AUTO-RECOVERABLE AUTO-CERTIFIABLE CRYPTOSYSTEMS

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 keys 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 efficient ly 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'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 interact ion 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 any schemes which require that the escrowing device be tamper-proof hardware.

Description: AUTO-RECOVERABLE AUTO-CERTIFIABLE CRYPTOSYSTEMS Background-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. Background-Description of Prior Art Public Key Cryptosystems (PKC's) allow secure commu- nications 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 authori- ty 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 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) . PKC'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's have a drawback, however. Criminals can use PKC'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. patents 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 '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 ' s prefered embodiment discloses how to convert the Diffie-Hellman PKC, and the RSA PKC into Fair PKC's. In the prefered embodiment of the Fair Diffie-Hellman PKC, each user submits five shares to five central trustees (also known as "trusted third par- ties") 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's). In Micali ' 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'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 '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's. These channels are used to display the public keys of the shadow PKC. The Kilian and Leighton paper discloses how to convert PKC's into Fail-safe Key Escrow (FKE) systems. Specifically, they disclose how to construct FKE systems for discrete-log based PKC's like Diffie-Hellman and DSS . In their expensive protocol, the user and the trusted authorities engage in a protocol to generate the user'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'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's seems remote. De Santis et al . teach an escrow system where trust- ees 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 '97, pages 119-1- 33, 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's public key, and then send the resulting ciphertext message using Binding ElGamal. In this case, the proof simply serves to show that the trust- ees 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'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 '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. 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", Euro- crypt '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 posses 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. Spread- ing the trust among too many parties increases the risk of security breaches and reduces scalability. (f) they require generation of cryptographic keys by TTP's. A corrupt or otherwise compromised TTP may put user security at risk by tampering or disclosing user'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's. This mechanism is called a key registration based escrow system. In comparison, in the prefered embodiment of Fair PKC'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's private keys will not be exposed if the database is exposed. This contrasts with Fair PKC'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's private key to be verified by anyone. The verification establishes that the private key is recoverable by the escrow authorities given the user's corresponding public key, the certif- icate, and public parameters. In comparison, in Fair PKC's, only the trustees perform this verification. This requirement of the new system is called universal verifi- ability . (f) the escrow system can be made shadow public key resistant. Fair PKC'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 '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 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 authori- ties, 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 recoverabili- ty which is a string of information which includes an implicit encryption of the user's private key under the escrowing public key. The signal information containing the user's public key, and the certificate of recoverabili- ty 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 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 constitutes a certified public key. 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 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'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'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'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 ' 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 Drawing The present invention will be described with refer- ence to the accompanying figures 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; Prefered Embodiments The following is a description of the first prefered embodiment of the present invention. 