US20100174904A1 - Use of modular roots to perform authentication including, but not limited to, authentication of validity of digital certificates - Google Patents

Abstract

Authentication of elements (e.g. digital certificates 140) as possessing a pre-specified property (e.g. being valid) or not possessing the property is performed by (1) assigning a distinct integer p_ito each element, and (2) accumulating the elements possessing the property or the elements not possessing the property using a P-th root u^1/P(mod n) of an integer u modulo a predefined composite integer n, where P is the product of the integers associated with the accumulated elements. Alternatively, authentication is performed without such accumulators but using witnesses associated with such accumulators. The witnesses are used to derive encryption and/or decryption keys for encrypting the data evidencing possession of the property for multiple periods of time. The encrypted data are distributed in advance. For each period of time, decryption keys are released which are associated with that period and with the elements to be authenticated in that period of time. Authentication can be performed by accumulating elements into data which are a function of each element but whose size does not depend on the number of elements, and transmitting the accumulator data over a network to a computer system which de-accumulates some elements as needed to re-transmit only data associated with elements needed by other computer systems. This technique is suitable to facilitate distribution of accumulator data in networks such as ad hoc networks.

Description

CROSS REFERENCE TO RELATED APPLICATIONS
• The present application is a continuation of U.S. patent application Ser. No. 11/454,262, filed Jun. 16, 2006, incorporated herein by reference, which is a division of U.S. patent application Ser. No. 11/304,200, filed on Dec. 15, 2005, incorporated herein by reference, which claims priority under 35 U.S.C. §119(e) to provisional U.S. patent application No. 60/637,177 filed Dec. 17, 2004, incorporated herein by reference.
BACKGROUND OF THE INVENTION
• The present invention relates to performing authentication as to whether or not an element possesses a pre-specified property. An example is authenticating validity of a digital revocation in a public key infrastructure, or authenticating validity of an entitlement to use a resource (e.g. to sign onto a World Wide Web site).
• FIG. 1 illustrates digital certificate validation and revocation in a public key infrastructure.Digital certificates104 are used in public key infrastructures (PKI) to facilitate secure use and management of public keys in a networked computer environment. Users U1, U2, . . . utilize their computer systems110.1,110.2, . . . to generate respective key pairs (PK, SK) where PK is the public key and SK is the secret key.FIG. 1 shows a key pair (PK_U1, SK_U1) for user U1. The users register their public keys PK, over a network, with a certification authority (CA)120. Alternatively, the key pairs can be generated byCA120 and sent to the users. CA120 is a secure, trusted computer system. For each public key PK,CA120 generates adigital certificate104.Certificate104 contains the public key PK and the user's name and/or email address or addresses, may also contain the certificate's serial number SN (generated by the CA to simplify the certificate management), the certificate issue date D1, the expiration date D2, an identification of algorithms to be used with the public and secret keys, an identification of theCA120, and possibly other data. The data mentioned above is shown at104D.Certificate104 also contains CA's signature104-Sig_CAon thedata104D. The signature is generated using CA's secret key SK_CA. CA120 sends thecertificate104 to the user's (key owner's)computer system110. Either the owner or theCA120 can distribute the certificate to other parties to inform them of the user's public key PK. Such parties can verify the CA's signature104-Sig_CAwith the CA's public key PK_CAto ascertain that the certificate's public key PK does indeed belong to the person whose name and email address are provided in the certificate.
• A certificate may have to be revoked prior to its expiration date D2. For example, the certificate owner U may change his affiliation or position, or the owner's private key SK_Umay be compromised. Other parties must be prevented from using the owner's public key if the certificate is revoked.
• One approach to prevent the use of public keys of revoked certificates is through a certificate revocation list (CRL). A CRL is a signed and time-stamped list issued by CA120 and specifying the revoked certificates by their serial numbers SN. These CRLs must be distributed periodically even if there are no new revoked certificates in order to prevent any type of replay attack. The CRL management may be unwieldy with respect to communication, search, and verification costs. The CRL approach can be optimized using so-called delta-CRLs, with the CA transmitting only the list of certificates that have been revoked in the previous time period (rather than for all time periods). The delta-CRL technique still has the disadvantage that the computational complexity of verifying that a certificate is currently valid is basically proportional to the number of time periods, since the verifier must confirm that the certificate is not in any of the delta-CRLs.
• Certificate revocation trees (CRTs) can be used instead of CRLs as described in [15] (the bracketed numbers indicate references listed at the end before the claims).
• Instead of CRLs and CRTs, CA120 could answer queries about specific certificates. InFIGS. 1, user U2 issues aquery150 with the serial number SN ofcertificate104 of user U1. CA120 responds with avalidity status information160 containing the serial number SN, a validity status field160VS (“valid”, “revoked” or “unknown”), and a time stamp “Time”. The response is signed by CA (field160-Sig_CA). This approach is used for Online Certificate Status Protocol (OCSP). See [23]. Disadvantageously, the CA's digital signature160-Sig_CAcan be quite long (over 1024 bits with RSA), especially since the CA must be very secure. In addition, if CA120 is centralized, the CA becomes a validation bottleneck. IfCA120 is decentralized (replicated), the security is weakened as the CA's signing key SK_CAis replicated.
• FIG. 2 illustrates a “NOVOMODO” approach, which allows CA120 to provide an unsigned validity status throughuntrusted directories210 at pre-specified time intervals (e.g. every day, or every hour, etc.).Directories210 are computer systems that do not store secret information. The system works as follows.
• Let f be a predefined public length-preserving function
• 
  f: {0,1}¹¹→{0,1}ⁿ
• where {0,1}ⁿis the set of all binary strings of a length n. Let f′ denote the f-fold composition; that is, fⁱ(x)=x for i=0, and fⁱ(x)—f(fⁱ⁻¹(x)) for i>0. Let f be one-way, i.e. given f(x) where x is randomly chosen, it is hard (infeasible) to find a pre-image z such that f(z)=f(x), except with negligible probability. “Infeasible” means that given a security parameter k (e.g. k=n), the pre-image z cannot be computed in a time equal to a predefined polynomial in k except with negligible probability. Let us assume moreover that f is one-way on its iterates, i.e. for any i, given y=fⁱ(x), it is infeasible to find z such that f(z)=y.
• We can assume, without loss of generality, that CA is required to provide a fresh validity status every day, and the certificates are valid for one year, i.e. 365 days (D2−D1=365 days). To create a certificate104 (FIG. 2),CA120 picks a random “seed” number x and generates a “hash chain” c₀, c₁, c₃₆₅wherein:
• 
  c₃₆₅=f(x), C₃₆₄=f(f(x)), . . . C₁=f³⁶⁵(x), c₀=f³⁶⁶(X).  (1)
• We will sometimes denote x as x(SN) for a certificate with a serial number SN, and similarly c_j=c_j(SN) where i=0, 1, . . . The value c₀is called a “validation target”.CA120 inserts c₀into thecertificate104 together withdata104D (FIG. 1).CA120 also generates a random revocation seed number N₀, computes the “revocation target” N₁=f(N₀), and inserts N₁intocertificate104. CA120 keeps all C_isecret for i>0. The values x and N₀are also secret. Clearly, all C_ican all be computed from x, and the validation target c₀can be computed from any c₁. CA120 stores in its private storage the values x and N₀for eachcertificate104, and possibly (but not necessarily) caches the c_jvalues.
• Every day i (i=1, 2, . . . 365), a certificate re-validation is performed for the valid certificates as follows. For eachcertificate104, CA distributes to directories210 a validation data structure which includes, in addition to a validity status indication (not shown inFIG. 2, can be “valid” or “revoked”):
• 1. the certificate's “i-token” c_jif the certificate is valid on day i;
  2. the revocation seed N₀if the certificate has been revoked.
  (We will call C_ia “validity proof”, and N₀a “revocation proof”.) This information is distributed unsigned. Eachdirectory210 provides this information, unsigned, to arequester system110 in response to a validity status request150 (FIG. 1). To verify, the requester (verifier)110 performs the following operations:
  1. If the validity status is “valid”, theverifier110 checks that fⁱ(c_j)=c₀.
  2. If the validity status is “revoked”, theverifier110 checks that f(N₀)=N₁.
  Despite the validity information being unsigned, the scheme is secure because given c_j, it is infeasible to compute the subsequent tokens C_i+1, C_i+2, . . . .
• To reduce the communication betweenCA120 anddirectories210, a hash chain (1) can be generated for a set ofcertificates104, and a single i-token C_ican be distributed for the set if the set is “unrevoked” (i.e. all the certificates are unrevoked in the set). The certificate140 will contain a separate target c₀for each set containing the certificate and associated with a hash chain (see [1]).
• Certificate revocation can also be performed using accumulators. See [37], An accumulator is a way to combine a set of values (e.g. a set of valid certificates) into a shorter value. A formal definition of a “secure” accumulator is given in Appendix A at the end of this disclosure before the claims. An accumulator example can be constructed as follows. Let us denote all possible values that can be accumulated as p₁, . . . , p_t. (For example, each p_ican be a unique number assigned to a certificate, and we want to accumulate the values corresponding to the valid certificates.) Suppose v₀is the accumulator value for the empty set. Let ƒ be a one-way function. To accumulate p_i, we compute the accumulator as follows:
• 
  v({p₁})=ƒ(v₀, p₁)  (2)
• Now to accumulate p₂, we set the accumulator to be v₂=ƒ(v₁,p₂), and so on, More generally, the accumulator value for some set {P_I1. . . , P_Iml } is
• 
  v({p_i1, . . .p_im})=ƒ(ƒ( . . .ƒ(v₀,p_i1) . . .),p_im)  (3)
• The function ƒ can be chosen such that the accumulation order does not matter, i.e.
• 
  ƒ(ƒ(v,p_i),p_j)=ƒ(ƒ(v,p_j),P_i)  (4)
• (this is the “quasi-commutative” property).
• In each period j,CA120 can send to the directories210 a pair (v_j, t) where v_jis the accumulator value for the set of the valid certificates, and t is a time stamp. The directories can respond toqueries150 with some proof that the accumulator value v_jaccumulates the value p_icorresponding to the certificate of interest.
• A common accumulator is an RSA accumulator defined as follows:
• 
  ƒ(v,p)=v^Pmodn  (5)
• where p is a positive integer, and n=q₁q₂is the product of large prime numbers q₁and q₂. In this case,
• 
  v({p_i1, . . . , p_im})=v₀P_I1. . . , P_Immodn  (6)
• The certificate validation is performed as follows. Without loss of generality, suppose that the values p₁, . . . , p_mcorrespond to the valid certificates in a period j. Then the accumulator value distributed byCA120 todirectories210 is
• 
  v_jv₀^p1pmmod n  (7)
• If averifier110 inquires adirectory210 of the status of a certificate corresponding to the value p_iwhich is one of p₁, . . . , p_m, the directory sends to the verifier the accumulator value v_jand a “witness” value
• 
  s_i,j=v₀^{p1. . . pi−1pi+1pm}modn  (8)
• The verifier checks that
• 
  s_i,j^Pi=v_jmod n  (9)
• If this equality holds, the certificate is assumed to be valid.
• The witness s_i,jis hard to forge provided that it is hard to compute the p_i-th root of v_j. The p_i-th root computation is hard if the adversary does not know the factorization of n and the strong RSA assumption is valid (this assumption is defined in Appendix A). However, it is possible to keep p_iand s_i,jsecret. For example, instead of providing the values s_i,jand p_ito the verifier, the verifier can be provided with a proof that such values exist and are known to the certificate owner.
• Accumulators can be used more generally to prove that an element satisfies some pre-specified property.
SUMMARY
• This section summarizes some features of the invention. Other features are described elsewhere in this disclosure. The invention is defined by the appended claims.
• In some embodiments of the present invention, accumulators are constructed using modular roots with exponents corresponding to the accumulated values. For example, suppose we need an accumulator to accumulate all the elements that possess some property (e.g. all the valid certificates) or all the elements that do not possess that property (e.g. all the revoked certificates). We will associate each element with an integer greater than 1. Let PP_m={p₁, . . . , p_m} be the set of integers associated with the elements to be accumulated (as in (7)), and denote the product of these integers as P_m:
• 
  P_m=II_l=1pl^m  (10)
• (By definition herein, the product of the empty set of numbers is 1; i.e. P_m=1 if PP_mis empty.) In some embodiments, the accumulator value represents the P_m-th root of some value u, e.g.
• 
  v_j=u^1/pmmodn  (11)
• In some embodiments, the following advantages are achieved.
• Suppose the value (11) accumulates valid digital certificates. The value (11) can be modified by exponentiation to de-accumulate all the values p_iexcept for some given value p_i, i.e. to compute
• 
  v_jp₁. . . p_I+1. . . p_m=u^1/pi  (12)
• This exponentiation can be performed by adirectory210 or by the certificate owner'ssystem110, without knowledge of factorization of the modulus n. The value (12) is the accumulator value as if p_iwere the only accumulated value. The witness value needed for verification can also be computed as if p_iwere the only accumulated value. The verification (the authentication) can be performed using the values that do not depend on accumulated values other than p_i. Alternatively, the verification can be performed with the accumulator and witness values that incorporate some other accumulated values but not necessarily all the accumulated values. Therefore, if adirectory210 or auser110 are responsible for providing validity proofs for less than all of the certificates, thedirectory210 or the user110 (the “validity prover”) can use an accumulator that accumulates less than all of the valid certificates. In some embodiments, this feature reduces the number of computations needed to be performed by all of the provers.
• In some embodiments, the value u=u(j) can depend on the time period j for which the authentication is provided (unlike the value v₀in (7)). Therefore, the time stamp t can be omitted.
• In some embodiments, the accumulator accumulates the revoked certificates rather than the valid certificates. In some embodiments, the witness values for an integer p_idepend on other p values.
• In some embodiments, the witness values are used as encryption keys for encrypting validity proofs. The validity proofs can be constructed using anon-accumulator based validation system, e.g. as inFIG. 2. For example, let c_j(i) denote the token c_jfor the period j for a certificate140.i, where c_jis formed as in (1). At the set-up time, when auser system110 joins the certificate validation system, the CA generates the tokens c_j(i) for all the periods j for the user. The CA also generates the witness values for all the periods j. The CA encrypts each token c_j(i) under a key equal to (or obtained from) the witness value for the period j, under a symmetric encryption system.CA120 transmits all the encrypted values to the user.
