Patent Application: US-201113194702-A

Abstract:
a system and method for cryptographically checking the correctness of outsourced set operations performed by an untrusted server over a dynamic collection of sets that are owned by a trusted source is disclosed . the system and method provides new authentication mechanisms that allow any entity to publicly verify a proof attesting the correctness of primitive set operations such as intersection , union , subset and set difference . based on a novel extension of the security properties of bilinear - map accumulators as well as on a primitive called accumulation tree , the system and method achieves optimal verification and proof complexity , as well as optimal update complexity , while incurring no extra asymptotic space overhead . the method provides an efficient proof construction , adding a logarithmic overhead to the computation of the answer of a set - operation query . applications of interest include efficient verification of keyword search and database queries .

Description:
we denote with k the security parameter and with neg ( k ) a negligible function 4 . 4 function ƒ : → is neg ( k ) if and only if for any nonzero polynomial p ( k ) there exits n such that for all k & gt ; n it is ƒ ( k )& lt ; 1 / p ( k ). the bilinear - map accumulator . let be a cyclic multiplicative group of prime order p , generated by element gε . let also be a cyclic multiplicative group of the same order p , such that there exists a pairing e : × → with the following properties : ( i ) bilinearity : e ( p α , q b )= e ( p , q ) ab for all p , qε and a , b ε ; ( ii ) non - degeneracy : e ( g , g )≠ 1 ; ( iii ) computability : for all p , q ε , e ( p , q ) is efficientl computable . we call ( p , , , e , g ) a tuple of bilinear pairing parameters , produced as the output of a probabilistic polynomial - time algorithm that runs on input i k . in this setting , the bilinear - map accumulator [ 28 ] is an efficien way to provide short proofs of membership for elements that belong to a set . let sε be a randomly chosen value that constitutes the trapdoor in the scheme . the accumulator primitive accumulates elements in −{ s }, outputting a value that is an element in . for a set of elements χ in −{ s } the accumulation value acc ( χ ) of χ is define as 5 πχεs i ( χ + s ) is called characteristic polynomial of set s i in the literature ( e . g ., see [ 25 ]). value acc ( χ ) can be constructed using χ and g , g s , g s 2 , . . . , g s q ( through polynomial interpolation ), where q ≧| χ |. subject to acc ( χ ) each element in χ has a succinct membership proof . more generally , the proof of subset containment of a set s ⊂ χ — for | s |= 1 , this becomes a membership proof — is the witness ( s , w s , χ ) where subset containment of s in χ can be checked through relation e ( w s , χ , g π χεs ( χ + s ) ) e ( acc ( χ ), g ) by any verifie with access only to public information . the security property of the bilinear - map accumulator , namely that computing fake but verifiabl subset containment proofs is hard , can be proved using the bilinear q - strong diffie - hellma assumption , which is slightly stronger than the q - strong diffie - hellma assumption [ 8 ] 6 6 however , the plain q - strong diffie - hellma assumption [ 28 ] suffice to prove just the collision resistance of the bilinear - map accumulator . assumption 1 ( bilinear q - strong diffie - hellman assumption ) let k be the security parameter and ( p , , , g ) be a tuple of bilinear pairing parameters . given the elements g , g s , . . . , g s q ε for some s chosen at random from , where q = poly ( k ) no probabilistic polynomial - time algorithm can output a pair ( a , e ( g , g ) 1 /( a + s ) ) ε × , except with negligible probability neg ( k ). we next prove the security of subset witnesses by generalizing the proof in [ 28 ]. subset witnesses also appeared ( independent of our work but without a proof ) in [ 10 ]. lemma 1 ( subset containment ) let k be the security parameter and ( p , , , g ) be a tuple of bilinear pairing parameters . given the elements g , g s , . . . , g s q ε for some s chosen at random from and a set of elements χ in −{ s } with q ≧| χ |, suppose there is a probabilistic polynomial - time algorithm that finds s and w such that s ⊂ χ and e ( w , g πχεs ( χ + s ) )= e ( acc ( χ ), g ). then there is a probabilistic polynomial - time algorithm that breaks the bilinear q - strong diffie - hellman assumption . proof : suppose there is a probabilistic polynomial - time algorithm that computes such a set s ={ y 1 , y 2 , . . . , y l } and a fake witness w . let χ ={ χ 1 , χ 2 , . . . , χ n } and y j ⊂ χ for some 1 ≦ j ≦ l . this means that e ( w , g ) π ν εs ( y + s ) = e ( g , g ) ( 1 + s )( χ 2 + s ) . . . ( χ n + s ) . note that ( y j + s ) does not divide ( χ 1 + s )( χ 2 + s ) . . . ( χ n + s ). therefore there exist polynomial q ( s ) ( computable in polynomial time ) of degree n − 1 and constant λ ≠ 0 , such that ( χ 1 + s )( χ 2 + s ) . . . ( χ n + s )= q ( s )( y j + s )+ λ . thus we have thus , this algorithm can break the bilinear q - strong diffie - hellma assumption . □ tools for polynomial arithmetic . our solutions use ( modulo p ) polynomial arithmetic . we next present two results that are extensively used in our techniques , contributing to achieve the desired complexity goals . the firs result on polynomial interpolation is derived using an fft algorithm ( see preparata and sarwate [ 34 ]) that computes the dft in a finit fiel ( e . g ., ) for arbitrary n and performing o ( n log n ) fiel operations . we note that an n - th root of unity is not required to exist in for this algorithm to work . lemma 2 ( polynomial interpolation with fft [ 34 ]) let π i = 1 n ( χ i + s )= σ i = 0 n b i s i be a degree - n polynomial . the coefficients b n = 0 , b n - 1 , . . . b 0 of the polynomial can be computed with o ( n log n ) complexity , given χ 1 , χ 2 . . . χ n . lemma 2 refers to an efficien process for computing the coefficient of a polynomial , given its roots χ 1 , χ 2 , . . . , χ n . in our construction , we make use of this process a numbers of times , in particular , when , given some values χ 1 , χ 2 , . . . , χ n to be accumulated , an untrusted party needs to compute g ( χ 1 + s )( χ 2 + s ) . . . ( χ n + s ) without having access to s . however , access to g , g s , . . . , g s n ( part of the public key ) is allowed , and therefore computing the accumulation value boils down to a polynomial interpolation . we next present a second result that will be used in our verificatio algorithms . related to certifying algorithms [ 21 ], this result states that if the vector of coefficient b =[ b n , b n - 1 , . . . , b 0 ] is claimed to be correct , then , given the vector of roots x =[ χ 1 , χ 2 , . . . , χ n ], with high probability , vector b can be certifie to be correct with complexity asymptotically less than o ( n log n ), i . e ., without an fft computation from scratch . this is achieved with the following algorithm : algorithm { accept , reject }← certify ( b , x , pk ): the algorithm picks a random κε . if σ i = 0 n b i κ i = σ i = 1 n ( χ i + κ ), then the algorithm accepts , else it rejects . lemma 3 ( polynomial coefficients verification ) let b =[ b n , b n - 1 , . . . , b 0 ] and x =[ χ 1 , χ 2 , . . . , χ n ]. algorithm certify ( b , x , pk ) has o ( n ) complexity . also , if accept ← certify ( b , x , pk ), then b n , b n - 1 , . . . , b 0 are the coefficients of the polynomial π i = 1 n ( χ i + s ) with probability ω ( 1 − neg ( k )). authenticated data structure scheme . we now defin our authenticated data structure scheme ( ads scheme ), as well as the correctness and security properties it must satisfy . definition 1 ( ads scheme ) let d be any data structure that supports queries q and updates u . let auth ( d ) denote the resulting authenticated data structure and d the digest of the authenticated data structure , i . e ., a constant - size description of d . an ads scheme a is a collection of the following six probabilistic polynomial - time algorithms : 1 . { sk , pk }← genkey ( 1 k ): on input the security parameter k , it outputs a secret key sk and a public key pk ; 2 . { auth ( d 0 ), d 0 }← setup ( d 0 , sk , pk ): on input a ( plain ) data structure d 0 and the secret and public keys , it computes the authenticated data structure auth ( d 0 ) and the respective digest do of it ; 3 . { d h + 1 , auth ( d h + 1 ), d h + 1 , upd }← update ( u , d h , auth ( d h ), d h , sk , pk ): on input an update u on data structure d h , the authenticated