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. The system setup of the prefered embodiment shown in FIG. 1 initiates the cryptosystem. In the prefered embodiment, the participants agree upon a large prime r such that q = 2r+l is prime and p = 2q+l 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 : fd90e33af0306c8bla9551ba0e536023b4d2965d3aa813587ccflae blba2da82489b8945e8899bc546dfded24c861742d2578764a9e70b 88alfe9953469c7b5b89blbl5blf3d775947a85e709fe97054722c7 8e31ba202379elel6362baa4a66c6da0a58b654223fdc4844963478 441afbbfad7879864feld5df0a4c4b646591 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 multi- precision 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&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't be 0 since then r wouldn'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't be 0 since then r would be divisible by 5. It can't be 2 since then q would be divisible by 5. It can'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 x 3 x 5 x...x 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 the elements in the set {l,2,3,...,p-l} and an odd value gl which generates all values less than 2q and relatively prime to 2q. Note that 2q is a multiplicative group and has a generator, g and s are odd in the prefered embodiment. The values r, q, p, g, and gl 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 gl and q are specified, the m authorities (m greater than or equal to 1) proceed to collectively compute an escrow authority public key (Y, gl , 2q) , also called the escrowing public key, and escrow authority private keys z_l, z_2 , ... , z_m. To do so, authority i, where i ranges from 1 to m, chooses a value z_i in {l, 2 , ... , 2r-l} at random and then sets Y_i to be gl raised to this value modulo 2q. At least one authority then receives all of the information of the Y_i ' s from the m-1 other escrow authorities. In the prefered embodiment, authority i, where i ranges from 2 to m, sends Y_i to authority 1. The sending of the Y_i ' s is depicted by step 11 in FIG. 1. Y is computed to be the product of the Y_i ' s modulo 2q by at least one of the authorities. In the prefered embodiment, Y is computed by authority 1. Authority 1 then verifies that (gl/Y) is a generator of all values less than 2q and relatively prime to 2q. If it isn'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 prefered embodiment, authority 1 chooses z_l 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 (gl/Y) is a generator of all values less than 2q and relatively prime to 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'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' 9' P) ror tne user. The signal Y may a priori have been included in the invention. The invention proceeds by choosing a value k in {l, 2 , ... , 2r-l} randomly. This is depicted by step 2004 in FIG. 2. In step 2005, the invention computes C = (gl raised to the k power) mod 2q. In step 2006 the invention computes the user's private key x to be ( (gl/Y) rasied to the k power) mod 2q. The invention also computes y to be (g raised to the x power) 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'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 ( (l/Y) raised to the k power) 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 prefered 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_l , 1 , e_l , 2 , ... , e_l , n, e_2 , 1 , e_2 , 3 , ... , e_2 , n, and e_3,l, e_3,2, e_3 , 3 , ... , e_3 , n in {l, 2 , ... , 2r-l} randomly. For i ranging from 1 to n, the system sets I_l,i to be gl raised to the e_l,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 (gl/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 concate- nating together the tuples (I_l,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_l,l, b_l,2,..., b_l,n, b_2,l, 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_l,i, I_2 , i , I_3,i), and (z_l,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 ad- dressed 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 certif- icates 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's private key is recoverable by the escrow authorities. To do so, the verifier uses the user'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 first verifies that y = v raised to the C power mod p. If y is not equal to v raised to the C power mod p, then the verification system returns a value of 0. The verifying system then verifies the three non- interactive proofs contained within the certificate of the user. The invention computes (b_l,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 gl raised to the z_l,i power equals C(I_l,i) mod 2q if b_l,i = 1, for i ranging from 1 to n. The verifying system also checks that gl to the z_l,i power equals I_l,i mod 2q if b_l,i = 0, for 1 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 l/Y to the z_2 , i power mod 2q. If any of these equalities fail, 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 (gl/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 (gl/Y) raised to the z_3 , i power mod 2q. If any of these equalities fails, then the verifying system returns a value 0. If all of the verifications 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 gener- ates 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 author- ities. 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. 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, and make the key along with the CA'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, which in turn 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'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 i computes t_i 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 raised to the k power mod 2q to be the product of the values for t_i where i ranges from 1 to m. Authority 1 then obcains the user's private key x by computing x = (C/ (Y raised to the k power) ) 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 FIG's 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 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's private key. In an alternative to the first 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 that the certificate of recoverability be sent along with the public key and encryption so that the user'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 first embodiment of the present invention. An alternate embodiment of this invention involves using an authority public key of the form (Y, g, 2 (q raised to the t power)), where t is some integer greater than 1. We chose t to be 1 in our prefered 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's