Abstract:
A method is provided for an escrow cryptosystem that is overhead-free, does not require a cryptographic tamper-proof hardware implementation (i.e., can be done in software), is publicly verifiable, and cannot be used subliminally to enable a shadow public key system. A shadow public key system is an unescrowed public key system that is publicly displayed in a covert fashion. The key generated by the method are auto-recoverable and auto-certifiable (abbrev. ARC). The ARC Cryptosystem is based on a key generation mechanism that outputs a public/private key pair, and a certificate of proof that the key was generated according to the algorithm. Each generated public/private key pair can be verified efficiently to be escrowed properly by anyone. The verification procedure does not use the private key. Hence, the general public has an efficient way of making sure that any given individual&#39;s private key is escrowed properly, and the trusted authorities will be able to access the private key if needed. Since the verification can be performed by anyone, there is no need for a special trusted entity, known in the art as a “trusted third party”. The cryptosystem is overhead free since there is no additional protocol interaction between the user who generates his or her own key, and the certification authority or the escrow authorities, in comparison to what is required to submit the public key itself in regular certified public key systems. Furthermore, the system is designed so that its internals can be made publicly scrutinizable (e.g., it can be distributed in source code form). This differs from many schemes which require that the escrowing device be tamper-proof hardware.

Description:
BACKGROUND 
     1. Field of Invention 
     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, but is also applicable to cryptosystems implemented in hardware. 
     2. Description of Prior Art 
     Public Key Cryptosystems (PKC&#39;s) allow secure communications between two parties who have never met before. The notion of a PKC was put forth in (W. Diffie, M. 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&#39;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 &#39;84, pages 10-18, Springer-Verlag, 1985). 
     PKC&#39;s are highly convenient in terms of use, and permit users to conduct private communications over insecure channels. They may be used to initiate symmetric key systems like DES (Data Encryption Standard). PKC&#39;s have a drawback, however. Criminals can use PKC&#39;s in the course of criminal activity, since no provision is made to supply law enforcement with the necessary decryption keys and untappable criminal communications may result. It is therefore desirable to permit private communications exclusively to law abiding citizens. A general solution to this problem is to have each user submit a representation of his or her private key to trusted escrow authorities, or trustees. The shares are taken out of escrow in the event of a court authorized wire tap. Alternatively, key escrow provides a way to recover lost private keys in an organization, or keys of a file system. 
     Let us review some of the key escrow systems and show that all require more than a PKC alone. U.S. Pat. Nos. 5,276,737, and 5,315,658 to Micali (1994) disclose a Fair Public Key Cryptosystem (FPKC) (see also, S. Micali, “Fair Public-Key Cryptosystems”, CRYPTO &#39;92, pages 113-138, Springer-Verlag, 1992) which satisfies the needs of law abiding citizens and law enforcement (and is based on P. Feldman, 28th annual FOCS). Micali&#39;s preferred embodiment discloses how to convert the Diffie-Hellman PKC, and the RSA PKC into Fair PKC&#39;s. In the preferred embodiment of the Fair Diffie-Hellman PKC, each user submits five shares to five central trustees (also known as “trusted third parties”) to register a public key. This solution is therefore not very scalable, since it requires the use of a small number of trusted authorities, and is thus very centralized. In the present invention, the user constructs a key pair such that the private key is provably escrowed automatically. Hence, no trusted third parties are needed whatsoever. The escrowed information can be sent to one of a multitude of decentralized certification authorities (CA&#39;s). In Micali&#39;s scheme each trustee verifies their respective shares. Provided the share is valid, the share is stored in a database. Each trustee then signs the values that were received and gives them to a key management center. The five authorities have the burden of securing and managing five private databases of shares. In the present embodiment, the key information is verified by a CA. Provided it has the correct form, the key is signed, and placed immediately in the database of public keys. There need only be one private database. Since only the CA is needed to manage user keys in the current embodiment, the least amount of communication overhead that is possible is achieved. In the Fair PKC&#39;s, only the trustees can verify that a key is escrowed properly. Verification is required since without it a user can easily generate keys which are not recoverable. In the current invention, everyone can verify this. This is particularly useful if, for example, a citizen suspects that a CA is failing to insure that its keys are properly escrowed. 
