Auto-Recoverable and Auto-certifiable cryptosystems with RSA or factoring based keys

[0002] This application is related to, and is a continuation-in part of, our applications Ser. No. 08/864,839, filed May 28, 1997 now U.S. Pat. No. 6,202,150, Ser. No. 08/878,189, filed Jun. 18, 1997 (now U.S. Pat. No. 6,122,742, granted Sep. 19, 2000), and Ser. No. 08/920,504 now U.S. Pat. No. 6,243,466, filed Aug. 29, 1997, and to Ser. No. 09/218,219, filed Dec. 22, 1997.

[0003] The field of this invention is cryptography. This invention relates to cryptosystems, and in particular to the escrowing and recovering of cryptographic keys and data encrypted under cryptographic keys. The escrow and recovery process assures that authorized entities like law-enforcement bodies, government bodies, users, and organizations, can when allowed or required, read encrypted data. The invention relates to cryptosystems implemented in software or in hardware. In particular, the invention relates to the generation of user public keys based on composite numbers and on the hardness of number factoring, and relates to implementing a hierarchical key escrow system.

[0004] Public Key Cryptosystems (PKC's) allow secure communications between two parties who have never met before. The notion of a PKC was put forth in (W. Diffie, M.

[0005] Hellman, “New directions in cryptography”, IEEE Transactions on Information Theory, 22, pages 644-654, 1976). This communication can take place over an insecure channel. In a PKC, each user possesses a public key E and a private key D. E is made publicly available by a key distribution center, also called certification authority (CA), after the registration authority verifies the authenticity of the user (its identification, etc.). The registration authority is part of the certification authority. D is kept private by the user. E is used to encrypt messages, and only D can be used to decrypt messages. It is computationally impossible to derive D from E. To use a PKC, party A obtains party B's public key E from the key distribution center. Party A encrypts a message with E and sends the result to party B. B recovers the message by decrypting with D. The key distribution center is trusted by both parties to give correct public keys upon request. A PKC based on the difficulty of computing discrete logarithms was published in (T. ElGamal, “A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms”, CRYPTO '84, pages 10-18, Springer-Verlag, 1985).

[0006] The current invention relates to key escrow systems. Prior methods for conducting key escrow are U.S. Pat. No. 5,276,737, and 5,315,658 which are due to Micali (1994). In these patents Micali discloses a Fair Public Key Cryptosystem (FPKC) which is based on the work of P. Feldman (28th annual FOCS). The FPKC solution is not as efficient in terms of use as Auto-Recoverable and Auto-Certifiable Cryptosystems. Furthermore, It has been shown that the Fair RSA PKC does not meet certain needs of law enforcement (J. Kilian, F. Leighton, “Fair Cryptosystems Revisited”, CRYPTO '95, pages 208-221, Springer-Verlag, 1995), since a shadow public key cryptosystem can be embedded within it. A shadow public key system is a system that can be embedded in a key escrow system that permits conspiring users to conduct untappable communications. Kilian and Leighton disclose a Fail-safe Key Escrow system. This system has the drawback that it requires users to engage in a costly multi-round protocol in order to generate public/private key pairs. Other key escrow systems with similar inefficiencies are by De Santis et al., Walker and Winston (TIS), and the IBM SecureWay document. These solutions propose session-level escrow which requires changes in communication protocols. A “Fraud-Detectable Alternative to Key-Escrow Proposals” based on ElGamal has been described in (E. Verheul, H. van Tilborg, “Binding ElGamal: A Fraud-Detectable Alternative to Key-Escrow Proposals”, Eurocrypt '97, pages 119-133, Springer-Verlag, 1997). This system, called Binding ElGamal, provides for session level key recoverability, and makes no provision for preventing users from encrypting messages using the provided unescrowed public key infrastructure prior to using the Binding ElGamal system. Hence, it permits conspiring criminals to conduct untappable communications. Binding ElGamal also imposes a large amount of communication overhead per communications session. An overview of key escrow schemes appears in (D. Denning, D. Branstad, “A Taxonomy for Key Escrow Encryption Systems,”Communications of the ACM, v. 39, n. 3, 1996). In (N. Jefferies, C. Mitchell, M. Walker, “A Proposed Architecture for Trusted Third Party Services”, Cryptography: Policy and Algorithms, LNCS 1029, Springer, 1996) and (R. Anderson, “The GCHQ Protocol and Its Problems”, Eurocrypt '97, pages 134-148, Springer-Verlag, 1997) a trusted third party approach to escrow is described where the trusted third parties of the participating users are involved in every session key establishment stage, and hence provides for another cumbersome solution as well.