• In each period j, if the certificate140.iis still valid,CA120 transmits the decryption key (the witness value) for the period j to the user, enabling the user to recover the token c_j(i). The user provides the token to the verifiers to proof the certificate validity as described above in connection withFIG. 2.
• Some embodiments of the invention are suitable for limited bandwidth or low reliability networks, for example in ad hoc networks, where thenodes110 may have low computational and transmission power, and where the nodes may have only incomplete information about the topology of the network. (An ad hoc network is a self-configuring wireless network of mobile routers.)
• Some embodiments are communication and storage efficient.
• The invention is not limited to the features and advantages described above. Other features are described below. The invention is defined by the appended claims.
BRIEF DESCRIPTION OF THE DRAWINGS
• FIGS. 1,2 are block diagrams illustrating prior art certificate revocation schemes.
• FIG. 3 illustrates a digital certificate according to some embodiments of the present invention.
• FIG. 4 is a flowchart of operations performed by a certification authority (CA) in initial certification according to some embodiments of the present invention.
• FIG. 5 is a block diagram illustrating certificate re-validation operations performed by a CA according to some embodiments of the present invention.
• FIG. 6 is a flowchart of witness derivation operations according to some embodiments of the present invention.
• FIG. 7 is a flowchart of authentication according to some embodiments of the present invention.
• FIG. 8 is a block diagram showing data transmitted over networks in de-accumulation operations according to some embodiments of the present invention.
• FIG. 9 is a state diagram illustrating accumulator transmissions according to some embodiments of the present invention.
• FIG. 10 is a block diagram showing data transmissions in determining the network paths according to some embodiments of the present invention.
• FIG. 11 is a flowchart of operations performed by a certification authority (CA) in initial certification according to some embodiments of the present invention.
• FIG. 12 is a block diagram illustrating certificate re-validation operations performed by a CA according to some embodiments of the present invention.
• FIG. 13 is a flowchart of witness derivation operations according to some embodiments of the present invention.
• FIG. 14 is a flowchart of authentication according to some embodiments of the present invention.
DESCRIPTION OF PREFERRED EMBODIMENTS
• The embodiments described in this section illustrate but do not limit the invention. The invention is defined by the appended claims.
• In the following description, numerous details are set forth. However, the present invention may be practiced without these details. Some portions of the detailed descriptions that follow are presented in terms of algorithms and symbolic representations of operations on data bits within a computer memory. These algorithmic descriptions and representations are the means used by those skilled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. An algorithm is here, and generally, conceived to be a self-consistent sequence of steps leading to a desired result. The steps are those requiring physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of electrical or magnetic signals capable of being stored, transferred, combined, compared, and otherwise manipulated. It has proven convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like.
• It should be borne in mind, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantities. Unless specifically stated otherwise as apparent from the following discussion, it is appreciated that throughout the description, discussions utilizing terms such as “processing” or “computing” or “calculating” or “determining” or “displaying” or the like, refer to the action and processes of a computer system, or some computing device, that manipulates and transforms data represented as physical (electronic) quantities within the computer system's registers and other storage into other data similarly represented as physical quantities within the computer system storage, transmission or display devices.
• The present invention also relates to apparatus for performing the operations herein. This apparatus may be specially constructed for the required purposes, or it may comprise a general-purpose computer selectively activated or reconfigured by a computer program stored in the computer. Such a computer program may be stored in a computer readable storage medium, such as, but is not limited to, any type of disk including floppy disks, optical disks, CD-ROMs, and magnetic-optical disks, read-only memories (ROMs), random access memories (RAMs), EPROMs, EEPROMs, magnetic or optical cards, or any type of media suitable for storing electronic instructions, and each coupled to a computer system.
• Some of the algorithms presented herein are not inherently related to any particular computer or other apparatus. Various general-purpose systems may be used with programs in accordance with the teachings herein, or it may prove convenient to construct more specialized apparatus to perform the required operations. The required structure for a variety of these systems will appear from the description below. In addition, the present invention is not described with reference to any particular programming language. A machine-readable medium includes any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computer). For example, a machine-readable medium includes read only memory (“ROM”); random access memory (“RAM”); magnetic disk storage media; optical storage media; flash memory devices; electrical, optical, acoustical or other form of propagated signals (e.g., carrier waves, infrared signals, digital signals, etc.); etc.
Preliminaries
• MODEL AND NOTATION. In our model, we have a certification authority CA, a certificate owner or holder Owner, and a certificate verifier VV. Here Owner has a certificate issued by CA. The concept of a certification authority may apply more generally to any authority responsible for issuing access control privileges or authorizations to a plurality of individuals. The concept of an owner may be tied to a specific human being or organization operating a computer or to the computer itself, e.g. a world wide web server. Similarly, the verifier may be tied to a specific human being or organization operating a computer or to the computer itself, e.g. an access control server determining if it should permit access to certain services.
• In a given transaction, VV wishes to ascertain that the certificate has not been revoked prior to its expiration date. To do so, VV must obtain a proof of validity or a proof of revocation that is or has been issued by the certification authority CA. VV may obtain this proof either from the CA directly, or through some other distribution mechanism.
• We let {0,1}* denote the set of all bit strings. For a bit string s, we denote its length by |s|. We let H denote a cryptographic compression function that takes as input a b-bit payload as well as a v-bit initialization vector or IV, and produces a v-bit output. We assume b≧2v, which holds for all well-known constructions in use. For the constructions we describe here, we typically take b=2v. We assume these cryptographic compression functions are collision resistant; that is, finding two distinct inputs m₁≠m₂such that H(IV, m₁)=H(IV, m₂) is difficult. We assume that the IV is fixed and publicly known. For notational simplicity, we will not always explicitly list IV as an argument in the hash function. A practical example of such a cryptographic compression function is SHA-1[26]. SHA-1′s compression function has an output and IV size of 20-bytes and a 64-byte payload size. In many embodiments, we will not need to operate on data larger than the compression function payload size; however there are numerous standard techniques such as iterated hashing or Merkle-trees [19] for doing so. For simplicity, we will use the term hash function instead of compression function, where it is understood that a hash function can take arbitrary length strings {0,1}* and produce a fixed length output in {0,1}^v.
• In practice, one often constructs a length preserving function that is one way on its iterates by starting with a hash function Hand padding part of the payload to make it length preserving.
• Finally, for a real number r, we set ┌r┐ to be the ceiling of r, that is, the smallest integer value greater than or equal to r. Similarly, └r┘ denotes the floor of r, that is, the largest integer value less than or equal to r.
• Let A be an algorithm. By A(·) we denote that A has one input. By A(·, . . . , ·) we denote that A has several inputs). By A_(·)we will denote that A is an indexed family of algorithms. Y←A(x) denotes that y was obtained by running A on input x. In case A is deterministic, then this y is unique; if A is probabilistic, then y is a random variable. If S is a set, then y←S denotes that y was chosen from S uniformly at random. Let b be a boolean function. The notation (y←A(x):b(y)) denotes the event that b(y) is true after y was generated by running A on input x. Finally, the expression
• 
  Pr[{x_i←A_i(y_i)}_1≦I≦n:b(x_n)]
• denotes the probability that b(x_n) is TRUE after the value x_nwas obtained by running algorithms A₁. . . , A_non inputs y₁, . . . y_n.
Accumulation of Valid Certificates
• Some accumulator schemes will now be described with respect to the digital certificate validation. These schemes can also be used for other authentication operations as described above.
• To simplify the description, we will assume that each certificate owner Ui operates just one system110.iand owns at most one certificate. This is not in fact necessary.
• CA SET-UP. TheCA120 generates a composite modulus n. In some embodiments, n is the product of two primes, but n can also be the product of three or more primes. In some embodiments, n is public, but n is hard to factor, and the factorization is known only to theCA120.
• INITIAL CERTIFICATION (FIG. 4): Suppose the i-th user u_ijoins the system in or after some time period j₀. Atstep410, the CA assigns a positive integer p_ito u_i. The numbers p_iare chosen to be mutually prime with the modulus n. Also, each p₁does not divide the LCM (lowest common multiple) of the remaining numbers p_i, l≠i. For example, the numbers p₁, . . . , p₁can be distinct prime numbers, or pairwise relatively prime.
• The CA computes a unique public value m_iε(Z/nZ)* for the user u_i. (Z/nZ is the ring of all residue classes modulo n, and (Z/nZ)* is the multiplicative group of the invertible elements of Z/nZ, i.e. of the elements defined by integers mutually prime with n.) In some embodiments,
• 
  m_i=H₁(u_i, p_i, j₀, w_i)ε(Z/nZ)*  (13)
• where H₁is a predefined function, e.g. a one-way collision resistant function, u_iis the user identification (e.g. user name), and w₁is an optional string (which may contain u_i's public key and/or some other information). The value m_ithus binds p_ito u_i's identity. If desired, any one or more of the values j₀, p_i, m_iand the identification of the function H₁can be made part of the certificate140. SeeFIG. 3 showing a certificate140 containing m_i, j₀, and specifying the functions H₁and H₂(H₂is described below).
• Atstep420, the CA computes an “initial certificate” value
• 
  s_i,j0=m_i^1/pi(modn)  (14)
• The CA keeps this value secret, and transmits it to the user u_i'ssystem110 in an encrypted form.
• CERTIFICATE RE-VALIDATION BY CA IN TIME PERIOD j (FIG. 5): At the start of, or shortly before, the period j, the CA performs the following operations. Let t denote the total number of users in the system (i.e. the total number of users for whom the initial certification ofFIG. 4 has been performed). Let UU denote the set of these users: UU={u₁, . . . , u_t}. PP denotes the set of the corresponding p numbers: PP={p₁, . . . , p_t}. Let UU_j⊂UU be the subset of users that are to be re-validated in period j. Let PP_j⊂PP be the corresponding set of numbers p, and let
• 
  P_j=Π_pkεPP_j^pk  (15)
• i.e., P_jis the product of the integers in PP_j. (By definition, the product of the empty set of numbers is 1.) The CA computes
• 
  h_j=H₂(j)ε(Z/nZ)*, h_j−1×H₂(j−1)ε(Z/nZ)*  (16)
• where H₂is the same as H₁or some other function. The CA also computes the accumulator
• 
  v_j=v_j(PP_j)=(h_j/h_j−1)^1/Pj(modn)  (17)
• Of note, in some embodiments, the root in (17) exists with a very high probability.
• The CA transmits the following data to thesystem110 of each user in UU_j(this data can also be broadcast to a larger set of users if such broadcast transmission is more efficient in terms of communication):
1. V_j;
• 2. the list of users in UU_j, which may be just the list of numbers p_iin PP_j. (Since there can be overlap among these lists for different periods j, the CA can simply transmit the information that has changed since the previous period; in this case, the CA transmits the list of numbers in PP_jto each new user atstep430 ofFIG. 4.)
• PERSONAL ACCUMULATOR AND WITNESS DERIVATION BY USERS (FIG. 6): Atstep610, each user u_iin UU computes its “personal accumulator” v_j(p_i), i.e. the accumulator which accumulates only the user's p_ivalue:
• 
  v_j(p_i)=(h_j/h_j−1)^1/pt(modn)  (18)
• This value is computed from the v_j; value (17) by exponentiation:
• $\begin{array}{cc}{v}_j\left(p_i\right)=v_j^{p_k\in P{\stackrel{\Pi }{P}}_j-{\left\{p_i\right\}}^{p_k}}\left(\mathrm{mod}n\right)& \left(19\right)\end{array}$
• Atstep620, the user aggregates this personal accumulator with previous personal accumulators to obtain a witness for the period j:
• 
  s_i,j=s_i,j−1v_j(p_i)=s_i,j−1(h_j/h_j−1)^1/pt(modn)  (20)
• The equations (20) and (14) imply that
• 
  s_i,j=s_i,j0(h_jh_j0)^1/pt(modn)  (21)
• USER AUTHENTICATION (FIG. 7): A user u_ican provide s_i,jto the verifier, along with (if necessary) the values for (u_i, p_i, j₀, w_i). Atstep710, the verifier:
• 1. computes m_ifrom (13) or obtains it from the certificate140 (FIG. 3);
  2. computes h_j=H₂(j) and h_j0=H₂(j₀) (see (16)); and
  3. confirms that
• 
  s_i,j^pi=m_ih_j/h_j0(modn)  (22)
• ALTERNATIVE USER AUTHENTICATION: Instead of giving the witness s_i,jto the verifier, the user can use s_i,jas a private key in an identity-based GQ (Guillou-Quisquater) signature scheme. This scheme is described in Appendix B at the end of this disclosure before the claims. The CA performs the functions of GQ's PKG (private key generator); GQ's parameters are set as follows: B=s_i,j, v=p_i, and J=m₁⁻¹mod n. The authentication proceeds as follows:
• The verifier sends to the user a random message m.
• The user generates a random number r and computes:
• 
  d=H(m∥r^pi), D=rs_i,j^d(modn)  (23)
• where His some predefined public function (e.g. H₁or H₂). The user sends the values m, r^pi, and D to the verifier.
• The verifier computes J=m_i⁻¹mod n and checks that the following equations hold:
• 
  d=H(m∥J^dD^pi)
• 
  J^dD^pi=r^pi(h_j/h_j0)^d(modn)  (24)
• This scheme may reduce total bandwidth because the verifier does not need the certificate140. Note: for the GQ signature scheme to be secure, it is desirable that each p_i>2¹⁶⁰.
• REMARKS. 1. First, s_i,j0can be computed in some other way, not necessarily as in (14).
• 2. ANONYMOUS AUTHENTICATION. If we want to allow theuser nodes110 to sign messages anonymously—i.e., to be able to authenticate themselves as users that have been certified by the CA, but without revealing their actual identities—we can handle initial certification differently. For example, instead of (13), the CA can set m_ito be independent of the user's identity, e.g.:
• 
  m_i=h_j0=H(j₀)ε(Z/nZ)*  (25)
• The m_ivalue is then provided to the user. To hide the fact that the m_ivalue is provided to the user, the m_ivalue may be transmitted to the user via a secure channel (e.g., by encrypting this value such that only u_ican decrypt it). Then, it follows from (14) and (21) that:
• 
  s_i,j=h_j^1/pi(modn)  (26)
• As mentioned below in Appendix A, there are efficient zero-knowledge (ZK) proofs through which user u_ican prove, in the j^thtime period, that it knows a p^throot of h_jmodulo n for some (unrevealed) number p that is contained in a specified interval I of integers, i.e.
• 
  PK{(α, β): α^β=h_jmodn