     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 &#39;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. 
     The flaw in the RSA FPKC lies in the fact that it is assumed that criminals will use the same secret keys that were provided to the escrow authorities. The shadow cryptosystems make use of what is known in the art as subliminal channels that exist in the public keys of the PKC&#39;s. These channels are used to display the public keys of the shadow PKC. The Kilian and Leighton paper discloses how to convert PKC&#39;s into Fail-safe Key Escrow (FKE) systems. Specifically, they disclose how to construct FKE systems for discrete-log based PKC&#39;s like Diffie-Hellman and DSS. In their expensive protocol, the user and the trusted authorities engage in a protocol to generate the user&#39;s public and private keys. In so doing, the authorities are convinced that no subliminal information is contained in the resulting public key. The user is also convinced that the keys are escrowed properly. This system is similar to the Fair Diffie-Hellman PKC, except for the added overhead of this protocol. It is thus subject to the same inefficiencies as the Fair Diffie-Hellman PKC. In the present invention, the user chooses his or her own keys independently. With respect to the threat of shadow PKC&#39;s, the present invention relies on the fact that there is no known way to inconspicuously embed a significant number of bits within a modular exponentiation in a finite field. Hence, the exploitation of shadow cryptosystems in discrete-log PKC&#39;s seems remote. 
     DeSantis et al. teach an escrow system where trustees are able to open only messages in session rather than opening the key of the party suspected of criminal activity. This refines the notion of Fair Cryptosystems. Other technologies that teach how to open the session key of users rather than their permanent public key is by Walker and Winston (TIS) and the IBM SecureWay document. These key recovery technologies require that users be aware of and use the keys of the set of trustees at any session initiation. These technologies may be overburdening each and every user since they require new protocol extensions which are used in every communication session and further require users to store many keys beyond what is needed for a PKI. 
     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 &#39;97, pages 119-133, Springer-Verlag, 1997). This system allows users to send encrypted information along with a short proof that the encrypted information can be recovered by a set of trustees. So, this system has the advantage that it does not depend on trusted third parties. However, this system requires an already existing Public Key Infrastructure (PKI). Therein lies the flaw in the Binding ElGamal approach: If the PKI is unescrowed then user A can public key encrypt a message using user B&#39;s public key, and then send the resulting ciphertext message using Binding ElGamal. In this case, the proof simply serves to show that the trustees can recover this ciphertext, and therefore prevents law-enforcement from being able to monitor the communications of users suspected of criminal activity. When this abuse is employed, fraud is not detectable. This abuse is made possible because user B&#39;s private key is not escrowed. Software that abuses the Binding ElGamal scheme could be readily distributed and could severely hamper attempts at law enforcement on a large-scale. The present invention discloses a method of establishing an escrowed PKI, and is hence not subject to this drawback. Like in Binding ElGamal, the present invention employs the general technique of non-interactive zero-knowledge proofs, though the proofs of the present invention involve new technology. A heuristic for how to construct such proofs was shown in (A. Fiat, A. Shamir, “How to Prove Yourself: Practical Solutions to Identification and Signature Problems”, CRYPTO &#39;86, pages 186-194, Springer-Verlag, 1987). 
     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. Jeffries, 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 &#39;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. 
     All key escrow solutions heretofore known suffer from some if not most of the following disadvantages. 
     (a) they require tamper-resistant implementation, or otherwise require hardware implementation. This imposes high implementation costs and slow establishment of use. 
     (b) they require use of classified or otherwise proprietary algorithms. This may be unacceptable to users who may be skeptical about the devices security or operation. 
     (c) they are implemented in software, and are therefore subject to alteration, resulting in improper operation and possibly untappable communications. This is however, an inherent problem of any software solution (all we can require in this case is that if users employ the software apparatus solely for achieving privacy then their plaintext or keys are recoverable). 
     (d) they require excessive protocol interaction in key generation and/or general use. In addition this interaction may be conducted with a small set of centralized entities, thus making traffic and communication delays a potential bottleneck. They may require users to possess the trustees keys and use them in every session initiation, and further require modifications to every communication protocol. 
     (e) they require excessive numbers of trusted third parties (TTP) to be involved in system operation. Spreading the trust among too many parties increases the risk of security breaches and reduces scalability. 