[0007] In the pending U.S. Pat. Nos. 08/864,839, 08/878,189, and “Auto-Recoverable And Auto-Certifiable Cryptosystems with Fast Key Generation” filed Aug. 29, 1997 (by Young and Yung), Auto-Recoverable and Auto-Certifiable public key cryptosystems were disclosed that have the following properties. Users of the system can generate a public/private key pair and a certificate of recoverability. This certificate of recoverability can be used to both recover the private key by the escrow authorities, and verify that the private key is recoverable. The present invention is a new Auto-Recoverable and Auto-Certifiable key escrow solution. The main restriction of the prior Auto- Recoverable and Auto-Certifiable cryptosystems that were proposed is that it is not possible to generate user public and private keys that are based on the difficulty of factoring. Such cryptosystems based on a multiple of two large prime numbers, for example, are popular. Inn particular the RSA system by Rivest et. al (Rivest 1983) is very popular. The previous Auto-Recoverable and Auto-Certifiable systems based users'keys on the hardness of the problem of computing discrete logarithms. Indeed, it could be the case that computing discrete logs is not as hard as is currently believed, but that factoring is hard. Thus for this reason, there is a significant advantage to an Auto-Recoverable and Auto-Certifiable cryptosystem based on the difficulty of factoring, as in the present invention. The present invention discloses an Auto-Recoverable and Auto-Certifiable cryptosystem in which the public key of the escrow authorities and the public keys of the users are based on the hardness of factoring composites. In another embodiment the escrow authorities keys are based on the discrete log problem.

[0008] The present invention also introduces a new notion, the notion of a key escrow hierarchy which is a method with new functionality. We disclose a method for implementing a key escrow hierarchy that takes the form of a tree. This tree has the property that any node has the ability to recover the communications of nodes of the subtree for which it is the root. Thus, the root node of the entire tree is able to recover the communications of all users of the system. Such a system is ideal for implementing national and multinational Public Key Infrastructures (PKI). For instance, a national PKI for the U.S. could be implemented as a depth-3 tree. The federal government could be the root. The state governments could be the middle nodes, and the residents of each state could be the leaves. This would allow the federal government to decrypt all communications, and would restrict the state governments to only be able to decrypt the communications of the states own residents.

[0009] Both Binding ElGamal and the Auto-Recoverable and Auto-Certifiable Cryptosystems solutions employ the use of non-interactive proofs. More specifically, they employ the Fiat Shamir heuristic which is disclosed in (A. Fiat, A. Shamir, “How to Prove Yourself: Practical Solutions to Identification and Signature Problems”, CRYPTO '86, pages 186-194, Springer-Verlag, 1987). It is known in the art how to replace non-interactive proofs by interactive proofs. The variant of proofs introduced by our mechanism is a proof that combines a zero-knowledge methodology with explicit encryption of values to a third party.

[0010] The present invention provides a method to verify that information that can be used to recover a user generated private key is contained within an encryption under the public key of the escrow authorities. Also, this method does not involve a lot of overhead. Furthermore, this verification can be performed by anyone in possession of the escrow authorities public key. The present invention can replace a public key infrastructure which is based on users' keys based on factoring (e.g., RSA keys as in Rivest 1983) without changing the encryption and decryption functions since the keys generated are compatible with factoring based keys. The present invention is also compatible with a usual Public Key Infrastructure and does not require other changes (like changes of communication protocols and computations). The present invention consists of a setting up process and three functions which process signals in different ways. The functions are key generation, key verification, and key recovery. In the setup process of the prefered embodiment, the participants agree upon a set of initial public parameters and the authorities generate an escrowing public key and corresponding private keys. The initial parameters and the escrowing public key are the public parameters of the system. The escrowing authorities, Certification Authority (CA), and users of the system all have access to the public parameters. In the key generation process, the method generates a user's public/private key pair, and a certificate of recoverability which is a string of information which includes encryptions of information that allows the recovery of the user's private key, encrypted under the escrowing public key. The signal information containing the user's public key, and the certificate of recoverability can be transmited to any entity. In the verification process, the user transmits this signal to the verifier. The verification process takes the input signal, processes it, and outputs either true or false. A result of true indicates that the user's private key is recoverable from the certificate of recoverability by the escrow authorities. A result of false indicates that the private key is not recoverable. The invention is designed such that it is intractable for the user to generate a public key, and certificate of recoverability such that the key is not escrowed and such that it passes the verification process with a result of true. In the prefered embodiment, the users certify their public keys with the registration authority of the certification authority (CA) who then signs their public key after successful verification. A public key together with a CA's signature on a string that contains the public key, the user's identity, and other information, constitutes a certified public key. The other information can include the certificate of recoverability in the present invention. It can also include the message digest of the certificate of recoverability. In more detail, upon receiving the user's public key, and certificate of recoverability, the CA verifies that the corresponding private key is recoverable. If it is, (namely, the verification process outputs true) the public key is certified and/or made publicly available by the CA. The CA keeps a copy of the certificate of recoverability, perhaps in encrypted form under its own key. The user is only required to keep his private key and to have access to the public key database containing public keys of other users as in a typical PKI. In the recovery process, the escrow authorities use the user's certificate of recoverability and public key, which is obtained from the CA, as an input signal. The escrow authorities process the certificate of recoverability, and the corresponding user's private key is the resulting output signal. When the escrow authorities are implemented in tamper-proof hardware, it is possible to output decryptions of messages encrypted under the private key rather than the private key itself.