data structure auth ( d h ), the digest d h , and the secret and public keys , it outputs the updated data structure d h + 1 along with the updated authenticated data structure auth ( d + 1 ), the updated digest d h + 1 and some relative information upd ; 4 . { d + 1 , auth ( d + 1 ), d h + 1 }← refresh ( u , d h , auth ( d h ), d h , upd , pk ): on input an update u on data structure d h , the authenticated data structure auth ( d h ), the digest d h , relative information upd ( output by update ), and the public key , it outputs the updated data structure d h + 1 along with the updated authenticated data structure auth ( d h + 1 ) and the updated digest d h + 1 ; 5 . { π ( q ), α ( q )}← query ( q , d h , auth ( d h ), pk ): on input a query q on data structure d h , the authenticated data structure auth ( d h ) and the public key , it returns the answer α ( q ) to the query , along with a proof π ( q ); 6 . { accept , reject }← verify ( q , α , h , d h , pk ): on input a query q , an answer α , a proof h , a digest d h and the public key , it outputs either accept or reject . let { accept , reject }← check ( q , α , d h ) be an algorithm that decides whether α is a correct answer for query q on data structure d h ( check is not part of the definitio of an ads scheme ). there are two properties that an ads scheme should satisfy , namely correctness and security ( intuition follows from signature schemes definitions ) definition 2 ( correctness ) let be an ads scheme { genkey , setup , update , refresh , query , verify }. we say that the ads scheme is correct if for all kε for all { sk , pk } output by algorithm genkey , for all d h , auth ( d h ), d h output by one invocation of setup followed by polynomially - many invocations of refresh , where h ≧ 0 , for all queries q and for all ø ( q ), α ( q ) output by query ( q , d h , auth ( d h ), pk ), with all but negligible probability , whenever algorithm check ( q , α ( q ), d h ) outputs accept , so does algorithm verify ( q , ø ( q ), α ( q ), d h , pk ). definition 3 ( security ) let be an ads scheme { genkey , setup , update , refresh , query , verify }, k be the security parameter ν ( k ) be a negligible function and { sk , pk }← genkey ( 1 k ). let also adv be a probabilistic polynomial - time adversary that is only given pk . the adversary has unlimited access to all algorithms of except for algorithms setup and update to which he has only oracle access . the adversary picks an initial state of the data structure d 0 and computes d 0 , auth ( d 0 ), d 0 through oracle access to algorithm setup . then , for i = 0 , . . . , h = poly ( k ), adv issues an update μ i in the data structure d i and computes d i + 1 , auth ( d i + i ) and d i + 1 through oracle access to algorithm update . finally the adversary picks an index 0 ≦ t ≦ h + 1 , and computes a query q , an answer α and a proof ø . we say that the ads scheme is secure if for all kε for all { sk , pk } output by algorithm genkey , and for any probabilistic polynomial - time adversary adv it holds that in this section we present an ads scheme for set - operation verification the underlying data structure for which we design our ads scheme is called sets collection , and can be viewed as a generalization of the inverted index [ 4 ] data structure . sets collection . referring now to fig5 , 6 , 11 and 12 , the sets collection data structure consists of m sets , denoted with s 1 , s 2 , . . . , s m , each containing elements from a universe μ . without loss of generality we assume that the universe is the set of nonnegative integers in the interval [ m + 1 , p − 1 ]−{ s } 7 where p is k - bit prime , m is the number of the sets in our collection that has bit size o ( log k ), k is the security parameter and s is the trapdoor of the scheme ( see algorithm genkey ). a set s i does not contain duplicate elements , however an element χε can appear in more than one set . each set is sorted and the total space needed is o ( m + m ), where m is the sum of the sizes of the sets . 