     (f) they require generation of cryptographic keys by TTP&#39;s. A corrupt or otherwise compromised TTP may put user security at risk by tampering or disclosing user&#39;s keys. 
     (g) they require the securing and management of database(s) of secret keys or secret shares on behalf of users. 
     (h) they can be used to establish a shadow public key infrastructure, thus defeating the purpose of the escrow system altogether. 
     Auto-Recoverable and Auto-Certifiable Cryptosystems 
     Due to the above disadvantages, what is required is a new mechanism incorporating the following advantages: 
     (a) a key escrow system that can be distributed in source code form with no loss of security, and hence provides a system that can be publicly scrutinized to insure that it operates properly. Furthermore, since the key escrow system can be available in software, it can be implemented on a large scale, quickly, and cost-effectively. This implies fast distribution of the system. 
     (b) in the case that a software solution is deemed unacceptable due to the possibility of modifying of the invention, it can be implemented directly in tamper-resistant hardware. This however adversely affects the gains from (a) (e.g., the easy distribution). 
     (c) the escrow system requires the least amount of protocol interaction between the escrow authorities, CA, and user, that is theoretically possible. To register a key, a message need only be sent to one of a multitude of CA&#39;s. This mechanism is called a key registration based escrow system. In comparison, in the preferred embodiment of Fair PKC&#39;s, five messages are sent from the user to the trustees, and then five more messages are sent to a key management center. 
     (d) only one private database is required to implement the escrow system. This database need only be authenticated and may be kept private to prevent a shadow PKC from being established. User&#39;s private keys will not be exposed if the database is exposed. This contrasts with Fair PKC&#39;s in which several databases must be maintained and if they are compromised, the users keys are compromised. This requirement makes the new system rely only on the CA in establishing and certifying users keys as in usual public key systems. 
     (e) the escrow system allows the user&#39;s private key to be verified by anyone. The verification establishes that the private key is recoverable by the escrow authorities given the user&#39;s corresponding public key, the certificate, and public parameters. In comparison, in Fair PKC&#39;s, only the trustees perform this verification. This requirement of the new system is called universal verifiability. 
     (f) the escrow system can be made shadow public key resistant. Fair PKC&#39;s were shown not to be shadow public key resistant, namely they can be abused to publish other PKC schemes (J. Kilian, F. Leighton, “Fair Cryptosystems Revisited”, CRYPTO &#39;95, pages 208-221). 
     The present invention is versatile enough so that either (a) or (b) can be chosen (namely, a software or hardware implementation). In each case requirements (c) through (f) are met. 
     SUMMARY OF THE INVENTION 
     In order to provide for the above and other objectives and features to be described below, the present invention introduces a new paradigm in cryptography. The present invention provides a method to verify that a user generated private key is contained within an encryption under the public key of the escrow authorities without excessive overhead. Furthermore, this verification can be performed by anyone in possession of the escrow authorities public key. 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 preferred 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&#39;s public/private key pair, and a certificate of recoverability which is a string of information which includes an implicit encryption of the user&#39;s private key under the escrowing public key. The signal information containing the user&#39;s public key, and the certificate of recoverability can be transmitted 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&#39;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 preferred embodiment, the users certify their public keys with registration authority of the certification authority (CA) who then signs their public key after successful verification. A public key together with a CA&#39;s signature on a string that contains the public key constitutes a certified public key. In more detail, upon receiving the user&#39;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 user is only required to keep his public 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&#39;s certificate of recoverability, which is obtained from the CA, as an input signal. The escrow authorities process the certificate of recoverability, and the corresponding user&#39;s private key or data encrypted using the corresponding public key is the resulting output signal. 
     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, etc. The successful escrowing of private keys implies the successful escrowing of public key encrypted information, and hence the present invention has many applications. 
     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 as communication-free as is theoretically possible. The only communication is the act of disseminating the software itself (or the hardware device itself) and the one-time transmission of a user&#39;s public key, certificate of recoverability, and additional information. 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&#39;s (e.g., session key establishment, key distribution, secure message transmission, etc.). The invention is therefore compatible with typical PKI&#39;s. The present invention thus provides a very efficient way of escrowing and recovering cryptographic keys. 