private key is recoverable by the escrow authorities. Note that by outputing k, however, a shadow public key cryptosystem may result . This follows from the fact that ((gl, 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's public keys. The CA chooses a k s.t. g' = (g raised to the k power) mod p is a generator, and sends the user (g', (y raised to the k power) mod p) . g' is the user's ElGamal generator and y' = (y raised to the k power) mcd 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's private key and let b be user B's private key. Let y_a = (g to the power a) mod p be user A's public key and let y_b = (g to the power b) mod p be user B'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 to it 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. The following is a description of a second primary embodiment of the present invention. The hashing algorithm selected is SHA (Schneier 2nd edition, pages 442-445) , though any cryptographic hashing algorithm will suffice in its place. We use the least significant bits of the hash results for convenience, but any subset of bits is possible. In the prefered embodiment, parameters are chosen uniformly at random from their respective groups or domains. 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. The system setup of this alternate embodiment shown in FIG. 1 initiates the cryptosystem. In the prefered embodiment, escrow authority i for 1 <- i -.-■ m generates a private share D_i, and corresponding public share E_i . The private shares D_i form the shared private key D. Escrow authorities 2 through m send their E_i to escrow authority 1. This is depicted by step 11. Escrow authority 1 com- bines all the public shares E_i and computes the shared public key E. The value for E is published, by escrow authority 1, as depicted in step 13. Each authority i keeps D__i private. As a concrete example, the escrow authorities can generate a strong prime p and a value g which generates {l, 2 , ... ,p-l} . Share D_i can be chosen uniformly at random from {l, 2 , .. ,p-l} , and E_i = (g raised to the D_i power) mod p. E is the product of all the values E_i modulo p. Variations on joint generation of keys are possible, as well as an implementation with a single escrow authority. A process similar to FIG. 2 illustrates how a user's system generates a public/private key pair and a certifi- cate of recoverability. Having obtained (and verified as much as possible) the signal E that is made available to users by the escrow authorities, the user system proceeds to generate an ElGamal public key (y,g,p) for the user (T. ElGamal, "A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms", CRYPTO '84, pages 10-18, Springer-Verlag, 1985) . The user system chooses a private key x uniformly at random from {l, 2 , .. ,p-l} , and computes y to be (g raised to the x power) modulo p. This key generation process corresponds to step 2006. The system then proceeds to step 2007 and computes a certificate that can be used by any interested party, in particular the CA, to verify that the user's private key x can be recovered from the certificate of recoverability P. Let ENC(a,s,E) denote the public key encryption of the message a under public key E using randomness s. Here ENC is a semantically secure probabilistic public key encryption, where the string s is used for the randomness in the probabilistic encryption. For example, ENC can be an ElGamal encryption, or an optimal asymmetric encryption (Bellare-Rogaway, "Optimal Asymmetric Encryption" , Eurocrypt '94). Let DEC be the corresponding public key decryption function which is performed in a shared fashion. Hence, DEC (ENC (a, s,E) ,D_1,D_2, ... ,D_m) = a. P is constructed according to the following algorithm: 1. P = () 2. for i = 1 to M do 3. choose r_i randomly from the domain {1,2, ..,p-l} 4. choose two random strings s_i,l and s_i,2 for use in ENC 5. Q_i = (g raised to the r_i power) mod p 6. C_i,l = ENC(r_i, s_i,l, E) 7. C_i,2 = ENC(r_i - x mod p-1, s_i,2, E) 8. add (Q_i, C_i,l, C_i,2) to the end of P 9. val = H(P) 10. set b_l, b_2,...,b_M to be the M least significant bits of val, where b_i is in {θ,l} 11. for i=l to M do 12. w_i = r_i - (b_i)x 13. Z_i = ( (w_i) , s_i, j ) where j = 1 + b_i 14. add Z_i to the end of P Thus, P = ((Q_l, C_l,l, C_l,2) , ... , (Q_M, C_M,1, C_M,2) , Z_l, ... , Z_M) . 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 and using them is analagous to the Fiat-Shamir Heuristic. The user system outputs (y,x,P) in step 2008. Note that the user has the option to interactively prove that his or her private key x 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) . 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 x 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. 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 y, the corresponding certificate P, and the escrowing public key E. The verifying system first checks that y < p. 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_i,j for all i and j , do not contain any repetitions . The verifying system checks that none of the Q_i for all i are repetitious. If any of these verifications fail, then false is returned. The verifying system then computes b_l , b_2,...,b_M in the same way as in the certificate genera- tion process. For i=l to M, the verifying system verifies the following things : 1. ENC(w_i, s_i,j, E) = C_i,j where j = 1 + b_i 2. (Q_i/ (y raised to the b_i power) ) mod p = ( g raised to the w_i power) mod p The verifying system returns true as long as all the verifications pass and as long as both 1 and 2 above 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) . 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 . 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 certifica- tion 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'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. 