  

  βεI}
• See [41], [42]. The interval I can be an interval containing all the numbers p_i. Using this scheme, the user can authenticate itself anonymously if and only if it has obtained an initial certificate s_i,j0and the subsequent personal accumulator values v_j(p_i) (see (18), (19)). The CA can revoke the user as before—i.e., by not re-validating it (not including p_iin the product P_jinFIG. 5).
• As mentioned above, the accumulator techniques are not limited to a user possessing a single certificate or to a digital certificate revocation. For example, a user u_imay or may not possess one or more of entitlements e₁, e₂, . . . , e_r. During initial certification, the user u_iis assigned a unique integer p_i,kif the user is to be certified for an entitlement e_k. The entitlement proof proceeds as described above with respect to equations (13)-(26), with numbers p_ireplaced by p_i,k, with s_i,jreplaced by S_i,k,jetc.
• In some of these embodiments, however, a value p_iis computed as the product of p_i,k. For example, if the user is initially certified for all the entitlements, then:
• 
  p_i=p_i,1. . . P_i,r  (27)
• In some embodiments, the integers p_i,kare mutually prime relative to each other and to the CA's modulus n. The user u_i's initial certificate is s_i,j0=m_i^1/Pi(mod n) as in (14), and u_ican use exponentiation to de-accumulate all the values p_i,lexcept a selected value p_i,kfor some k:
• $\begin{array}{cc}{s}_{i,k,j_o}=m_i^{1/p_{i,k}}\left(\mathrm{mod}n\right)=s_{i,j_o}^{\prod _{l\ne k}p_{i,l}}\left(\mathrm{mod}n\right)& \left(28\right)\end{array}$
• The user can use this value to demonstrate that it possesses the entitlement that corresponds to prime p_i,kwithout revealing the user's other entitlements. The CA can revoke a specific entitlement e_kfor the user without revoking the user's other entitlements, simply by issuing a validation accumulator in (17) that does not accumulate p_i,k(i.e. the product P_iis computed as in (15) except that p_iis replaced with p_i/p_i,k(the number p_i,kis not included in the product)). If the user wants to sign anonymously and unlinkably, then the user cannot reveal its specific value of p_i,k, but it would seem to make it difficult for a verifier to determine what entitlement the user is claiming (if the verifier cannot see the value of p_i,k). We can get around this problem by, for example, associating each entitlement e_kto a range of integers I_i. This range will contain all the p_i,kvalues for all the users u_iinitially certified to have the entitlement e_k. The ranges I_kdo not overlap in some embodiments. Then the user can prove in ZK that it possesses a modular p-th root (for some unrevealed number p in the range I_k) of the appropriate value, i.e. the user can provide the following proof of knowledge:
• 
  PK {(α,β): α^β=Cmodn

  

  βεI_k}  (29)
• where C=m_ih_i/h_j−1or C=h_j; (see (21), (26)). See [41], [42]. The user can also prove that it possesses multiple entitlements simultaneously, e.g., by proving (in ZK if desired) its possession of a p^throot for some p that is the product of (for example) two integers p₁and p₂in ranges corresponding to the claimed entitlements:
• 
  PK{(α,β): α^β=Cmodb

  

  β=β₁β₂

  

  β₁εI_k1

  