    
    
     THE DRAWING 
     The present invention will be described with reference to the accompanying FIGS. 1-7. 
     FIG. 1 is a data flow diagram of the process of setting up the method of the invention for use with m escrow authorities. 
     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. 
     FIG. 3 is a data flow diagram of the process of verifying the recoverability of a private key. 
     FIG. 4 is a data flow diagram of the process of registering a key using the invention. 
     FIG. 5 is a data flow diagram of the process of private key recovery by the escrow authorities. 
     FIG. 6 describes a generic public key system and its main components and operations 
     FIG. 7 describes the escrowable public key system which results by employing the present invention and its main components and operations. 
    
    
     DESCRIPTION OF THE INVENTION 
     The following is a description of the preferred embodiment of the present invention. Variations on the preferred embodiment will accompany the description of the preferred 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 preferred 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. 
     The system setup of the preferred embodiment shown in FIG. 1 initiates the cryptosystem. In the preferred embodiment, the participants agree upon a large prime r such that q=2r+1 is prime and p=2q+1 is prime. Examples of values for r that satisfy this relation are 5 and 11, though they are small values. The following is a 1024 bit value for r in hexadecimal: 
     fd90e33af0306c8b1a9551ba0e536023b4d2965d3aa813587ccf1aeb1ba2da82489b8945e8899bc546dfded24c861742d2578764a9e70b88a1fe9953469c7b5b89b1b15b1f3d775947a85e709fe97054722c78e31ba202379e1e16362baa4a66c6da0a58b654223fdc4844963478441afbbfad7879864fe1d5df0a4c4b646591 
     An r of size 1024 bits is large enough for use in cryptographic systems. Such values of r, q, and p are not as easy to find as merely finding a prime number, but doing so is not intractable. What is needed is highly efficient algorithms which can be implemented using, say, a multiprecision library. Such algorithms include Karatsuba multiplication, Montgomery reduction, addition chains, and the Rabin-Miller probabilistic primality test (J. Lacy, D. Mitchell, W. Schell, “CryptoLib: Cryptography in Software,” AT&amp;T Bell Laboratories, cryptolib@research.att.com). 
     The following method can be used to find large values for r, q, and p efficiently. Note that r mod 3 must be 2. It can&#39;t be 0 since then r wouldn&#39;t be prime. It can be 1 since then q would be divisible by 3. Also, r mod 5 must be 1 or 4. It can&#39;t be 0 since then r would be divisible by 5. It can&#39;t be 2 since then q would be divisible by 5. It can&#39;t be 3 since then p would be divisible by 5, etc. We call this method “trial remaindering”. By performing trial remaindering, we can throw out values for r, q, and p quickly prior to performing trial divisions and probabilistic primality tests. Once we perform trial remaindering up to, say, 251, we perform trial divisions on r, q, and p. If r, q, and p are not thrown out we then do the Rabin-Miller primality test on r, then q, then p, then r, then q, etc. alternating between the three. We do so using small potential witnesses of compositeness that are fixed in advance. If any of r, q, or p are found to be composite, we set r to be equal to r+2×3×5×. . . ×251 and repeat starting with trial divisions and the same set of potential witnesses. This way we need not perform trial remaindering again, since the prior conditions on r are assured. Once r, q, and p are found, we perform additional primality tests using potential witnesses that are found using a strong random number generator. If r, q, and p pass these tests, then they are assumed to be prime and are declared as systems parameters. 