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 y and P, as depicted in step 5015 of FIG. 5. In an alternate embodiment the CA transmits y and P and/or other parameters to one or more of the authorities. Thus they are already in possession of y and P. At this point escrow authorities use a subset of their shares D_l, D_2 , ... , D_m to decipher P to open all of the unopened C_i,j (using DEC for example. 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,j from P and decrypts them using D_i . 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,j from P. Thus all of the plaintexts corresponding to all C_i,j are known to the escrow authorities. There are alternative methods in the art for recovering the plaintext corresponding to the unopened C_i,j, so that the unopened plaintext is represented distributively among the authorities. The escrow authorities check the plaintext of each pair C_i,l and C_i,2 for a pair of values that when subtracted together mod p-1, are equal to the exponent x in y = (g raised to the x power) mod p. Also, the quantity (g raised to the x power) mod p can be matched against the public y to assure correctness. Once such a pair is found, the private key of the user has been found. We will now describe a third primary embodiment of this invention. In this embodiment, the users of the system generate composite public keys. The user system generates n and s in the same way as described in the pending U.S. Patent 08/920,504 (by Young and Yung) . Recall that n is the product of two (preferably strong) primes p and q, and s is a string that can be used in conjunction with a public one-way function to derive the upper order bits of n. Let e and d denote the public and private exponents (e.g., for RSA) , respectively. The following is how P is constructed: 1. P = 0 2. choose a string t_0 randomly mod n 3. add t_0 to the end of P . for i = 1 to M do 5. choose a_i,l randomly from the set {1,2, .., (p-1) (q-l)-l} 6. compute a_i,2 = d - a_i,l mod (p-1) (q-1) 7. choose two random strings s_i,l and s_i,2 for use in ENC 8. t_i = H(t_(i-1) ) 9. v_i,l = (t_i raised to the a_i,l power) mod n 10. v_i,2 = (t_i raised to the a_i,2 power) mod n 11. Q_i = (t_i, v_i,l, v_i,2) 12. C_i,l = ENC(a_i,l, s_i,l, E) 13. C_i,2 = ENC(a_i,2, s_i,2, E) 14. add (Q_i, C_i,l, C_i,2) to the end of P 15. val = H(P) 16. set b_l, b_2,...,b_M to be the M least significant bits of val, where b_i is in {θ,l} 17. for i=l to M do 18. Z_i = (a__i,j, s_i,j) where j = 1 + b_i 19. add Z_i to the end of P 20. add s to the end of P Thus, P = (t_0, (Q_l, C_l,l, CAL, 2 ),..., (Q_M, C_M,1, C_M,2) , Z_l, ... , Z_M, s) . H above can be based on SHA or on concatenations of a few SHA applications to generate the t_i of appropriate size. It is most likely that the t_i will be in the set of elements less than n and relatively prime to n. The verifying system is a bic different than before. The verifying system first checks that n was chosen from the correct set of values. Let u denote the integer corresponding to the k/2 upper order bits of n. The verifying system makes sure that either H(s) = u or that H(s) = u+1, as described in the pending U.S. patent 08/920,504. The verifying system checks that all of the values in P lie in the correct sets. For example, the verifying system checks that the t_i fall within the range of H, and that a_i , j < n (or some function of n) where j is 1 or 2. The verifying system also checks that t_i = H(t_(i-1)) for i ranging from 1 to M. The verifying system checks that elements of the tuple Q_i for each i does not contain repetitions, and also that the elements in the pair Z_i for all i does not have repetitions. If any of these verifications fails, then false is returned. The verifying system then computes b_l, b_2 , ... , b_M in the same way as in the certificate generation process. For i ranging from 1 to M, the verifying system verifies the following things: 1. ((v_i,l multiplied by v_i,2) raised to the e power) mod n = t_i 2. (t_i raised to the a__i,j power) mod n v_i , j , where j = 1 + b_i 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. The escrow authorities recover the user's private key as follows. For i ranging from 1 to M, the authorities compute w_i to be the sum of the plaintexts corresponding to C_i,l and C_i,2. If a value w_i is found such that (t_i raised to the e(w__i) power) mod n equals t_i, then w_i constitutes a valid RSA private key corresponding to e . It is well known in the art how to factor n given such a value w_i . Note that the RSA function is a homomorphic function and the above embodiment is applicable to homomorphic functions similar to RSA. We remark that from the above embodiment it is clear that this 'proof technique 1 for showing that a value is recoverable by the escrow authorities can be generalized to any homomorphic function. 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's messages or keys and exchange them through bilateral agreement. This is appli- cable 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 previous embodiments, in particular based on the preceding para- graph. 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 above description of our first embodiment of our cryptosystem makes novel uses of number theory in cryptog- raphy. 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+l and p = 2q+l. 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 prefered embodiment. Another relation can be p = 2q+l 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+l, q = 2r+l, and r = 2s+l 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 first embodiment 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 crypto- systems. Applications of this fact along the lines of the present invention, are readily available co those who are skilled in the art. 