  β₂εI_k2}  (30)
See [41], [42].
• USERS MOVING FROM NETWORK TO NETWORK: Suppose the CA has users (certificate owner systems)110 in different networks, and the users move from a network to a network (e.g. some of the networks may be ad hoc networks). The CA can calculate a separate accumulator (say, (17)) for each network. Each accumulator will accumulate only the valid users in the corresponding network based on the CA's knowledge of the current composition of users in each network. Each user will compute its personal accumulator value and/or witness value (e.g., as in (18), (20), (26), and/or (28)). The user can move to another network and use the same personal accumulator value and/or witness value in the other network.
• AGGREGATION OF PERSONAL ACCUMULATORS: Multiple personal accumulators can be aggregated into a single value, in order to save bandwidth; from this aggregated value, a verifier can batch-verify that multiple users are indeed certified. For example, in the scheme of equations (17), (22), user u_i(if it is still valid) possesses a personal value s_i,jthat satisfies
• 
  s_i,j^pi=m_i(h_j/h_j0)(modn)  (31)
• where m_iis as in (13), and j_o,iis the period j_ofor the user u₁. Denote
• 
  z_i=m_i)h_j/h_j0,i)(modn).  (32)
• Then, for multiple users in period j, their personal values can simply be multiplied together
• 
  S=Π_i=1^t′S^i,j(modn)  (33)
• where t′ is the number of users which take part in the aggregation (or the number of certificates or entitlements belonging to a user if each u₁is a certificate or an entitlement). A verifier can use the value (33) to confirm that users (or certificates or entitlements) (u₁, . . . , u_t′) are valid by confirming that:
• $\begin{array}{cc}{S}^{\prod _{l=1}^{t^{\prime }}p_i}=\prod _{i=1}^{t^{\prime }}z_i^{\frac{1}{p_i}\prod _{l=1}^{t^{\prime }}p_i}\left(\mathrm{mod}n\right).& \left(34\right)\end{array}$
• DE-ACCUMULATION. We will now discuss concrete approaches to the de-accumulation of values from the accumulator computed by the CA; however, we note that the CA does not need to compute a single accumulator value that accumulates all of the values associated to all of thevalid nodes110 in the network. Instead, the CA can use a tradeoff, which we now describe. De-accumulation is somewhat computationally expensive. Procedure Split(v,P) below performs de-accumulation on an accumulator v that accumulates t k-bit numbers P={p_i}; the procedure's output is t personal accumulators; the execution time is ◯(t log²t) (where we consider k to be a constant). Let us denote the accumulator accumulating an empty set of values as u (in equation (17), u=h(j)/h(j−1)). Then
• $\begin{array}{cc}u=v^{\prod _{p_i\in P}p_i}\left(\mathrm{mod}n\right).& \left(35\right)\end{array}$
Procedure Split(v,P):
• 1. Split the set P into two disjoint halves P₁and P₂.
• 2. Compute
• ${\mathrm{<figure-callout\; label="v"\; filenames="US20100174904A1-20100708-D00000.png,US20100174904A1-20100708-D00002.png"\; state="\{\{state\}\}">v</figure-callout>}}_1=v^{\prod _{\mathrm{pi}\in P_2}\mathrm{pi}}\left(\mathrm{mod}n\right);$
• this is the accumulator for the set P_i.
• 3. Compute
• $v_2=v^{\prod _{\mathrm{pi}\in {\mathrm{<figure-callout\; label="P"\; filenames="US20100174904A1-20100708-D00000.png,US20100174904A1-20100708-D00002.png"\; state="\{\{state\}\}">P</figure-callout>}}_1}\mathrm{pi}}\left(\mathrm{mod}n\right);$
• this is the accumulator for the set P₂.
• 4. If P₁has only one member, output (v₁,P₁), otherwise call Split (v₁,P₁).
• 5. If P₂has only one member, output (v₂,P₂), otherwise call Split (v₂,P₂).
• End of Split (v,P)
• As a rule of thumb, exponentiating a number modulo n by a product of t′ numbers takes time proportional to t′; thus, the first split operation (steps2 and3 in procedure Split(v,P)) is the most expensive one in the recursion above. (Actually, since there are t distinct numbers, the (p₁, . . . , p_t), the p_iare ◯(log t) bits apiece on average, so that the exponentiation is proportional to t′ log t′; since the Split algorithm can recurse to a depth of log t′, the overall computation complexity is t log²t.) To reduce the amount of de-accumulation that needs to be performed, the CA can (for example) compute two accumulators—one for each of two halves P₁and P₂of P—and transmit these two accumulators to the users, thereby allowing the users to skip the first split operation and thereby reducing their computation. The CA can reduce their computation further by transmitting even more accumulators, each for an even smaller subset of the users. In computing these multiple accumulators, it is advantageous for the CA to use its knowledge of the current network topology to enhance the performance of the scheme. For example, suppose that the users are divided into several (say, 10) topological areas. Then, instead of using one product of integers P_jas in (15), the CA can compute10 such products P_j,1, . . . , P_j,10, and 10 area-specific accumulators (h_j/k_j−1)^1/Pj,k. The CA then transmits each one of these 10 area-specific accumulators to the respective one of the 10 topological areas (along with a list of the re-validated users in that area). Users in a given area can compute their personal accumulators and witnesses from the area-specific accumulators.
• De-accumulation is also relevant to how the personal accumulators (18)-(19) are computed. Clearly, in terms of computation, it is non-optimal for each of the t users to perform de-accumulation (18), (19) independently; doing it this way would entail ◯(t²) computation (actually, worse, since the size of the p_imust grow at least as fast as log t). In some embodiments, the users compute their personal accumulators cooperatively, as illustrated, for example, inFIG. 8 and described below.
• If the performance metric is minimum communication, a simple broadcast by the CA of the accumulator (17) to t users (seeFIG. 5) is a good way of accumulator distribution. This results in overall communication of ◯(t). However, overall computation is at least ◯(t²) because each of t users has to perform t−1 exponentiations on the same accumulator.
• Referring toFIG. 8, let PP_jbe the set of valid users (i.e. the set of the p numbers corresponding to the valid users). At the stage ofFIG. 5 (re-validation), the CA sends the accumulator v_j(PP_j) (see (17) for example) and the numbers p in the set PP_jto a designated node110 (also marked D inFIG. 8). Node D divides the set PP_j; into sub-groups (sub-sets). In one example, the sub-group Pp_j-0 includes one or more of nodes110.0,110.00,110.01; the sub-group PP_j-1 includes one or more of nodes110.1,110.10,110.11; the third sub-group consists ofnode110 itself if this is a valid node. Node D computes the accumulator v_j(PP_j-0) for the sub-group PP_j-0, and the accumulator v_j(PP_j-1) for the sub-group PP_j-1. This computation can be performed by de-accumulating the nodes which are not sub-group members. See e.g. equation (19) and Procedure Split(v,P), steps1 and2. If node D is valid (i.e. its number p is in PP_j), node D also computes its personal accumulator as inFIG. 6. Node D transmits the accumulator v_j(PP_j-0), and the p numbers in the set PP_j-0, to node110.0. Node D transmits the accumulator v_j(PP_j-1), and the p numbers in the set PP_j-1, to node110.1. The nodes110.0,110.1 perform the same operations as, node D. In particular, node110.0 divides the valid users PP_j-0 into sub-groups. In one example, the nodes110.0,110.00,110.01 are valid; the sub-group PP_j-0-0 consists of node110.00; the sub-group PP_j-0-1 consists of node110.01; the third sub-group consists of node110.0. The sub-groups are determined by node110.0 from the p numbers in PP_j-0. Node110.0 computes accumulators v_j(PP_j-0-0), v_j(PP_j-0-1) for the respective sub-groups PP_j-0-0, PP₃-0-1, and also computes its personal accumulator as inFIG. 6. Node110.0 transmits the accumulator v_j(PP_j-0-0), and the corresponding p number, to node110.00. Node110.0 transmits the accumulator v_f(PP_j-0-1), and the corresponding p number, to node110.01. The accumulators v_j(PP_j-0-0), v_j(PP_j-0-1) are personal accumulators, so the nodes110.00,110.01 do not need to derive their personal accumulators.
• Node110.1 performs operations similar to those for node110.0.
• This procedure can be extended recursively to any number of nodes. The procedure is not limited to the “binary-tree” type ofFIG. 8. A node can split a set of users into any number of sub-groups in addition to itself, possibly resulting in a non-binary-tree type operation. Also, in some embodiments, the accumulators are transmitted only to valid nodes. For example, ifnode110 is invalid, thenCA120 transmits the accumulator v_j(PP_j) to a valid node instead. For communication efficiency, the sub-groups are selected so that the communications are efficient within each sub-group. For example, in case of ad hoc networks, the sub-groups can be selected based on geographic proximity, with the proximate nodes being grouped together.
• Now, we describe one approach for selecting the sub-groups, where nodes only have local topology information. Roughly speaking, the idea is as follows. To partition (a portion of) the network, the designated node D chooses two “group heads” in the network, such as nodes110.0,110.1 inFIG. 8. (More than two group heads can also be selected.) These group heads can be chosen randomly. Alternatively, if the designated node has some topological information, it can use that information to attempt to choose the two group heads in a way that they are topologically distant—preferably on opposite sides of the network from each other. After D picks the two group heads, the designated node tells the two group heads to transmit different messages; for example, the designated node may tell the first group head to transmit ‘0’ and the second to transmit ‘1’. Then, each group head (e.g.110.0,110.1) “floods” the network with its particular message. When this occurs, the other nodes in the network will either receive a ‘0’ first or the ‘1’ first. Nodes that receive a ‘0’ first (e.g. nodes110.00,110.01) report back to the designated node D (or to the group head110.0) that they are a member of the 0-group; the members of the 1-group report similarly. The designated node D (or the respective group heads) receives this information, and it de-accumulates the accumulator into two halves—one accumulator for the 0-group and one for the 1-group. If the designated node performed this de-accumulation (as in the procedure illustrated inFIG. 8), it passes along the appropriate accumulators to their corresponding group heads (110.0,110.1). Then, each group head becomes the designated node for its corresponding sub-network, and the entire process described above is repeated for the sub-networks. Now, we will provide more detail for one embodiment.
• At the beginning, there is only a single group PP of users which covers the whole network. Each recursion step is carried out in four phases as depicted inFIG. 9. The first recursion step is started by a node which is preferably located in the center of the network. We call this node the current designated node D. Node D executes the following four phases of recursion:
• SEARCH GROUP HEADS (STATE910): Node D selects randomly two nodes out of its group as group heads GH0 and GH1. (For the sake of illustration, assume that node GH0 is node110.0 ofFIG. 8 or10, and node GH1 is node110.1). Node D sends out a broadcast message M_searchGHwith its group ID, the addresses of GH0 and GH1, and a path variable containing D's address. All receiving nodes within the same group obtain a copy of the path and check if they are GH0 or GH1. If not, they add their address to the path variable in the message and forward it (broadcast it) once. The path variable provides each node with the path to D.FIG. 10 shows the messages m_searchGHtransmitted by nodes D,110.01,110.10,110.11.