     The participants agree upon, or the CA chooses, a value g which generates (1, 2, 3, . . . , p−1)=Z p , and an odd value g 1  which generates Z* 2q . Note that 2q is a multiplicative group and has a generator. g and s are odd in the preferred embodiment. The values r, q, p, g, and g 1  are the systems initial parameters and are made publicly available with no loss of security. They can be chosen by the authorities themselves and/or anyone else. Once g 1  and q are specified, the m authorities (m≧1) proceed to collectively compute an escrow authority public key (Y, g 1 , 2q), also called the escrowing public key, and escrow authority private keys z 1 , z 2 , . . . , z m . To do so, authority i, where i ranges from 1 to m, chooses a value z 1  in Z 2r  at random and then sets Y 1  to be g 1  raised to this value modulo 2q. At least one authority then receives all of the information of the Y 1 &#39;s from the m−1 other escrow authorities. In the preferred embodiment, authority i, where i ranges from 2 to m, sends Y 1  to authority  1 . The sending of the Y 1 &#39;s is depicted by step  11  in FIG. 1. Y is computed to be the product of the Y 1 &#39;s modulo 2q by at least one of the authorities. In the preferred embodiment, Y is computed by authority  1 . Authority  1  then verifies that (g 1 /Y) is a generator of Z* 2q . If it isn&#39;t then step  12  is executed. In step  12  the other m−1 authorities are told to choose new values for z, hence the procedure is restarted from the beginning of step  11 . In the preferred embodiment, authority  1  chooses z 1  over again also. In an alternative embodiment, at least 1 and less than m of the authorities generate new values for z. This process is continued as many times as necessary until (g 1 /Y) is a generator of Z* 2q . Y is then published, or otherwise made available to the users and the CA, by one or more of the escrow authorities. This is depicted by step  13  in FIG.  1 . 
     FIG. 2 is a diagram showing the process of how a user&#39;s system generates a public/private key pair and a certificate of recoverability. Having obtained the signal Y that is made available to users by the escrow authorities, the user system proceeds to generate an ElGamal key (y, g, p) for the user. The signal Y may a priori have been included in the invention. The invention proceeds by choosing a value k in Z 2r  randomly. This is depicted by step  2004  in FIG.  2 . In step  2005 , the invention computes C=g 1   k  mod 2q. In step  2006  the invention computes the user&#39;s private key x to be (g 1   k /Y k ) mod 2q. The invention also computes y to be g x  mod p. 
     The system then proceeds to step  2007  and computes a certificate that can be used by any interested party to verify that the user&#39;s private key is properly encrypted within C. The certificate contains the value v, which is computed by the system to be g raised to the power w mod p, where w is 1/Y k  mod 2q. The public key parameter y can be recovered from g and v by computing v raised to the C power mod p. The system also processes three non-interactive zero-knowledge proofs as they are called in the art and includes them in the certificate. Let n denote the number of repetitions in each non-interactive proof. In the preferred embodiment, n is set to be 40. The first proof is designed so that the user can prove that he or she knows k in C. The second proof is designed so that the user can prove that he or she knows k in v. The last proof is designed so that the user can prove that he or she knows k in v raised to the C power mod p. By saying “the user knows value x” we mean that the system has value x in its state. 
     In more detail, to construct the non-interactive proofs, the system proceeds as follows. It chooses the values e 1,1 , e 1,2 , . . . , e 1,n , e 2,1 , e 2,2 , e 2,3 , . . . , e 2,n  and e 3,1 , e 3,2 , e 3,3 , . . . , e 3,n  in Z 2r  randomly. For i ranging from 1 to n, the system sets I 1,i  to be g 1  raised to the e 1,i  power mod 2q. For i ranging from 1 to n, the invention sets I 2,i  to be v raised to the d i  power mod p, where d i  is Y raised to the −e 2,i  power modulo 2q. For i ranging from 1 to n, the invention sets I 3,i  to be y to the t i  power mod p, where t i  is (g 1 /Y) raised to the e 3,i  power mod 2q. The invention then computes the value rnd to be the SHA hash of the set formed by concatenating together the tuples (I 1,i , I 2,i , I 3,i ) for i ranging from 1 to n. Note that rnd is a function of all of the I values, using a suitably strong cryptographic hash function. In alternate embodiments, the hash function can have an effective range of size different than 160 bits. A greater range of the hash function allows for significantly larger values for n. The system sets each of the bit-sized values b 1,1 , b 1,2 , . . . , b 1,n , b 2,1 , b 2,2 , . . . , b 2,n , b 3,1 , b 3,2 , . . . , b 3,n  to be each of the corresponding 3n least significant bits of rnd. There are a multitude of ways in which an embodiment can securely assign the bits of rnd to the values for b. The values for b are the challenge bits, and this method of finding them is known as the Fiat-Shamir Heuristic. The system then proceeds to compute the responses to these challenge bits. For i ranging from 1 to 3 and for j ranging from 1 to n, the invention sets z i,j  to be e i,j +b i,j  k mod 2r. This completes the description of step  2007  of FIG.  2 . 