[0011] The present invention is useful in any environment that demands the recovery of private keys, or keys encrypted under these keys, or information encrypted under these keys. Such environments arise in law enforcement nationally and internationally, in the business sector, in secure file systems, on the Internet, in Certified Mail services, etc. The successful escrowing of private keys implies the successful escrowing of public key encrypted information, and hence the present invention has many applications.

[0012] The present invention is robust with respect to any underlying technology since it can be implemented in both hardware and software. When implemented in software it can be easily scrutinized to insure that it functions as desired and to insure that it does not compromise the security of its users. The software implementation allows for fast and easy dissemination of the invention, since it can be disseminated in source code form over diskettes or over a computer communication network. The present invention is also communication efficient. At most, one, two, or three message exchanges suffice for the various embodiments of the present invention. The signals can be processed quickly and the signals themselves constitute a small amount of information. The invention does not require changes in communication protocols used in typical unescrowed PKI's (e.g., session key establishment, key distribution, secure message transmission, etc.). The invention is therefore compatible with typical PKI's. The present invention thus provides a very efficient way of escrowing and recovering cryptographic keys. The present invention does not require changes in header information of messages, content of messages, or additional messages outside the PKI protocols, and as mentioned above does not require changes of cryptographic encryption and decryption in systems based on RSA or factoring based keys.

[0013] The present invention will be described with reference to the accompanying FIGS. 1-7.

[0014] FIG. 1 is a data flow diagram of the process of setting up the method of the invention for use with m escrow authorities.

[0015] FIG. 2 is a flow chart of the basic steps of the process of generating a public/private key pair and certificate of recoverability using the invention.

[0016] FIG. 3. is a data flow diagram of the process of verifying the recoverability of a private key.

[0017] FIG. 4 is a data flow diagram of the process of registering a key using the invention.

[0018] FIG. 5 is a data flow diagram of the process of private key recovery by the escrow authorities.

[0019] FIG. 6 describes a generic public key system and its main components and operations

[0020] FIG. 7 describes the escrowable public key system which results by employing the present invention and its main components and operations.

[0021] FIG. 8 is a schematic representation of the procedure by which users are informed of failed certification requests.

[0022] FIG. 9 is a schematic representation of a procedure for checking for subliminal channel abuse.

[0023] FIG. 10 is a schematic representation illustrating the decryption capabilities of a Root Escrow Authority and levels of nodes below.

[0024] The following is a description of the prefered embodiment of the present invention.

[0025] Variations on the prefered embodiment will accompany the description of the prefered embodiment wherever applicable. For convenience in the presentation, the hashing algorithm selected is SHA (Schneier 2nd edition, pages 442-445), though any cryptographic hashing algorithm will suffice in its place. In the prefered embodiment, parameters are chosen uniformly at random from their respective groups. Alternate embodiments include alterations of the probability distributions from which such values are chosen. Such choices based on random number generators or pseudo-random generators are available in the art.