• BUILD GROUP (STATE920): If the receiver of m_{searchGH is}either GH0 or GH1, it starts building a sub-group by broadcasting a message m_buildGcontaining its address. If a receiver of m_buildGwhich belongs to the same group hasn't already received a message from a group head and therefore, joined its sub-group, he joins the sub-group announced in m_buildGand forwards the message once.FIG. 10 illustrates the m_buildGmessages broadcast by nodes GH0, GH1,110.01,110.10,110.11, assuming that the node110.01 received the m_buildGmessage from GH0 first, and nodes110.10,110.11 received the m_buildGmessage from GH1 first.
• REPORT GROUP (STATE930): After some prescribed amount of time, all members of both sub-groups start reporting their membership status to D. Therefore, they include their addresses together with theirsub-group IDs 0 or 1 in the message m_reportGand send it over the path obtained inphase1 to D. To make the reporting more efficient, nodes which are more distant from D start the reporting first. Nodes on the path of m_reportGadd their sub-group membership information to the message and forward it according to the path to D. Nodes having forwarded a m_reportGdon't initiate a report themselves.
• TRANSMIT ACCUMULATOR (STATE940): After D has received all reports, it looks up the paths p0 to GH0 and p1 to GH1 included in the received reports, exponentiates the accumulator v_jfor its group to obtain the accumulators v_j(0) and v_j(1) for the respective sub-groups, and sends the accumulators in respective messages m_transinitu0over p0 to GH0 and m_transmitu1over p1 to GH1, respectively. Additionally to the accumulators v_j(0) and v₁(1), these messages contain a list of the corresponding sub-group members. After reception of m_transmitu0or M_transmitu1respectively, GH0 and GH1 become designated nodes D of their sub-groups and enterphase910 of the next recursion step. The recursion ends if the sub-group accumulator is a personal accumulator.
• Considering a set of all users UU with cardinality |UU|=2^R=t, and equal distribution of each group PP into two groups of identical size |PP|/2, the recursion will end after R steps with a total number of t sub-groups ofsize 1. Here, we mention a few aspects of the scheme's computational and communication complexity, assuming that, at each step, the split is into approximately equal halves.
• SEARCH GROUP HEADS: Within a subnetwork of size t′, this step requires ◯(t′) messages, each of size log t (since it requires log t bits on average to specify a single member of a group of t elements). Each message contains a path with average length C_I√{square root over (t′)}. Therefore, communication for this subnetwork is of ◯(t′√{square root over (t′)} log t). If we assume that the partitions are always equally-sized, the total communication of the entire recursive procedure is also ◯(t′√{square root over (t′)} log t).
• BUILD GROUP: This step involves a total of t messages, each of constant size, for communication complexity ◯(t).
• REPORT GROUP: Group membership information of the nodes is transmitted over C₃*√{square root over (t)} hops, where C₃is a constant depending on the network density and reflecting the number of reports initiated: this results in communication of ◯(t′√{square root over (t′)} log t).
• TRANSMIT ACCUMULATOR: Accumulators together with group membership information of t nodes is transmitted over C₄*√{square root over (t)} hops, where C₄is a constant depending on the network density: this results in communication of ◯(t′√{square root over (t′)} log t) and computation of ◯(tR log t)=◯(tlog²t).
• Overall, the communication is ◯(t√{square root over (t)} log t) and the computation is ◯(tlog²t).
• If we assume that there is an entity E of the network with less stringent limitations in terms of computational power and energy such as the certification authority itself, another interesting opportunity arises. By acquiring some topology knowledge this entity may pre-de-accumulate the accumulator into accumulators for constant size groups. In detail, such a scheme works as follows: Some node C which holds the current accumulator for all users and has connectivity to E initiates a distributed clustering algorithm ofFIG. 9. In a second step, C provides the clustering information to E and lets E perform the pre-de-accumulation for all clusters. Then C distributes the accumulators to the clusters. Suitable clustering algorithms have communication complexity of ◯(t√{square root over (t)} log t), but perhaps with a smaller constant factor than the approach described above.
• Briefly, we mention a distributed approach that splits a connected network into two connected subnetworks, where neither subnetwork is more than twice the size of the other. As before, we begin with a sub-network that has a designated node. We assume that the sub-network is connected, and that its topology is basically constant during the accumulator distribution (though it may change dramatically from period to period). The first step in the preferred algorithm for achieving ◯(tlogt) communication and ◯(tlogt) computation is that, in a distributed fashion, the nodes of the sub-network establish an enumeration of themselves. This can be accomplished as follows. The designated node initiates the procedure by broadcasting a message. The nodes that receive the message transmit an acknowledgement message back to the designated node, and they log the designated node as the node from which they received the message. The designated node logs the acknowledging nodes as the nodes that received its message. This process recurses throughout the network. Specifically, a node that received the message from the designated node broadcasts the message, and nodes that have not sent or received the message before log the identity of the node that they received the message from and they send back an acknowledgment, after which the sending node logs the acknowledgers. If a node receives the message more than once, it only logs and responds back to the first node that sent it. In this fashion, since the subnetwork is connected, every node in the subnetwork (except the designated node) has a unique node from which it received the message, as well as a list of the nodes that received the message directly from it. Each node that has a nonempty list of acknowledgers chooses an arbitrary enumeration of those acknowledgers. In effect, from this procedure, we have constructed a tree (a graph with no loops) from the network, as well as an enumeration of all of the tree nodes given by the depth-first pre-order traversal of the tree. If the subnetwork has t′ nodes, this procedure can be accomplished with ◯(t) communication and computation.
• The next stage of the algorithm is to use the enumeration to approximately bisect the subnetwork. There are a variety of different ways of doing this. One method is that each node, beginning with the nodes with no acknowledgers, could back-transmit (backwards according to the enumeration) the number of nodes in its subtree including itself; in this fashion, each node computes the number of nodes in its subtree. There must be exactly one node that has at least half of the nodes in its subtree, but such that none of its children have at least half. This node is designated to be the midpoint of the subnetwork. Now, viewing the midpoint point as the root of the tree, it divides its children into two groups, such that the number of nodes that are in a subtree emanating from one of the children in the first group is approximately equal to the number of nodes that are in a subtree emanating from one of the children in the second group. (This can always be accomplished such that ratio between the two numbers is at most two.) Thus, all of the network nodes except the midpoint become members of one of the two groups; the midpoint is considered to be a member of both groups. The step of computing how many nodes are in each subtree requires ◯(t′logt′) communication, since there are t transmissions, where the size of each transmission (which is a number between 1 and t′ representing how many nodes are in the given subtree) is log t′ bits. Viewing the midpoint node as the root node, a new enumeration of the nodes is established with the midpoint node as the initiator of the message, beginning with the nodes in the first group. (This new enumeration could be computed as before, or it could actually be derived indirectly from the previous enumeration. Either way, it does not add to the asymptotic communication complexity of the protocol, which is t log²t overall.)
Accumulator-Based Encryption of Validity Proofs
• The personal accumulators v_j(p_i) and/or the witness values s_i,j(see e.g. equations (17), (20), (21), (26)) can be used to obtain symmetric keys that users can use to decrypt validation proofs under other validation systems. An example will now be given for the validation system obtained by combining the systems ofFIGS. 2,4-7, but this is not limiting,
• INITIAL CERTIFICATION: When a user u_ijoins the system, the CA:
• 1. Generates the validation and revocation seeds x, N₀as inFIG. 2.
  2. Generates all the tokens c_jfor all the periods j (seeFIG. 2 and equation (1)).
  3. Generates the certificate140 as inFIG. 2 and transmits the certificate to the user u_i.
  4. Performs the steps ofFIG. 4, i.e. generates the values p_i, m_is_i,joand transmits s_i,joto the validity prover (which can be the user u_iand/or the directories210). The values p_i, m_i, can also be transmitted to the prover.
  5. Generates all the witness values s_i,j(for all the periods j) for the user u_i. Also generates the encryption keys K_i,jeach of which is s_i,jor some function of s_i,j.
  6. Encrypts each token c_j(generated at step2 in this procedure) with the encryption key K_i,junder some encryption scheme (possibly a symmetric encryption scheme). Let us denote the encrypted c_jvalue as E_i,j.
  7. Transmits all the values E_i,j(for all j) to the prover (i.e. thedirectories210 and/or the certificate owner u_i).
• RE-VALIDATION BY CA: At the start of, or shortly before, each period j, the CA:
• 1. Performs the procedure ofFIG. 5, i.e. computes and broadcasts the accumulator value v_jand the list of valid p numbers to the provers.
  2. If the user u_i's certificate is invalid, the CA transmits the revocation seed N₀to the prover corresponding to the certificate (e.g. to the user u_iand/or the directories210).
• PROOF DERIVATION BY THE PROVER: If the user u_i's certificate is valid (as indicated by the transmission of number p_iin the RE-VALIDATION procedure at step1), the prover:
• 1. Performs the procedure ofFIG. 6 to derive the witness value S_i,j.
  2. Obtains the decryption key K_i,jfrom s_i,j.
  3. Decrypts E_i,jto recover c_j.
• AUTHENTICATION (VALIDITY PROOF): As inFIG. 2.
• Many variations are possible. For example, the decryption keys can be some different function of s_i,jthan the encryption keys.
Accumulation of Revoked Certificates
• Above, we have described an approach in which an accumulator accumulates the valid certificates; an alternative approach is to accumulate revoked certificates. The valid certificates' owners (or validity provers) then use the “dynamic” feature of dynamic accumulators to compute a new accumulator for the valid nodes, and to compute their personal accumulators with respect to this new accumulator.
• As before, we will assume for the sake of illustration that each user operates acorresponding computer system110 and owns at most one certificate140. This is not limiting, as a user may own multiple certificates and/or operatemultiple systems110. The scheme is also applicable to controlling resource access and other kinds of authentication. We will use the word “user” to denote both thesystem110 and the system's operator where no confusion arises. As described below, each user u_iwill be assigned a positive integer p_iwith the same properties as in the scheme ofFIGS. 4-7 (e.g. p_iare mutually prime relative to the CA's public composite modulus n and to each other). The symbols n, PP, p_iwill be as in the scheme ofFIGS. 4-7. In particular, PP is the set of the p_inumbers.