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 three. 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. At any time, the leaf of the tree can form another subtree and act as an escrow agent (s) . 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 . 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. Yet another application of this invention is a certified electronic mailing system. When the users regis- ter into the system, they register an auto-recoverable encryption public key and a certificate of recoverability to the CA, and they also register a signature public key. To send a certified piece of mail, the following is done. The sender sends a packet which includes the following: an encryption of an e-mail key under his own auto-certified public key, the receiver's name, an encryption of the e-mail message encrypted under the e-mail key, a header indicating a certified e-mail message, his own certified public key, and the CA's certificate on his certified public key, and other information. The packet is signed using the senders signature private key. Both the packet and the signature on the packet are sent to the receiver. The receiver forms a return receipt packet that consists of a fixed return receipt header, the received message (or the hash of the received message) , and additional information. This packet is signed using the receivers private signature key and is sent to the original sender. The original sender verifies the signature on the return receipt packet. If the signature is valid, the original sender sends the receiver the e-mail key encrypted under the receiver's certified public key. This message is sent along with a signature on it using the original sender's private signing key. The receiver verifies the signature on the encrypted e-mail key. If the signature is valid, the receiver decrypts the e-mail key using his private decryption key. The receiver then encrypts the result using the original senders certified public key. If the result matches the ciphertext in the first packet that the original sender sent, then the e-mail key is regarded as authentic. This key is then used to decrypt and obtain the actual information that the original sender sent. If the receiver is for some reason unable to contact the original sender after the first packet is received, the receiver sends the return receipt and the first packer. to the escrow authorities. The escrow authorities will recover the e-mail key, provided the packet and return receipts are authentic and provided that the packet contains the corrects receivers name. The escrow authorities retain the return receipt and the packet. Provided the checks pass, the e-mail key is sent to the receiver. This constitutes a certified e-mail system based on auto-recoverable keys and signature keys, and where user registration is analogous to user registra- tion in a typical public key system with a CA. Also, it is known in the art how to employ certified e-mail systems as above for contract signing between two parties. The above application can be used as such. 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 ar- rangements can be readily devised by those who are skilled in the art without departing from the scope of the present invention.

Claims: What we claim is: 1. A method and apparatus comprising a crypto- system which can be used for generating, verifying, using, and recovering cryptographic keys. Said method and appara- tus is comprised of a subset of the following steps: (1) establishing a set of system parameters; (2) having the escrow authorities generate crypto- graphic keys and having said escrow authorities publish escrow authorities public keys; (3) having each certification authority publish unique certification authority parameters; (4) having each user generate a public/private key pair based on said system parameters using a speci- fied user algorithm; (5) having each user generate a proof which is a proof -transcript claiming that said user's public/private key pair was generated using said specified user algorithm and that said user ' s private key is recoverable by the escrow authorities; (6) having a certification authority employ verifica- tion to check that said proof is valid; (7) having a certification authority certify said user's public key and uiake a corresponding certif- icate available to the public only if proof is valid; (8) having users employ users keys and certificates in use of said cryptosystem; (9) having the escrow authorities recover the private key or information enciphered under said private key's corresponding public key, of a user when given proper authorization to do so. 2. A method and apparatus as in Claim 1 wherein the escrow authorities are a multitude of entities and where: each entity publishes a public key; user key generation and recovery of user's private information can be performed using the keys of a subset of said multitude of entities. 3. A method and apparatus as in Claim 1 where parts of said proof and verification are conducted interactively between said user and said certification authority. 4. A method and apparatus as in Claim 1 wherein said user's certificate is comprised of a signature of a certification authority on a record comprising of at least a string associated with said user, and said user's public key. 5. A method and apparatus as in Claim 1 wherein said user's certificate is comprised of a signature of a certification authority on a record comprising of at least a string associated with said user, and a modification of said user's public key. 6. A method and apparatus as in Claim 1 wherein said use of said cryptosystem comprises the employment of said public/private key pairs and said certificates in said cryptosystem for the performance of any subset of the following: public key encryption/decryption, signing and digital signature verification, key exchanges, and identi- fication protocols. 