[0026] The system setup of the prefered embodiment shown in FIG. 1 initiates the cryptosystem. In the prefered embodiment, the escrow authorities generate a shared integer N=p′q′, where p′ and q′ are prime such that gcd(E,(p′−1))(q′−1))=1. Here E is the public RSA exponent. N is a k-bit integer, where k is even. The method described in (A. De Santis, Y. Desmedt, Y. Frankel, M. Yung, “How to Share a Function Securely”, ACM STOC '94, pages 522-533, 1994) can be used to employ the shared values D₁, D₂, . . . , which are the shares of the private exponent, corresponding to the public exponent E. It is preferred that p′ and q′ be strong primes, to thwart known attacks. The valued N and E are made public and N's factorization is kept private.

[0027] FIG. 2 is a diagram showing the process of how a user's system generates a public/private key pair and a certificate of recoverability. Having obtained (and verified as much as possible) the signal N and E that is made available to users by the escrow authorities, the user system proceeds to generate a public key n and public encryption exponent e for the user. The user system chooses a random string s' such that s′,s′+MAX−1ε{0,1}^k/2. Here MAX is a small integer, say 32768. The user system computes U_w=H(s′+w) for 0≦w≦MAX. H is a pseudo-random one-way function with a domain of {0,1}^k/2. The user system then chooses a k/2 bit strong prime p randomly such that gcd(e,p−1)=1. The user system also chooses a value ε_R{0,1}^k/2. The user system solves for q_win the equation [U_w][rnd]=pq_w+r for 0≦w≦MAX. The left side of this equation is the integer formed by concatenating the two bitstrings U_wand rnd. For each q_wthat is a strong prime such that gcd(e,q_w−1)=1, the user system checks that pq_w<N. If this holds for one or more of the q_w, the system sets q=q_wfor the least w and s=s′+w. Otherwise, s′ is chosen again and the algorithm is repeated. This algorithm is similar to the SETUP attack against RSA keys (A. Young, M. Yung, “Kleptography: Using Cryptography Against Cryptography”, Eurocrypt '97, pages 62-74, Springer-Verlag, 1997) and actually intends to prevent attacks of these types by minimizing what is known in the art as “subliminal channels”. This process of generating n, p, and s is depicted in step 2006 of FIG. 2.

[0028] The system then proceeds to step 2007 and computes a certificate that can be used by any interested party to verify that the user's “private key” p (or q) can be recovered from the certificate of recoverability P. P is constructed according to the following algorithm:

[0029] 1. P=( )

[0030] 2. for i=1 M do

[0031] 3. choose r_i,1ε_RZ_n*

[0032] 4. Q_i=r_i,1²mod n

[0033] 5. compute r_i,2to be a root of Q_imod n s.t. r_i,2≠r_i,1mod n

[0034] 6. C_i,l=r_i,1^Emod N

[0035] 7. C_i,2=r_i,2^Emod N

[0036] 8. add (Q_i,C_i,1,C_i,2) to the end of P

[0037] 9. val=H(P)

[0038] 10. set b₁,b₂, . . . , b_Mto be the M least significant bits of val, where b_iεZ₂

[0039] 11. for i=1 to M do

[0040] 12. Z_i=r_i,jwhere j=1+b_i

[0041] 13. add Z_ito the end of P

[0042] 14. add s to the end of P

[0043] Thus, P =((Q₁,C_1,1,C_1,2), . . . , (Q_M,C_M,1,C_M,2),Z₁, . . . , Z_M,s). Here s is the seed used to find the upper order bits of n. Note that Q_iis a quadratic residue mod n, and the C_i,j's are the RSA encryptions under N of pairs of ambivalent roots of Q_i(a pair which is neither identical nor a pair in which one is the negation of the other modulo n). The encryption is actually a deterministic commitment to the roots and the encryption may include some preprocessing of the value (using cryptographic hash functions, reversible operations, and stretching the encryption block into several blocks) which are known in the art; we will assume encryption only with no preprocessing in the following description. H is a suitable public one-way hash function (e.g., SHA), so the b_i's can be recovered from P. The values for b are the challenge bits, and this method of finding them is known as the Fiat-Shamir Heuristic. The user system outputs (n,p,P) in step 2008. Note that the user has the option to interactively prove that his or her private key p is recoverable by the escrow authorities. This will be addressed in more detail later. M is a large enough parameter of security (e.g., M=50).

[0044] The description of the embodiment has thus far explained how the system is setup for use by the CA and authorities, and how the system is used by users (potential receivers) to generate public/private key pairs and certificates of recoverability. These certificates are strings showing to anyone presented with them that the private key corresponding to the public key generated is recoverable by the escrow authorities using P. The following describes how the invention is used by the user to prove to a verifier that p is recoverable from P. This process is depicted in FIG. 3. The verifier can be the CA, an escrow authority, or anyone else who knows the system parameters.