• For each period j, the symbol QQ_jwill denote the set of the revoked p_inumbers (i.e. the set of the p_inumbers corresponding to the certificates to be declared as revoked in period j). Q_jdenotes the product of the numbers in QQ_j:
• 
  Q_j=Π_pkεQQj^pk  (36)
• The symbol a will denote an accumulator seed, which is an integer mutually prime with the modulus n. The accumulator of the values in Q_jis:
• 
  v_j=a^1/Qjmodn  (37)
• Let RR_j=QQ_j−QQ_j−1, i.e. RR_jis the set of the p numbers corresponding to the certificates declared as revoked in period j but not in period j−1; in period j−1 these certificates were either declared as valid or were not yet part of the validation system. Let R_jdenote the product of the numbers in RR_j:
• 
  R_j=Π_pkRRj^pk  (38)
• It is easy to see from (37) that if each certificate cannot be “unrevoked” (i.e. cannot be made valid once revoked), then:
• $\begin{array}{cc}{v}_j=v_{j-1}^{1/R_j}\mathrm{mod}n& \left(39\right)\end{array}$
• CA SET UP: The CA generates its modulus n and the accumulator seed a. The CA sets the initial accumulator value v_j=a for the initial period j.
• INITIAL CERTIFICATION (FIG. 11): Suppose a user u_iwants to join the system in a period j_o, to be validated starting the next period j_o+1. Atstep1110,CA120 generates p_i, m_ias atstep410 inFIG. 4. These numbers can be made part of the user's certificate140 (FIG. 3). Atstep1120, the CA computes the following secret numbers:
• 
  t_i=m_i^1/pi(modn)  (40)
• 
  s_i,j0=v_j0^1/pi(modn)  (41)
• Here, v is the accumulator value for the period j_o(v_jo=a if QQ_jois empty). Atstep1130, the CA transmits the values t_i, s_i,jto the user u_iin an encrypted form. As will be explained, the user will be able to derive the value
• 
  s_i,j=v_j^1/pi(modn)  (42)
• for each period j>j_ofor which the certificate is valid.
• CERTIFICATE RE-VALIDATION (FIG. 12): In period j, suppose that v_j−1εZ/nZ is the accumulator value for period j−1. For validation in period j, the CA:
• 1. Computes v_j(using the equation (37) or (39) for example);
  2. Transmits the value v_jtogether with a list of the newly-revoked certificates (perhaps represented by the set RR_j) to the users. If desired, the CA may also sign the pair (v_j, j) and transmit the signature to the users.
• WITNESS DERIVATION (FIG. 13): If a user u_ivalid for the period j, then the user has the value s_i,j−1=v_j−1^1/pi(mod n). See (41), (42). Atstep1310, the user computes s_i,j=v_j^1/pias follows:
• 1. The user applies the Extended Euclidian Algorithm to compute integers a and b such that
• 
  ap_i+bR_j=1  (43)
• (this is possible because p_i∉RR_j, so p_iand R_jare mutually prime
  2. The user sets s_i,jto the value:
• 
  v_j^as_i,j−1^bmod n  (44)
• This value is indeed a p_i-th root of v_./modulo n, because (note equation (39)):
• ${\left(v_j^as_{i,j-1}^b\right)}^{p_i}=v_j^{{\mathrm{ap}}_i}v_{j-1}^{\frac{1}{p_i}{\mathrm{bp}}_i}=v_j^{{\mathrm{ap}}_i}v_j^{R_jb}=v_j\mathrm{mod}n$
• Atstep1320, the user computes the witness
• 
  w_i,j=t_is_i,j  (45)
• USER AUTHENTICATION (FIG. 14): The user u_iprovides t_is_i,j(mod n) to the verifier, along with (if necessary) the values for (u_i, p_i, j₀, w_i). In addition, the user provides to the verifier the accumulator value v_jand the CA's signature on (v_j,j). Atstep1410, the verifier checks the CA's signature and confirms that:
• 
  w_i,j^pi=m_iv_j(modn)  (46)
• ALTERNATIVE USER AUTHENTICATION: The authentication can be performed using the identity-based GQ signature scheme as described above in connection with equations (23), (24), using w_i,jas a private key. The CA performs the functions of GQ's PKG (private key generator); GQ's parameters are set as follows: B=w_i,j, v=p_i, and J=m_i⁻¹mod n. The authentication proceeds as follows:
• The verifier sends to the user a random message m.
• The user generates a random number r and computes:
• 
  d=H(m∥r^pi), D=rw_i,j^d(modn)  (47)
• where H is some predefined public function. The user sends the values m, m₁, r^pi, and D to the verifier.
• The verifier computes J=m_i⁻¹mod n and checks that the following equations hold:
• 
  d=H(m∥J^dD^pi)
• 
  J^dD^Pi=r^Piv_j(modn)  (48)
• This scheme may reduce total bandwidth because the verifier does not need the certificate140. Note: for the GQ signature scheme to be secure, it is desirable that p_i>2¹⁶⁰.
• The use of accumulator v_j/allows a user to revoke itself, without the CA's help. To revoke itself in a period j, user u_isimply broadcasts v_j^1/pI(mod n), after which every other user u_kcan update the accumulator to be v_j←v_j^1/pi(mod n) and can recompute its personal s_k,jand w_k,jvalues as in (43)-(45). Of note, R_jcan be re-computed by multiplying the previous R_jvalue by p_i.
• In some embodiments, this scheme allows efficient distribution (e.g. broadcast) for the CA at the stage ofFIG. 12, since the CA transmits the same information to all the users. This information requires only ◯(t) transmissions. On the other hand, each transmission is proportional in size to the size of RR_j. Clients of the CA may include many users that are not in the particular t-member network that we are concerned about. If the “super-network” of clients of the CA is very large in proportion to t, this may not be a very good solution. Even if the CA only manages our t-member network, the CA's size of each transmission is still, strictly speaking, proportional to t, since the number of revocations within the network in a given period will tend to be a constant fraction of the total number of users. In this sense, the communication complexity of certificate distribution in this scheme is analogous to the communication complexity of a delta-CRL scheme. Recall that in a delta-CRL scheme, the CA transmits a list of users revoked in the given period, together with a signature on that list, to all users. On the other hand, since a personal accumulator's size is independent of the number of time periods, the scheme has better communication complexity for authentication than delta-CRLs, because in the delta-CRL scheme the verifier must separately check the delta-CRLs for all time periods to confirm that a given user has not been revoked during any of these periods.
• The invention is not limited to the embodiments described above. The invention is not limited to secure or dynamic accumulators. An accumulator can be any data that accumulate some elements. Further, the invention is not limited to the accumulators described above. For example, the accumulator seed h(j)/h(j−1) in equation (17) can be replaced with a value independent of the period j, and the accumulator seed a in the accumulator (37) can be replaced with a function of j. The accumulator methods can be used to prove (authenticate) membership in a set or possession of some property. Examples include authentication of valid entitlements, or authentication of people as being members of some organization.
• In some embodiments, theCA120, thedirectories210, and thesystems110 are computer systems communicating with each other over a network or networks. Each of these systems may itself be a computer system having components communicating over networks. Each computer system includes one or more computer processors executing computer instructions and manipulating computer data as described above. The term “data” includes “computer data” and covers both computer instructions and computer data manipulated by the instructions. The instructions and data can be stored on a data carrier such as a computer storage, i.e. a computer readable medium (e.g. a magnetic or optical disk, a semiconductor memory, and other types of media, known or to be invented). The data carrier may include an electromagnetic carrier wave transmitted over a network, e.g. through space, via a cable, or by some other means. The instructions and data are operable to cause the computer to execute appropriate algorithms as described above.
• The invention is not limited to any particular hash functions, or to cryptographic functions (which are easy to compute but are one-way or collision resistant). In some embodiments, it is desirable that a function for H be collision resistant not in the sense that it is difficult to find different x and y with the same image but in the sense that if x and y are uniformly drawn from the function's domain, the probability is small that they both will have the same image:
• 
  P{H(x)=H(y)}≦α
• where α is a small constant (e.g. 1/10, or 1/100, or 2⁻²⁵, or 2⁻⁵⁰, or 2⁻⁸⁰, or 2⁻¹⁶⁰, or some other value). Some or all of the techniques used for validity proofs can also be used for invalidity proofs and vice versa. The CA, the directories, and thesystems110 may include software-programmable or hardwired computer systems interconnected via a network or networks. Each function for H represents an evaluation method performed by a computer system. The invention is not limited to the step sequences shown in the flowcharts, as the step order is sometimes interchangeable and further different steps may be performed in parallel. Other embodiments and variations are within the scope of the invention, as defined by the appended claims.
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APPENDIX AAccumulators and Proofs of Knowledge
• Definition (Secure Accumulator). A secure accumulator for a family of inputs {XX_k} is a family of families of functions GG={FF_k} with the following properties:
• Efficient generation: There is an efficient probabilistic algorithm G that oninput1^kproduces a random element f of {FF_k}. G also outputs some auxiliary information about f, denoted by aux_f.
• Efficient evaluation: fε{FF_k} is a polynomial-size circuit that, on input (u,x)εUU_f×XX_k, outputs a value vεUU_fwhere UU_fis an efficiently-samplable input domain for the function f, and {XX_k} is the intended input domain whose elements are to be accumulated.
• Quasi-commutative: For all k, for all fεFF_k, for all uεUU_ffor all X₁,X₂E XX_k, f(f(u,x₁),X₂)=f(f(u,X₂),X₁). If X={x₁, . . . , x_m}⊂XX_kthen by f(u,X) we denote f(f( . . . (u,x₁), . . . , x_m).
• Witnesses: Let vεUU′_fand ×XX′_k. A value wεUU_fis called a witness for x in v under f if v=f (w,x).
• Security: Let UU′_f×XX′_kdenote the domains for which the computational procedure for function f εFF_kis defined (thus UU_f⊂UU′_f, XX_k⊂XX′_k). For all probabilistic polynomial-time adversaries A_k,
• 
  Pr[f←G(1^k); u←UU_f, (x,w,X)←A_k(f, UU_f, u):
• 
  X⊂XX_k; WεUU_f′;xεXX_k; x∉X; ƒ(w,x)=ƒ(u,X)]−neg(k).
• Camenisch and Lysyanskaya ([39]) define the notion of a dynamic accumulator:
• Definition (Dynamic Accumulator). A secure accumulator is dynamic if it has the following property:
• Efficient Deletion: There exist efficient algorithms D and W such that, if v=f(u,X), x, x′εX, and f(w,x)=v, then:
• D(aux_f, v, x′)=v′ such that v′=f(u, X−{x′}) and
• W(f, v, v′, x, x′)=w′ such that f (w¹, x)=v′.
• ZERO-KNOWLEDGE PROOFS. An advantage of accumulators (at least, RSA-based accumulators, which are described later) is that it is possible to construct efficient zero-knowledge proofs (ZK proofs) that a value has been accumulated. It has been proven that any statement that is in NP (nondeterministic polynomial-time) can be proven in ZK, but some statements can be proven in ZK much more efficiently than others. Briefly, we describe the concept of a ZK proof, which was introduced independently by Brassard, Chaum, and Crepeau and by Goldwasser, Micali, and Racko, and further refined by Bellare and Goldreich.
• Let x be an input, and let R be a polynomially computable relation. Roughly speaking, a zero-knowledge proof of knowledge of a witness w such that R(x,w)=1 is a probabilistic polynomial-time protocol between a prover P and a verifier V such that, after the protocol, V is convinced that P knows such a witness w, but V does not obtain any explicit information about w. In other words, apart from “proving” that it knows a witness w such that R(x, w)=1, P imparts “zero knowledge” to V.
• In the sequel, we may use the notation introduced by Camenisch and Stadler for various proofs of knowledge of discrete logarithms and proofs of the validity of statements about discrete logarithms. For instance,
• 
  PK{(α,β,γ): y=g^αh^β