     The system proceeds to step  2008 . In step  2008 , the invention outputs the parameters C, v, y, (I 1,i , I 2,i , I 3,i ), and (z 1,i , z 2,i , z 3,i ) for i ranging from 1 to n. In an alternate embodiment the value k is output by the invention to the user. The user then has the option to later interactively prove that his or her private key x is recoverable by the escrow authorities. This will be addressed in more detail later. Also, the values b can be made a part of the certificate. This step is however, not necessary, since the values for b can be derived from the values for I alone. 
     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 key generated has the publicly specified properties. The following describes how the invention is used by the user to prove to a verifier that x is recoverable from C. This process is depicted in FIG.  3 . The verifier can be the CA, an escrow authority, or anyone else who is part of the system. 
     The verification process of FIG. 3 is as follows. In step  3009 , the user generates a public/private key pair, encryption of x, 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&#39;s private key is recoverable by the escrow authorities. To do so, the verifier uses the user&#39;s public key, the encryption C, the corresponding certificate, and the escrowing public key Y. 
     The way in which the users signal is processed will now be described in detail. The verifying system outputs a 0 if the public key and/or certificate are invalid, and a 1 otherwise. The invention may take subsequent action and indicate to the verifier that the public key is invalid in the event that 0 is returned. Similarly, the verifying system may inform the verifier of a validation that passes. 
     To perform the verification, the verifying system verifies the three non-interactive proofs contained within the certificate of the user. The invention computes (b 1,i , b 2,i , b 3,i ) for i ranging from 1 to n in the same way as performed during the certificate generation process. Recall that this process was described in regards to FIG.  2 . 
     For the first non-interactive proof, the verifying system checks that g 1  raised to the z 1,i  power equals CI 1,i  mod 2q if b 1,i =1, for i ranging from 1 to n. The verifying system also checks that g 1  to the z 1,i  power equals I 1,i  mod 2q if b 1,i =0, for i ranging from 1 to n. If any of these equalities fails, then the verifying system returns a value of 0. This completes the verification of the first non-interactive proof. 
     For the second non-interactive proof, the verifying system checks that g raised to the w i  power equals I 2,i  mod p if b 2,i =1, for i ranging from 1 to n. Here w i  is 1/Y raised to the z 2,i  power mod 2q. The verifying system also checks that v to the v i  power equals I 2,i  mod p if b 2,i =0, for i ranging from 1 to n. Here v i  is 1/Y to the z 2,i  power mod 2q. If any of these equalities fails, then the verifying system returns a value of 0. This completes the verification of the second non-interactive proof. 
     For the third non-interactive proof, the invention checks that g raised to the w i  power equals I 3,i  mod p if b 3,i =1, for i ranging from 1 to m. Here w i  is (g 1 /Y) raised to the z 3,i  power mod 2q. The invention also checks that y to the v i  power equals I 3,i  if b 3,i =0, for i ranging from 1 to m. Here v i  is (g 1 /Y) raised to the z 3,i  power mod 2q. If any of these equalities fails, then the verifying system returns a value of 0. If all of the checks pass, then the value 1 is output by the verifying system. 
     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&#39;s private key is recoverable by the escrow authorities. 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&#39;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. 
     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&#39;s public key, and make the key along with the CA&#39;s signature of that key available on request. If the CA is not trusted, 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 intern can insure recoverability. This completes the description of the public key certification process. 
     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 n escrow authorities to recover the user&#39;s private key based on C. In this process, all m of the escrow authorities obtain C, as depicted in step  5015  of FIG.  5 . In an alternate embodiment the CA transmits C and/or other parameters to one or more of the authorities. Thus they are already in possession of C. At this point escrow authority  1  computes t 1  to be C raised to the z i  power mod 2q. Recall that z i  is the private key of the ith escrow authority. This is done for i ranging from 1 to m. Authorities  2  through m then send their respective values for t to authority  1 , as depicted in step  5016 . Authority  1  then computes Y k  mod 2q to be the product of the values for t i  where i ranges from 1 to m. Authority  1  then obtains the user&#39;s private key x by computing x=(C/Y k ) mod 2q. There are alternative methods in the art for computing x so that x is represented distributively among the authorities. These methods also allows the authorities to decrypt messages encrypted under the public key corresponding to x, without revealing x itself. 