[0045] The verification process of FIG. 3 is as follows. In step 3009, the user generates a public/private key pair, and a certificate using the invention as described above. In step 3010, the user transmits a signal containing these parameters to a verifier. In step 3011 the verifier uses this signal to verify whether or not the user's private key is recoverable by the escrow authorities. In this process, the verifying system takes n, the corresponding certificate P, and the escrowing public key N and E. The verifying system first checks that n<N (this assumption is not necessary since one can stretch blocks in the commitment encryptions as explained above, but this assumption simplifies the design). Let y denote the integer corresponding to the k/2 upper order bits of n. The verifying system makes sure that either H(s)=y or that H(s)=y+1. The verifying system checks that all of the values in P lie in the correct sets. The verifying system also checks that the values C_ijare not equal for all i and j. The verifying system checks that none of the Q_iare the same for all i. If any of these verifications fail, then false is returned. This check for subliminal channel abuse is depicted in FIG. 9. In step 1 of FIG. 9 the values n and s are input into the subliminal channel certification method. Step 2 is reached if and only if the verifying method concludes that there is no significant subliminal channel present. This corresponds to H(s) equaling y or y+1. Step 3 is reached if and only if the verifying method concludes that a subliminal channel is present. This corresponds to H(s) not equaling y nor y+1. The verifying system then computes b₁, b₂, . . . , b_Min the same way as in the certificate generation process. For i=1 to M, the verifying system verifies the following things:

[0046] The verifying system returns true as long as all the verifications pass and as long as all three criterion are satisfied for 1≦i≦M. The invention may take subsequent action and indicate to the verifier that the public key is invalid in the event that false is returned. Similarly, the verifying system may inform the verifier of a validation that passes (the verifying system returns true).

[0047] In FIG. 4, the user certifies his or her public key with the CA. In step 4012 of this process, the user generates his or her public key and certificate of recoverability, as previously described. The user transmits this signal to the CA. This corresponds to step 4013 of FIG. 4. In step 4014 the CA acts as a verifier and verifies that the user's private key is recoverable by the escrow authorities.

[0048] So far, steps 4012 through 4014 are identical to steps 3009 through 3011 in the key verification process of FIG. 3. However the CA, in addition, will make keys that pass the verification process available to others upon request and/or certify them. If the user's public key fails the verification process, then either the certification attempt is ignored, or alternatively the user is notified of the failed certification attempt.

[0049] Depending on the demands of the environment in which the invention is used, users may be required to submit extra information in order to register a public key and to certify that they know the private key portion without divulging it. Such information could be a password, social security number, previously used private key, etc. In the case that the CA is a trusted entity, the CA can simply digitally sign the user's public key together with the user's name and additional information, and make the key along with the CA's signature on this information available on request. If the CA is not trusted (which is not the typical assumption in PKI), then the certificate should be stored in the public file and the certificate together with the certificate of recoverability should be given to the escrow authorities, who in turn can insure recoverability. This completes the description of the public key certification process. We note that the CA keeps the certificate of recoverability, possibly in encrypted form under its own key with authentication information for integrity.

[0050] The last process to describe is the private key recovery process. This process is depicted in FIG. 5. In this process, the invention is used by the m escrow authorities to recover the user's private key based on P. In this process, all m of the escrow authorities obtain n and P, as depicted in step 5015 of FIG. 5. In an alternate embodiment the CA transmits n and P and/or other parameters to one or more of the authorities. Thus they are already in possession of n and P. At this point escrow authorities use a subset of their shares D₁, D₂, . . . , D_mto decipher P to open all of the unopened C_i,j. This is accomplished by having escrow authority i recover the ith shares of the user's private key. In this process, escrow authority i extracts the M values for the unopened C_i,jfrom P and decrypts them using D_j. The resulting values are pooled with the values from the other escrow authorities, as depicted in step 5016 of FIG. 5. The pool is then used by the authorities to decrypt all of the unopened values C_i,jfrom P. Thus all of the plaintexts corresponding to all C_i,jare known to the escrow authorities. There are alternative methods in the art for recovering the roots corresponding to the unopened C_i,j, so that the unopened roots are represented distributively among the authorities. The escrow authorities check each C_i,1and C_i,2for a pair of ambivalent roots of Q_i. Once such a pair is found, the gcd algorithm is used to output a nontrivial factor of n, using a method well known in the art.