  

  y′=g′^αh′^γ

  

  (u≦α≦v)}
• denotes a zero-knowledge Proof of Knowledge of integers α, β, and γ such that y=g′^αh′^β and y′=g′^αh′^γ, where u≦α≦v and where g, g′, h, h′, y, and y′ are elements of some groups G=<g>=<h>and G′ g′>=<h′>. The convention is that Greek letters denote quantities the knowledge of which is being proved, while all other parameters are known to the verifier. Using this notation, a proof-protocol can be described by just pointing out its aim while hiding all details.
• Often, these proofs of knowledge are instantiated by a three-pass protocol, in which the prover first sends the verifier a commitment to certain values, after which the verifier sends the prover a challenge bit-strings, and the prover finally sends a response that incorporates both the “known value”, the committed values and the challenge value in such a way that it convinces the verifier is convinced of the prover's knowledge.
• These proofs of knowledge can be turned into signature schemes via the Fiat-Shamir heuristic. That is, the prover determines the challenge c by applying a collision-resistant hash function H to the commitment and the message m that is being signed and then computes the response as usual. We denote such signature proofs of knowledge by the notation, e.g., SPK{α: y=ƒ(α)} (m). Such SPK's can be proven secure in the random oracle model, given the security of the underlying proofs of knowledge.
• ZK proofs are often accomplished with the help of a commitment scheme. A commitment scheme consists of the algorithms Commit and VerifyCommit with properties as follows. The commitment algorithm Commit takes as input a message m, a random string r and outputs a commitment C C=Commit(m,r). The (commitment) verification algorithm VerifyCommit takes as input (C, m, r) and outputs1 (accept) if C is equal to Commit(m,r) and0 (reject) otherwise. The security properties of a commitment scheme are as follows. The hiding property is that a commitment C=Commit(m, r) contains no (computational) information on m. The binding property is that given C, m, and r, where 1=VerifyCommit(C,m,r), it is (computationally) impossible to find a message m₀and a string r₀such that 1=VerifyCommit(C,m₀,r₀).
• To prove, in ZK, knowledge of a witness w of a value x that has been accumulated—i.e., that f(w,x)=v, where v is the accumulator value—the usual method is to choose a random string r and construct a commitment c=Commit(x,r) and then provide the following proof of knowledge:
• 
  PK{(α,β,γ):c=Commit(α,γ)

  