     What has been described is an Auto-Recoverable and Auto-Certifiable (ARC) cryptosystem. The users of such a cryptosystem employ the public key system in a way that is identical to a typical PKI 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&#39;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&#39;s database and the certificate is verified. (4) Then, the message is encrypted under new 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&#39;s private key. 
     In an alternative embodiment any large enough subset of the authorities can recover the private key x or messages encrypted under the public key corresponding to x without revealing x itself. This is done independently by receiving the appropriate values for t by the other authorities. This adds robustness in the case that some or all of the authorities cannot be completely trusted or are otherwise unavailable. Also, the authorities can require that the certificate of recoverability be sent along with the public key and encryption so that the user&#39;s parameters can first be verified using the verification process. This completes the description of the private key recovery process. 
     The following are a few alternate embodiments of the present invention. An alternate embodiment of this invention involves using an authority public key of the form (Y, g, 2q t ), where t is some integer greater than 1. We chose t to be 1 in our preferred embodiment, though other values can be used instead and still operate based on primitive roots. Another alternate embodiment is to use the product of two or more large primes as part of the public parameters. Clearly, the exact structure of the moduli used can vary significantly without departing from the scope of the invention. In another embodiment, the interactive versions of the three non-interactive proofs can be used. Such an embodiment requires that the system output k to the user during key generation. This value k is used during the interactive protocol, so that the verifier can be convinced that the user&#39;s private key is recoverable by the escrow authorities. Note that by outputting k, however, a shadow public key cryptosystem may result. This follows from the fact that ((g 1 , C, 2q), k) is a valid ElGamal public/private key pair mod 2q. 
     In yet another embodiment, the CA, or other trusted entity, takes the further action of blinding the user&#39;s public keys. The CA chooses a k s.t. g′=g k  mod p is a generator, and sends the user (g′, y k  mod p). g′ is the user&#39;s ElGamal generator and y′=y k  mod p is part of the users final key (g′, y′, p). This prevents users from exploiting subliminal channels in y. 
     In another variant the users publish their public keys which are used for key exchanges in a Diffie-Hellman like “key-exchange”. For example, the following method can be used. Let a be user A&#39;s private key and let b be the user B&#39;s private key. Let y a =(g to the power a) mod p be user A&#39;s public key and let y b =(g to the power b) mod p be user B&#39;s public key. To establish a random session key, user B chooses a random string s. User A then sends m=(y b  to the a power)s mod p to user B. User B recovers s by computing m/(y a  to the power b) mod p. Users A and B derive a session key from s using a known public function (e.g., applying it to a one-way hash function). Later, when the session key is required to be taken out of escrow, the trustees can use either a or b to recover s, and hence the session key. 
     An application of this invention is an multi-escrow authority system where each escrow authority has its own CAs and users. When users from two different escrow authorities conduct secure communication the two escrow authorities can retrieve the user&#39;s messages or keys and exchange them through bilateral agreement. This is applicable to international multi-country scenarios. 
     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 precious 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. 
     Thus, there has been described a new and improved key escrow system, its variants, and applications. It is to be understood that the preferred 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. 
     The above description of our cryptosystem makes novel uses of number theory in cryptography. It shows how to design a cryptosystem based on three primes with direct arithmetic relations between all them. That is r, q, and p are primes such that q=2r+1 and p=2q+1. The usage of three or more primes with relations between them can produce various cryptosystems of a similar nature to the one described above. Some of them are described in the variation on the preferred embodiment. Another relation can be p=2q+1 and q=2rs+1 where p, q, r, and s are all prime and r is 160 bits in length. Another example is p=2q+1, q=2r+1, and r=2s+1 where p, q, r, and s are all prime numbers. Furthermore, another innovative use of number theory is performing cryptographic operations in the exponent, where the operations are, for example, modular exponentiations. For example, the second zero-knowledge proof in step  2007  of the invention involves proving knowledge of k in v where v is equal to g raised to the w power mod p, where w is (Y raised to the −k power) mod 2q. The use of three or more domains in successive exponentiations adds flexibility and power to cryptosystems. Applications of this fact along the lines of the present invention, are readily available to those who are skilled in the art.