by generating first and second cyclic groups    1  and    2  of elements. In block  804 , a function ê is selected, such that the function ê is capable of generating an element of the second cyclic group    2  from two elements of the first cyclic group    1 . The function ê preferably is an admissible pairing, as described above. A root generator P 0  of the first cyclic group    1  is selected in block  806 . In block  808 , a random root key generation secret s 0  associated with and known only to the root PKG  302  is selected. Preferably, s 0  is an element of the cyclic group  /q . A root key generation parameter Q 0 =s 0 P 0  is generated in block  810 . Preferably, Q 0  is an element of the first cyclic group    1 . In block  812 , a first function H 1  is selected such that H 1  is capable of generating an element of the first cyclic group    1  from a first string of binary digits. A second function H 3  is selected in block  814 , such that H 3  is capable of generating a second string of binary digits from an element of the second cyclic group    2 . The functions of blocks  802  through  814  are part of the HIDS Root Setup algorithm described above, and preferably are performed at about the same time. By way of example, functions such as those disclosed in Boneh-Franklin may be used as H 1  and H 3 . In fact, the functions H 1  and H 3  may be exactly the same function. However, there is a potential pitfall. An attacker may try to get the signer to sign M=ID t , wherein ID t  represents an actual identity. In this case, the signer&#39;s signature may actually be a private key, which thereafter may be used to decrypt messages and forge signatures. This pitfall may be avoided, however, by using some expedient—such as a bit prefix or a different function for H 3 —that distinguishes between signing and private key extraction. 
   The next series of blocks (blocks  816  through  824 ) show the functions performed as part of Lower-level Setup algorithm. In block  816 , a public element P yi  is generated for each of the sender&#39;s m ancestral lower-level PKGs. Each of the public elements, P yi =H 1 (ID 1 , . . . , ID yi ) for 1≦i≦m, preferably is an element of the first cyclic group    1 . Although represented in a single block, generation of all the public elements P yi  may take place over time, rather than all at once. 
   A lower-level key generation secret s yi  is selected (block  818 ) for each of the sender&#39;s m ancestral lower-level PKGs  304   a,b,d . The lower-level key generation secrets s yi  preferably are elements of the cyclic group  /q  for 1≦i≦m, and each lower-level key generation secret s yi  preferably is known only to its associated lower-level PKG. Again, although represented in a single block, selection of all the secrets s yi  may take place over time, rather than all at once. 
   A lower-level secret element S yi  is generated (block  820 ) for each of the sender&#39;s m ancestral lower-level PKGs. Each lower-level secret element, S yi =S y(i−1) +s y(i−1) P yi  for 1≦i≦m, preferably is an element of the first cyclic group    1 . Although represented in a single block like the public elements P yi  and the secrets s yi , generation of all the secret elements S yi  may take place over time, rather than all at once. For purposes of these iterative key generation processes, S 0  preferably is defined to be the identity element of    1 . 
   A lower-level key generation parameter Q yi  also is generated (block  824 ) for each of the sender&#39;s m ancestral lower-level PKGs. Each of the key generation parameters, Q yi =s yi P 0  for 1≦i≦m , preferably is an element of the first cyclic group    1 . Again, although represented in a single block, generation of all the key generation parameters Q yi  may take place over time, rather than all at once. 
   The functions of the next two blocks (blocks  824  and  826 ) are performed as part of the Extraction algorithm described above. A sender public element P y(m+1)  associated with the sender y is generated in block  824 . The sender public element, P y(m+1) =H 1 (D y1 , . . . , ID y(m+1) ), preferably is an element of the first cyclic group    1 . A sender secret element S y(m+1)  associated with the sender y is then generated in block  826 . The sender secret element 
               S     y   ⁡     (     m   +   1     )         =         S   ym     +       s   ym     ⁢     P     y   ⁡     (     m   +   1     )             =       ∑     i   =   1       m   +   1       ⁢       s     y   ⁡     (     i   -   1     )         ⁢     P   yi             ,         
also preferably is an element of the first cyclic group    1 .
 
   For convenience, the first function H 1  optionally may be chosen to be an iterated function so that, for example, the public points P i  may be computed as H 1 (P y(i−1) , ID yi ) rather than H 1  (ID 1 , . . . , ID yi ). 
   The last two blocks shown in  FIG. 8  (blocks  828  and  830 ) represent the Signing and Verification algorithms described above. In block  828 , the message M is signed to generate a digital signature Sig. The signing preferably uses at least the sender secret element S y(m+1) . The digital signature Sig is then verified in block  830 . The verification preferably uses at least the root key generation parameter Q 0  and the lower-level key generation parameters Q yi . The specific use of these parameters and elements in the signing of the message M and verification of the digital signature Sig will now be discussed with reference to  FIG. 9 . 
     FIG. 9  shows a flow diagram illustrating a method of generating and verifying a digital signature Sig of a digital message M communicated between a sender y and a recipient z according to another presently preferred embodiment of the invention. In this scheme the Root Setup, Lower-level Setup, and Extraction algorithms are the same as for the embodiment shown in blocks  802  through  826  of  FIG. 8 . Accordingly, the flow diagram of  FIG. 9  begins with the selection of a sender key generation secret s y(m+1) , known only to the sender y, in block  927   a . A sender key generation parameter Q y(m+1)  associated with the sender is generated in block  927   b  using the formula Q y(m+1) =s y(m+1) P 0 . The Signing algorithm then begins with the sender generating a message element P M =H 3 (ID y1 , . . . , ID y(m+1) , M) in block  928   a.  The message element P M  preferably is a member of the first cyclic group    1 . The digital signature Sig itself is generated in block  928   b  using the formula Sig=S y(m+1) +s y(m+1) P M . The recipient verifies the digital signature Sig (block  930 ) by confirming that the formula 
                 e   ^     ⁡     (       P   0     ,   Sig     )             e   ^     ⁡     (       Q     y   ⁡     (     m   +   1     )         ,     P   M       )       ⁢       ∏     i   =   2       m   +   1       ⁢           ⁢       e   ^     ⁡     (       Q     y   ⁡     (     i   -   1     )         ,     P   yi       )             =       e   ^     ⁡     (       Q   0     ,     P   1       )             
is satisfied.
 
   The invention has been described in detail with particular reference to preferred embodiments thereof and illustrative examples, but it will be understood that variations and modifications can be effected within the spirit and scope of the invention.