  ƒ(γ,α)=v}.
• Above α represents the (hidden) x value, while β represents r and γ represents w.
• RSA-BASED ACCUMULATORS. Here we describe a common concrete instantiation of accumulators, which uses mathematics related to the well-known RSA public-key cryptosystem, invented by Rivest, Shamir and Adleman in 1977. Above, our description focused on some RSA-based instantiation of accumulators, but this description should not be considered limiting; our accumulator-based certificate revocation schemes could be used with any type of accumulators. An accumulator structure has an advantage that its size does not depend on the number of accumulated elements. An RSA-based accumulator makes use of a composite integer n, called a modulus, that should be chosen in such a way that it is hard to factor. In some embodiments of the schemes defined above, the modulus is an RSA modulus, which is defined as follows:
• Definition (RSA modulus). A 2k-bit number n is called an RSA modulus if n=pq, where p and q are k-bit prime numbers.
• Of course, one can choose n in a different way—e.g., as the product of three primes, or as the product of two primes of different sizes.
• Definition (Euler totient function). Let n be an integer. The Euler totient function Φ(n) is the cardinality of the group Z_n* (the multiplicative group of elements having an inverse in the ring Z_nof the integers modulo n; Z_n* is the set of all elements mutually prime with n).
• If n=pq is the product of two primes, then Φ(n)=(p−1)(q−1).
• The security of RSA-based accumulators is based on the following assumption.
• Definition (Strong RSA Assumption) The strong RSA assumption is that it is “hard,” on input an RSA modulus n and an element uεE_n*, to compute values e>1 and v such that v^e=u(mod n). By “hard”, we mean that, for all polynomial-time circuit families {A_k}, there exists a negligible function neg(k) such that
• 
  Pr[n ←RSAmodulus(1^k); u←Z_n*; (v,e) A_k(n,u):v^e=u(modn)]=neg(k),
• where RSAmodulus(1^k) is an algorithm that generates an RSA modulus as the product of two random k-bit primes, and a negligible function neg(k) is a function such that for all polynomials p(k), there is a value k₀such that neg(k)<1/p(k) for all k>k₀. The tuple (n,u) generated as above, is called a general instance of the strong RSA problem.
• Corollary 1. Under the strong RSA assumption, it is hard, on input a flexible RSA instance (n,u), to compute integers e>1 and v such that v^e=u(mod n).
• The most common concrete instantiation of accumulators is based on the above strong-RSA assumption. Roughly speaking, the idea is as follows: Given a fixed base u(mod n), one can compute an accumulator of values x₁and x₂(for example) as v=u^x1x2(mod n). To prove that x₁(for example) has been accumulated, one can forward the witness w=u^x2(mod n) and a verifier can confirm that indeed w^x1(mod n).
• Now, we relate the formal description of accumulators to the concrete RSA-based construction. A secure RSA-based accumulator for a family of inputs X_kis a family of functions FF_k, where the particular function fεFF_kdepends on what the modulus n is. For reasons that will become clear later, we assume that elements of X_kare pairwise relatively prime integers. Then, aux_fis the (secret) factorization of n. As alluded to above, given an initial accumulator value v′, an additional value x is added to the accumulator by computing a new accumulator value v=v′^x(mod n). Notice that the computational complexity of this algorithm is independent of the number of prior values that have been accumulated. The RSA-based accumulator possesses the quasi-commutative property; e.g., regardless of the order in which x₁and x₂are incorporated into an initial accumulator v′, the result is v=v′^x1x2(mod n). Given an accumulator v(mod n), the witness that a value x has been accumulated is w=v^1/x(mod n), which can readily be verified by confirming that v=w^x(modn). To reiterate, the security of the construction is based on the assumption that the strong RSA problem is infeasible to solve.
• RSA-based accumulators can be made dynamic. Recall that an accumulator is dynamic if, given an accumulator v that accumulates values of the set X and given the secret information aux_f, one can “de-accumulate” a value x′εX—i.e., compute a new accumulator v′ that accumulates the values of the set X-{x′}. Moreover, given a witness w that a value x has been accumulated (with respect to accumulator v that accumulates members of the set X) and given the accumulator v′ that only accumulates members of X-{x′}, one can compute a new witness w for x with respect to the accumulator v′. Specifically, for RSA-based accumulators, one can use the factorization of n to de-accumulate x′ by computing v′=v^1/x′(mod n). And given a witness w for x with respect to v—i.e., w^x=v(mod n)—and given the value of v′, one can compute a witness w′ for x with respect to v′—i.e., w′^x=v′=v′^1/xx′(mod n)→w′=v′^1/xx′(mod n) as follows. Assuming that x and x′ are relatively prime, one can compute integers (a,b) such that ax+bx′=1, by using the Extended Euclidean Algorithm. Then w′=v′^aw^b=v^a/x′v^b/x=v^{(ax+bx′)/xx′}v^1/xx′(mod n). Notice that the computation of w′ is efficient, and (after v′ is computed) it doesn't require any secret knowledge (e.g., the factorization of n).
• Given a witness w that a value x is accumulated (with respect to an accumulator v(mod n)), it also well-known in the art how to construct a ZK proof of knowledge of a pair (w,x) that satisfies w^x=v(mod n) (that hides the values of both w and x from the verifier).
End of Appendix AAPPENDIX BGuillou-Quisquater (GQ) ID-based Signature Scheme
• Set-Up:
• 1. A public key generator (PKG, a trusted party), publishes its public key (v,n) where n=q₁q₂(a product of two primes) is such that its factorization is hard to find, and v is a prime less than Φ(n)=(p−1)(q−1).
• 2. For a user with an identity ID (e.g., an email address), the PKG computes the secret signing key B such that JB^v≡1 mod n, where J=R(ID), where R is a predefined public function, e.g. a redundancy function. In some embodiments, the function R maps ID into an element of Z_n. The function R is such that the number of elements in Z_nthat correspond to mappings from valid ID′S is small. The PKG sends B to the user via a secure channel (e.g. encrypted).
• SIGNING: To sign a message M, the user:
1. Computes J=R(ID)
• 2. Generates a random number r and computes
• 
  d=H(M∥r^v), D=rB^d  (B-1)
• where H is a predefined public function (e.g. a hash function).
  3. Sends the values M, r^v, d, D to the verifier.
• VERIFICATION: The verifier:
1. Computes J=R(ID);
• 2. Checks that d=H(M∥r^v);
  3. Checks that J^dD^v=r^v.
  End of Appendix B

Claims (5)

1. A manufacture for use in connection with a computer-implemented method for providing authentication as to whether or not elements possess a pre-specified property, each element being operable to acquire the property and/or to lose the property, wherein the authentication is to be provided in each of a plurality of consecutive periods of time, the method comprising:

generating, by a computer system, a plurality of pieces of authentication data c_c(E1) for said periods of time for an element E1 which is one of said elements, each piece c_j(E1) being evidence that the element E1 possesses said property in the respective period j;

assigning an integer p_E1to the element E1;

generating, by the computer system, an encryption key K_i(E1) for each period j for the element E1, the encryption key being generated from a p_E1-th root u(j)¹/P_E1(mod n) of an integer u(j) modulo a predefined composite integer n, where u(j) depends on j;

encrypting, by the computer system, at least part of each piece c_j(E1) with the respective key K_j(E1);

the manufacture being encoded with computer-readable data comprising at least one said key K_j(E1) and/or an encryption of at least part of at least one said piece c_j(E1) under the at least one said key K_j(E1), the computer-readable data associating the at least one said key K_i(E1) and/or the encryption of the at least part of the at least one said piece c_j(E1) with the period j.

2. A manufacture for use in connection with a computer-implemented method for providing authentication as to whether or not elements possess a pre-specified property, each element being operable to acquire the property and/or to lose the property, wherein the authentication is to be provided in each of a plurality of consecutive periods of time, the method comprising:

for each period j of said plurality of consecutive periods of time, determining, by a first computer system, if an element E1 is to be authenticated in the period j as possessing said property;

if the element E1 is to be authenticated in the period j as possessing said property, then providing, by the first computer system, a decryption key DK_j(E1) to each second computer system which is to provide evidence data c_j(E1) in the period j that the element E1 possess said property, wherein DK_i(E1) is generated from a p_E1-th root u(j)¹/P_E1(mod n) of an integer u(j) modulo a predefined composite integer n, where u(j) depends on j, where p_E1is an integer assigned to the element E1, the decryption key DK_j(E1) being for decrypting at least a portion of the evidence data c_j(E1);

if the element E1 is not to be authenticated in the period j as possessing said property, then not providing the decryption key DK_j(E1) to at least one of said second computer systems;

the manufacture being encoded with computer-readable data comprising at least one said decryption key DK_j(E1), the computer-readable data associating the at least one said decryption key DK_i(E1) with the period j.

3. A network transmission method comprising transmitting, over a computer network, a computer program for performing a computer-implemented authentication method for providing authentication as to whether or not elements possess a pre-specified property, each element being operable to acquire the property and/or to lose the property, wherein the authentication is to be provided in each of a plurality of consecutive periods of time, the authentication method comprising:

generating, by a computer system, a plurality of pieces of authentication data c_c(E1) for said periods of time for an element E1 which is one of said elements, each piece c_j(E1) being evidence that the element E1 possesses said property in the respective period j;

assigning an integer p_E1to the element E1;

generating, by the computer system, an encryption key K₃(E1) for each period j for the element E1, the encryption key K_ibeing generated from a p_E1-th root u(j)¹/P_E1(mod n) of an integer u(j) modulo a predefined composite integer n, where u(j) depends on j;

encrypting, by the computer system, at least part of each piece c_j(E1) with the respective key K_j(E1).

4. A network transmission method comprising transmitting, over a computer network, a computer program for performing a computer-implemented authentication method for providing authentication as to whether or not elements possess a pre-specified property, each element being operable to acquire the property and/or to lose the property, wherein the authentication is to be provided in each of a plurality of consecutive periods of time, the authentication method comprising:

for each period j of said plurality of consecutive periods of time, determining, by a first computer system, if an element E1 is to be authenticated in the period j as possessing said properly;

if the element E1 is to be authenticated in the period j as possessing said property, then providing, by the first computer system, a decryption key DK_j(E1) to each second computer system which is to provide evidence data c_j(E1) in the period j that the element E1 possess said property, wherein DK_j(E1) is generated from a p_E1-th root u(j)¹/P_E1(mod n) of an integer u(j) modulo a predefined composite integer n, where u(j) depends on j, where p_E1is an integer assigned to the element E1, the decryption key DK_j(E1) being for decrypting at least a portion of the evidence data c_j(E1);

if the element E1 is not to be authenticated in the period j as possessing said property, then not providing the decryption key DK_j(E1) to at least one of said second computer systems.

6. A network transmission method comprising transmitting, over a computer network, a computer program for performing a computer-implemented authentication method for providing authentication as to whether or not elements possess a pre-specified property, each element being operable to acquire the property and/or to lose the property, wherein the authentication is to be provided in each of a plurality of consecutive periods of time, the authentication method comprising:

a computer system receiving a plurality of data pieces EN_j(c_j(E1)) for consecutive periods of time j, each data piece EN_j(c_j/E1)) being an encryption of at least a portion of authentication data c_j(E1) which evidences that an element E1 possesses said property in a period j which is one of consecutive periods of time, the element E1 being one of said elements;

after receiving said plurality of data pieces EN(c_j(E1)) for each period j, if the element E1 is to be authenticated in the period j as possessing said property, then:

(a) the computer system receiving a decryption key DK_j(E1) generated from a P_E1-th root u(j)^1/PE1(mod n) of an integer u(j) modulo a predefined composite integer n, wherein p_E1is an integer associated with the element E1; and

(b) the computer system decrypting the data piece EN(c_j(E1)) with the key DK_j(E1) to recover c_j(E1).

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