7 this choice simplifie the exposition ; however , by using some collision - resistant hash function , universe can be set to −{ s }. in order to get some intuition , we can view the sets collection as an inverted index . in this view , the elements are pointers to documents and each set s i corresponds to a term w i in the dictionary , containing the pointers to documents where term w i appears . in this case , m is the number of terms being indexed , which is typically in the hundreds of thousands , while m , bounded from below by the number of documents being indexed , is typically in the billions . thus , the more general terms “ elements ” and “ sets ” in a sets collection can be instantiated to the more specifi “ documents ” and “ terms ”. the operations supported by the sets collection data structure consist of updates and queries . an update is either an insertion of an element into a set or a deletion of an element from a set . an update on a set of size n takes o ( log n ) time . for simplicity , we assume that the number m of sets does not change after updates . a query is one of the following standard set operations : ( i ) intersection . given indices i 1 , i 2 , . . . , i t , return set i = s i 1 ∩ s i 2 ∩ . . . s i t ; ( ii ) union : given indices i l , i 2 , . . . , i t , return set u = s i 1 ∪ s i 2 ∪ . . . ∪ s it ; ( iii ) subset query : given indices i and j , return true if s i ⊂ s j and false otherwise ; ( iv ) set difference : given indices i and j , return set d = s i − s j . for the rest of the paper , we denote with δ the size of the answer to a query operation , i . e ., δ is equal to the size of i , u , or d . for a subset query , δ is o ( 1 ). we next detail the design of an ads scheme for the sets collection data structure . this scheme provides protocols for verifying the integrity of the answers to set operations in a dynamic setting where sets evolve over time through updates . the goal is to achieve optimality in the communication and verificatio complexity : a query with t parameters and answer size δ should be verifie with o ( t + δ ) complexity , and at the same time query and update algorithms should be efficien as well . referring to fig7 - 9 , 14 and 15 , we describe an ads scheme ={ genkey , setup , update , refresh , query , verify } for the sets collection data structure and we prove that its algorithms satisfy the complexities of table 1 . we begin with the algorithms that are related to the setup and the updates of the authenticated data structure . algorithm { sk , pk }← genkey ( 1 k ): bilinear pairing parameters ( p , , , e , g ) are picked and an element ε is chosen at random . subsequently , an one - to - one function h (•): → is used . this function simply outputs the bit description of the elements of according to some canonical representation of . finally the algorithm outputs sk = s and pk ={ h (•), p , , , g , g }, where vector g contains values where q ≧ max { m , max i = 1 , . . . , m {| s i |}}. the algorithm has o ( 1 ) access complexity . algorithm { d 0 , auth ( d 0 ), d 0 }← setup ( d 0 , sk , pk ): let d 0 be our initial data structure , i . e ., the one representing sets s 1 , s 2 , . . . , s m . the authenticated data structure auth ( d ) is built as follows . first , for each set s i its accumulation value acc ( s i )= g πχεs i ( χ + s ) is computed ( see section 6 . 1 ). subsequently , the algorithm picks a constant 0 ≦ ε ≦ 1 . let t be a tree that has l =[ 1 / e ] levels and m leaves , numbered 1 , 2 , . . . , m , where m is the number of the sets of our sets collection data structure . since t is a constant - height tree , the degree of any internal node of it is o ( m ε ). we call such a tree an accumulation tree , which was originally introduced ( combined with different cryptography ) in [ 32 ]. for each node of the tree ν , the algorithm recursively computes the digest d ( ν ) of ν as follows . if ν is a leaf corresponding to set s i where 1 ≦ i ≦ m , the algorithm sets d ( ν )= acc ( s i ) ( i + s ) ; here , raising value acc ( s i ) to exponent i + s , under the constraint that i ≦ m , is done to also accumulate the index i of set s i ( and thus prove that acc ( s i ) refers to s i ). if node ν is not a leaf , then where ( ν ) denotes the set of children of node ν . the algorithm outputs all the sets $ as the data structure d 0 , and all the accumulation values acc ( s i ) for 1 ≦ i ≦ m , the tree t and all the digests d ( ν ) for all νεt as the authenticated data structure auth ( d 0 ). finally , the algorithm sets d 0 = d ( r ) where r is the root of t , i . e ., d 0 is the digest of the authenticated data structure ( define similarly as in a merkle tree ). 