[0051] Instead of executing the last algorithm which recovers the factors in software, it can be done in a tamper-proof environment. The environment gets the key n (and the public exponent which is typically a companion of n), and a message encrypted under that key. The above algorithm is executed and the decryption key is derived from the factors of n (in one of the ways known in the art). The decryption key is applied to the ciphertext, and the cleartext is output by the environment.

[0052] What has been described is an Auto-Recoverable and Auto-Certifiable (ARC) cryptosystem where the public key of the escrow authorities and the public key of the user are composite integers. The users of such a cryptosystem employ the public key system in a way that is identical to a typical PKI with composite number based keys, for secure communications. This is demonstrated schematically in FIGS. 6 and 7. FIG. 6 is a typical public key cryptosystem in a PKI environment. The following are the steps that are followed by the users. (1) The user first reads the CA's information and address. (2) The user generates a public/private key pair and sends the public key to the CA. The registration of the authority in the CA verifies the identity of the user, and publishes the public key together with the CA certificate on that key, identifying the user as the owner of that key. (3) For another user to send a message to that user, the public key is read from the CA's database and the certificate is verified. (4) Then, the message is encrypted under the public key and sent. FIG. 7 schematically describes the ARC cryptosystem. The additional operations are as follows. (0) The authority generates the escrowing public key and gives it to the CA. Steps 1 and 2 are analogous, except that a proof is sent along with the public key. Steps 3 and 4 are the operation of the system and are identical. Steps 5 and 6 describe the case in which keys are recovered from escrow. (5) The escrow authority gets information from the CA. (6) The escrow authority recovers the user's private key.

[0053] The method by which users are informed of failed certification requests is depicted in FIG. 8. The steps are as follows: (1) the user reads the CA's address and information. (2) The user (or receiver in this case) generates and publishes a key to the CA. (3) The CA verifies the key and the corresponding certificate and publishes the key only if the certificate is valid. (4) The CA informs the receiver in the event of a failed certification request. (5) The sender reads the receiver key from the CA and verifies the receivers certificate. (6) The sender encrypts a message and sends it to the receiver.

[0054] The authorities can require that that the certificate of recoverability be sent along with the public key so that the user's parameters can first be verified using the verification process by the authorities as well. This completes the description of the primary embodiment and variations thereof.

[0055] We will now describe our secondary embodiment of this invention. In this embodiment, the users of the system generate composite public keys as before, except the escrowing public key of the escrow authorities is an ElGamal public key. In this system, the escrow authorities have the shared public key Y=g^Xmod Q. The private key X is shared additively among the escrow authorities (other sharing or a single authority are possible variations known in the art, as well as multiple authority keys where each user can choose a subset of authorities by multiplying their public keys). Q is a large strong prime and g is a generator of the domain Z_Q. The user system generates n and s in the same way as before, except that n <Q instead of n<N. The construction of P is the same as before, except that we don't use RSA to encrypt r_i,1and r_i,2. Note that straightforward ElGamal encryption using (Y,g,Q) won't suffice to form the commitments. To see this, note that unlike RSA, ElGamal is a probabilistic PKCS and does not constitute a permutation. The prover could thus encrypt the same root of Q_iin C_i,1and C_i,2, and there would be no way to know this without opening both values. The solution to this problem is quite subtle. The following is the modified algorithm for computing the Q_i's, C_i,j's, and Z_i's in the user system (its security will be based on both discrete logarithm and factoring which are needed to be secure for this embodiment to work).

[0056] 1. P=( )

[0057] 2. for i=1 to M do

[0058] 3. choose a large prime r_i,1ε_RZ_n*

[0059] 4. Q_i=r_i,1²mod n

[0060] 5. Let t and t′ be roots of Q_is.t. they are ambivalent roots (i.e., t,t′≠r_i,1, −r_i,1mod n)

[0061] 6. set r_i,2=t if t or t/2 is a large strong prime

[0062] 7. set r_i,2=t′ if t′ or t′/2 is a large strong prime

[0063] 8. if none of t,t′,t/2,t′/2 are large strong primes go to step 3

[0064] 9. choose k_i,1ε_RZ*_Q−1s.t. Y raised to the k_i,lpower mod Q is a large strong prime

[0065] 10. A_i,1=g raised to the k_i,1power mod Q

[0066] 11. B_i,1=(Y raised to the k_i,1power mod Q)(r_i,1)

[0067] 12. C_i,1=(A_i,1,B_i,1)