7. A method and apparatus as in Claim 1 where said user's private key or information encrypted under the public key corresponding to said user's private key is recovered in order to monitor communication of user's suspected of criminal activity while protecting the privacy of other users . 8. A method and apparatus as in Claim 1 where said proper authorization is a court order given to the escrow authorities on behalf of a government agency or a court order recognized by a group of governments or agencies corresponding to a group of governments. 9. A method and apparatus as in Claim 1 with the further step of : characterizing the user's activities as unlawful if the escrow authorities are unable to monitor the user's communications. 10. A method and apparatus as in Claim 1 where the functionality of at least one of the escrow authorities, the user, and a certification authority, in at least one of the steps is implemented in hardware. 11. A method and apparatus as in Claim 1 where use of said user's public key is for the encryption of files. 12. A method and apparatus as in Claim 1 whereby the key recovery is of information between two users, user 1 and user 2, where the first subset of said escrow authori- ties recovers the private key or information enciphered under the public key corresponding to user l's said private key, and another subset recovers the private key or infor- mation enciphered under the public key corresponding to user 2's said private key. 13. A method and apparatus as in Claim 1 where said proof includes a transcript which is a zero-knowledge proof of knowledge of said user's private key. 14. A method and apparatus as in Claim 1 where said proof includes a transcript claiming that the escrow authorities are capable of recovering the private key of said user or information encrypted under said private key. 15. A method and apparatus as in Claim 1 where proper authorization is generated by following a proper process within said user's organization. 16. A method and apparatus as in Claim 1 which can be used for generating, using, verifying, and recovering cryptographic keys wherein said set of system parameters includes at least three domains FI, F2 , and F3 such that FI is the exponent domain of F2 , and F2 is the exponent domain of F3. 17. A method and apparatus as in Claim 1 which can be used for generating, using, verifying, and recovering cryptographic keys wherein said set of system parameters includes at least three domains 2r, 2q, and p such that p = 2q+l = 4r+3, where p, q, and r are prime. 18. A method and apparatus as in Claim 1 where said user's key is y, where y equals g raised to the x power modulo p, where g is a generator modulo the prime p; x is said user's private information. 19. A method and apparatus as in Claim 1 where said user's key is based on a number n where only said user knows the factorization of n into prime numbers. 20. A method and apparatus as in Claim 1 where said user's key is a homomorphic function. 21. A method and apparatus as in Claim 1 where said proof includes encryptions using said escrow authorities public keys . 22. A method and apparatus as in Claim 13 where said zero-knowledge proof of knowledge of said user's private key employs trapdoor one-way functions to generate en- crypted values . 23. A method and apparatus as in Claim 14 where said transcript claiming that the escrow authorities are capable of recovering the private key of said users or information encrypted under said private key employs trapdoor one-way functions to generate encrypted values. 24. A method and apparatus as in Claim 1 where said user generates a public/private key pair based on system parameters which include said escrow authorities public keys . 25. A method and apparatus as in Claim 1 where steps (2) , (5) , (6) , (9) are added to already existing steps (1), (3), (4), (7), (8) which by themselves are typical steps in an existing apparatus comprising a public key infrastructure. 26. A method and apparatus as in Claim 1 where the escrow authority recovers the private key or information enciphered under said private key's corresponding public key with the further help of the certification authority. 27. A method and apparatus as in Claim 1 with the additional steps of having a user generate a private/public pair constituting a signature key which is different from the public/private key pair of step (4) , and having a certification authority certify said public portion of said signature key. 28. A method and apparatus as in claim 27 where use of said cryptosystem is for electronic mail with assured delivery. 29. A method and apparatus as in Claim 1 where the escrow authorities public key and public keys generated by users are from different key domains. 30. A method and apparatus as in Claim 1 where the escrow authorities are a multitude of elements with an extended step (2) where the escrow authorities are orga- nized in a hierarchy where each element is able to open keys in its sub-hierarchy. 31. A method and apparatus as in Claim 1 where the escrow authorities are a multitude of elements and where said user in step (5) generates a proof that said user's private key is recoverable by a subset of escrow authori- ties. 32. A method and apparatus as in Claim 30 where user in step (5) generates a proof that said user's private key is recoverable by a subset of escrow authorities.

Inventors: YOUNG ADAM LUCAS, YUNG MARCEL MORDECHAY, YOUNG, ADAM, LUCAS, YUNG, MARCEL, MORDECHAY
Applicants: YOUNG ADAM LUCAS, YUNG MARCEL MORDECHAY, YOUNG, ADAM, LUCAS, YUNG, MARCEL, MORDECHAY
Application year: 1998
Main classification: H04L
IPC codes: G09C1, H04L9