8 the access complexity of the algorithm is o ( m + m ) ( for postorder traversal of t and computation of acc ( s i )), where m = σ i = 1 m | s i |. the group complexity of auth ( d 0 ) is also o ( m + m ) since the algorithm stores one digest per node of t , t has o ( m ) nodes and there are m elements contained in the sets , as part of auth ( d 0 ). algorithm { d h + 1 , auth ( d h + 1 ), d h + 1 , upd }← update ( u , d h , auth ( d h ), d h , sk , pk ): we consider the update date “ insert element χε into set s i ” ( note that the same algorithm could be used for element deletions ). let ν 0 be the leaf node of t corresponding to set s i . let ν 0 , ν 1 , . . . , ν l be the path in t from node ν 0 to the root of the tree , where l =┌ 1 / ε ┐. the algorithm initially sets d ( ν 0 )= acc ( s i ) ( χ + s ) , i . e ., it updates the accumulation value that corresponds to the updated set ( note that in the case where χ is deleted from s i the algorithm sets d ′( ν 0 )= acc ( s i ) ( χ + s )- 1 ). then the algorithm sets 8 digest d ( r ) is a “ secure ” succinct description of the set collection data structure . namely , the accumulation tree protects the integrity of values acc ( s i ), 1 ≦ i ≦ m , and each accumulation value acc ( s i ) protects the integrity of the elements contained in set s i . d ′( ν j )= d ( ν j ) ( h ( d ′( ν j - 1 ))+ s )( h ( d ( ν j - 1 ))+ s ) - 1 for j = 1 , . . . , l , ( 4 ) where d ( ν j - 1 ) is the current digest of ν j - 1 and d ′( ν j - 1 ) is the updated digest of ν j - 1 . 9 all these newly computed values ( i . e ., the new digests ) are stored by the algorithm . the algorithm then outputs the new digests d ′( ν j - 1 ), j = 1 , . . . , l , along the path from the updated set to the root of the tree , as part of information upd . information upd also includes χ and d ( ν l ). the algorithm also sets d h + 1 = d ′( ν l ), i . e ., the updated digest is the newly computed digest of the root of t . finally the new authenticated data structure auth ( d h + 1 ) is computed as follows : in the current authenticated data structure auth ( d h ) that is input of the algorithm , the values d ( ν j - 1 ) are overwritten with the new values d ( ν j - 1 ) ( j = 1 , . . . , l ), and the resulting structure is included in the output of the algorithm . the number of operations performed is proportional to 1 / ε , therefore the complexity of the algorithm is o ( 1 ). 9 note that these update computations are efficien because update has access to secret key s . algorithm { d h + 1 , auth ( d h + 1 ), d h + 1 }← refresh ( u , d h , auth ( d h ), d h , upd , pk ): we consider the update “ insert element χε into set s i ”. let ν 0 be the node of t corresponding to set s i . let ν 0 , ν 1 , . . . , ν l be the path in t from node ν 0 to the root of the tree . using the information upd , the algorithm sets d ( ν j )= d ′( ν j ) for j = 0 , . . . , l , i . e ., it updates the digests that correspond to the updated path . finally , it outputs the updated sets collection as d h + 1 , the updated digests d ( ν j ) ( along with the ones that belong to the nodes that are not updated ) as auth ( d h + 1 ) and d ′( ν l ) ( contained in upd ) as d h + 1 . 10 the algorithm has o ( 1 ) complexity as the number of performed operations is o ( 1 / e ). 10 note that information upd is not required for the execution of refresh , but is rather used for efficien y . without access to upd , algorithm refresh could compute the updated values d ( ν j ) using polynomial interpolation , which would have o ( m ε log m ) complexity ( see lemma 2 ). referring to fig1 , so far we have described the authenticated data structure auth ( d h ) that our ads scheme will use for set - operation verifications overall , auth ( d h ) comprises a set of m accumulation values acc ( s i ), one for each set s i , i = 1 , . . . , m , and a set of o ( m ) digests d ( ν ), one for each internal node ν of the accumulation tree t . our proof construction and verificatio protocols for set operations ( described in section 6 . 2 . 