[0068] 13. choose k_i,2ε_RZ*_Q−1s.t. Y raised to the k_i,2power mod Q is a large strong prime

[0069] 14. A_i,2=g raised to the k_i,2power mod Q

[0070] 15. B_i,2=(Y raised to the k_i,2power mod Q)(r_i,2)

[0071] 16. C_i,2=(A_i,2,B_i,2)

[0072] 17. add (Q_i,C_i,1,C_i,2) to the end of P

[0073] 18. val=H(P)

[0074] 19. set b₁,b₂, . . . , b_Mto be the M least significant bits of val, where b_iεZ₂

[0075] 20. for i=1 to M do

[0076] 21. Z_i=k_i,j, where j=1+b_i

[0077] 22. add Z_ito the end of P

[0078] 23. add s to the end of P

[0079] Note that the C_i,jare not ElGamal encryptions or RSA encryptions. They are a hybrid of both. The value A_i,jis the same as the A in an ElGamal encryption, but B is quite different. The value B is not reduced modulo Q. Instead, it is a composite, as in RSA. We make sure that Y raised to the k_i,jpower mod Q is a strong prime. We also make sure that r_i,jis either a strong prime or two times a strong prime. Thus, B is either an RSA modulus or two times an RSA modulus. The plaintext r_i,jin this encryption is in fact one of the large prime factors of B, and is also a root of Q_imod n. We may relax the requirement regarding r_i,2(the root ambivalent to r_i,1) and allow it to be a large prime multiplied by a small number smaller than some bound (say, 1000). We may also relax the need for strong primes, when the numbers chosen are of size large enough (say of size 2048 bits).

[0080] The verifying system is a bit different than before. The verifying system first checks that n<Q. Let y denote the integer corresponding to the k/2 upper order bits of n. The verifying system makes sure that either H(s)=y or that H(s)=y+1. The verifying system checks that all of the values in P lie in the correct sets. For example, the verifying system checks that B_i,j<Qn. The verifying system also checks that the values A_i,jare not equal for all i and j. The verifying system computes the gcd for each pair of values B_i,j, and makes sure that none of the gcd's exceeds 2. The verifying system also checks that all of the B_i,1's are odd. The verifying system checks that none of the Q_iare equal for all i. If any of these verifications fails, then false is returned. The verifying system then computes b₁,b₂, . . . , b_Min the same way as in the certificate generation process. The verifying system computes r_i,j=B_i,j/(Y raised to the k_i,jpower mod Q), where j=1+b_i, and returns false if the remainder isn't zero or if r_i,jwhere j=1+b_iisn't less than n. If r_i,1is opened, the verifying system checks that it is a large strong prime. If r_i,2is opened, the verifying system checks that it is a large strong prime or two times a large strong prime. For i=1 to M, the verifying system verifies the following things:

[0081] 1. r_i,j²=Q_imod n, where j=1+b_i

[0082] 2. −r_i,jmod n, where j=1+b_idoesn't divide B_i,j, where j=2−b_i

[0083] The verifying system returns true as long as all the verifications pass and as long as all both criterion are satisfied for 1≦i≦M.

[0084] In case we relax the requirement on r_i,2to be a prime multiplied by a small number, we can take the gcd of all the B_i,j's and making sure it is smaller than the allowed bound.

[0085] The escrow authorities recover the unopened r_i,jby computing B_i,j/(Y raised to the k_i,jpower mod Q) using the private shares of X and A_i,j. In the recovery process, the opened r_i,jare found in the same way as in the verifying system. The escrow authorities operate in the same way as before using these pairs of roots and the gcd algorithm to factor n.

[0086] In another embodiment, the interactive version of the proof for P can be used. Note that by outputting P, a shadow public key cryptosystem may result. This follows from the fact that the Z_ican be chosen explicitly by a malicious user.

[0087] A primary PKI using our system can be the RSA system (Rivest 1983). In yet another embodiment, the public exponent of the escrow authorities E, is equal to 2. This implies that the commitments are performed using Rabin's system which is known in the art, thereby weakening the cryptographic assumptions necessary for security.

[0088] An application of this invention is a hierarchical public key escrow system. A hierarchical public key escrow system is an escrow system that takes the form of a tree data structure. The escrow authorities at the root of the tree are able to decrypt the communications of all entities corresponding to the nodes of the rest of the tree. Recursively, the escrow authorities at any given node i in the tree are able to decrypt the communications of all entities corresponding to the nodes in the rest of the subtree for which node i is root. Any time, the leaf of the tree can form another subtree and act as an escrow agent(s). For example, in FIG. 10 the Root Escrow Authority (EA) can decrypt all of the communications of EA₁, EA₂, EA₃, . . . EA_i, and below (all nodes below it). More specifically, in step 1 the root escrow authority decrypts data encrypted by EA₁. Also, EA₁can decrypt all of the communications of EA₁, EA₂, EA₃, . . . , EA_i, and below, etc. So in step 2 EA₁decrypts the communications of EA_i. By ordering the size of the moduli properly, it is possible to have multiple escrow agents for any node of the tree. All that is necessary is to do the commitments of the roots starting with the smallest modulus and ending with the largest.

[0089] As a concrete example, the root of a depth-3 tree can be the federal government. Let Nus denote the escrowing public key of the federal government. It can be generated using (A. De Santis, Y. Desmedt, Y. Frankel, M. Yung, “How to Share a Function Securely”, ACM STOC '94, pages 522-533, 1994), so that the private exponent is shared among m escrow authorities. For concreteness, suppose that |N_US|=1028 bits. In this system, each of the fifty states has two escrow agents. So, let N_CT1and N_CT2be the public keys of Connecticut, where |N_CT1|=|N_CT2|=1026 bits. Let us assume that Alice knows the factorization of N_CT1and Bob knows the factorization of N_CT2, and that N_US>N_CT1>N_CT2. The primary embodiment of the present invention explains exactly how Alice and Bob can generate their keys using N_US. Let Carol be a US citizen and a Connecticut state resident. Carol can generate a public key n using both N_CT1and N_CT2as follows. She generates n s.t. n<N_CT2, and s.t. |n|=1024. She then commits to a root of Q_iby first encrypting with N_CT2and then encrypting the result with N_CT1. Thus, the commitment aspect of the user system uses both N_CT2and N_CT1. This way, Carol is certain the Alice needs Bob and vice-versa to factor n. Note that rather than using function sharing, we could have used two different keys N_US1and N_US2at the root level, just as we did for Alice and Bob.

[0090] Similarly, rather than a fixed tree which determines an order, the user can decide on a subset of escrow agents and generate its own preferred tree which is the chosen subset of escrow agents ordered by the relative size of their public keys in a line where the largest key is the root. This enforces a structure of the commitment, and assures that the subset needs to work together to recover a key or information encrypted under the key.

[0091] Also, the previous Auto-Recoverable and Auto-Certifiable cryptosystem inventions (pending patent applications by Young and Yung) can be combined together to implement hierarchical key escrow systems. For example, the solution of the pending application from Aug. 29, 1997 can be combined with the solution of pending application Ser. No. 08/864,839 as follows. The root of the hierarchy uses the escrowing public key n. The children of the root use the escrowing public key Y mod 2(2tn+1). The next level down in the tree uses escrowing public key y mod 2(2tn+1)+1. Here n is a composite, 2tn+1 is prime, and 2(2tn+1)+1 is prime. Similarly, the present invention can be combined with the invention from Aug. 29, 1997. In this case, the root of the hierarchy uses the escrowing public key N. The children of this root use the escrowing public key n. Finally, the next level down in the hierarchy uses the public keys (y,g,2tn+1), where 2tn+1 is prime. Clearly, these auto-recoverable and auto-certifiable cryptosystem solutions can be combined in many different ways to implement hierarchical key escrow systems.

[0092] Another application of key escrow systems is a secure file system or file repository system with recoverable keys. Such a system can be implemented based on the previous embodiments, in particular based on the preceding paragraph. For example, user A can be the owner of the file, user B can be the file server, and the trustees can be file recovery agents. An example of a file could be a password, in which case, the file recovery agents are password recovery agents. The present invention can also be used for certified mail where the escrow authorities are certified mail servers, when a certified mail message does not reach the destination, but only a commitment to its encryption value is available (together with a signature of the sender) then the escrow authority can “open the mail” as disclosed by Young and Yung in pending application from Aug. 29, 1997. The hierarchical system above can assure better availability of such mail servers which can back lower level servers.

[0093] Thus, there has been described a new and improved key escrow system, its variants, and applications. It is to be understood that the prefered embodiment is merely illustrative of some of the many specific embodiments which represent applications of the principles and paradigms of the present invention. Clearly, numerous and alternate arrangements can be readily devised by those who are skilled in the art without departing from the scope of the present invention; some of these variations were discussed above.