3 ) make use of the accumulation values acc ( s i ) ( subject to which subset - containment witnesses can be defined ) and therefore it is required that the authenticity of each such value can be verified tree t serves this exact role by providing short correctness proofs for each value acc ( s i ) stored at leaf i of t , this time subject to the ( global ) digest d h stored at the root of t . we next provide the details related to proving the authenticity of acc ( s i ). the correctness proof ø i of accumulation value acc ( s i ), 1 ≦ i ≦ m , is a collection of o ( 1 ) bilinear - map accumulator witnesses ( as define in section 6 . 1 ). in particular , ø i is set to be the ordered sequence ø =( π 1 , π 2 , . . . , π l ), where π j is the pair of the digest of node ν j - 1 and a witness that authenticates ν j - 1 , subject to node νj , in the path ν 0 , ν 1 , . . . , ν l define by leaf ν 0 storing accumulation value acc ( s i ) and the root ν l of t . conveniently , π j is define as π j =( β j , γ j ), where β j = d ( ν j - 1 ) and γ j = w ν j - 1 ( ν j ) = g øwεn ( ν j )−{ ν j - 1 } ( h ( d ( w )+( s )). ( 5 ) note that π j is the witness for a subset of one element , namely h ( d ( ν j - 1 )) ( recall , d ( ν 0 )= acc ( s i ) ( i + s ) ). clearly , pair π j has group complexity o ( 1 ) and can be constructed using polynomial interpolation with o ( m ε log m ) complexity , by lemma 2 and since ν j has degree o ( m ε ). since ø i consists of o ( 1 ) such pairs , we conclude that the proof ø i for an accumulation value acc ( s i ) can be constructed with o ( m ε log m ) complexity and has o ( 1 ) group complexity . the following algorithms querytree and verifytree are used to formally describe the construction and respectively the verificatio of such correctness proofs . similar methods have been described in [ 32 ]. algorithm { ø i , α i }← querytree ( i , d h , auth ( d h ), pk ): let ν 0 , ν 1 , . . . , ν l be the path of t from the node storing acc ( s i ) to the root of t . the algorithm computes ø i by setting ø i =( π 1 , π 2 , . . . , π l ), where π j =( d ( ν j - 1 ), w ν j - 1 ( ν j ) ) and w ν j - 1 ( ν j ) is given in equation 5 and computed by lemma 2 . finally , the algorithm sets α i = acc ( s i ). algorithm { accept , reject }← verifytree ( i , α i , ø i , d h , pk ): let the proof be ø i =( π 1 , π 2 , . . . , π l ), where π j =( β j , γ j ). the algorithm outputs reject if one of the following is true : ( i ) e ( β 1 , g )≠ e ( α i , g i g s ); or ( ii ) e ( β j , g )≠ e ( γ j - 1 , g h ( β j - 1 ) g s ) for some 2 ≦ j ≦ 1 ; or ( iii ) e ( d h , g )≠ e ( γ l , g h ( β l ) g s ). otherwise , it outputs accept . we finall provide some complexity and security properties that hold for the correctness proofs of the accumulated values . the following result is used as a building block to derive the complexity of our scheme and prove its security ( theorem 1 ). lemma 4 algorithm querytree runs with o ( m ε log m ) access complexity and outputs a proof of o ( 1 ) group complexity . moreover algorithm verifytree has o ( 1 ) access complexity . finally , for any adversarially chosen proof ø i ( 1 ≦ i ≦ m ), if accept ← verifytree ( i , α i , ø i , d h , pk ), then α i = acc ( s i ) with probability ω ( 1 − neg ( k )). referring to fig1 and 17 , with the correctness proofs of accumulation values at hand , we complete the description of our scheme by presenting the algorithms that are related to the construction and verificatio of proofs attesting the correctness of set operations . these proofs are efficientl constructed using the authenticated data structure presented earlier , and they have optimal size o ( t + δ ), where t and δ are the sizes of the query parameters and the answer . in the rest of the section , we focus on the detailed description of the algorithms for an intersection and a union query , but due to space limitations , we omit the details of the subset and the set difference query . we note , however , that the treatment of the subset and set difference queries is analogous to that of the intersection and union queries . the parameters of an intersection or a union query are t indices i 1 , i 2 , . . . , i t , with 1 ≦ t ≦ m . to simplify the notation , we assume without loss of generality that these indices are 1 , 2 , . . . , t . let n i denote the size of set s i ( 1 ≦ i ≦ t ) and let n = σ i = 1 t n i . note that the size δ of the intersection or union is always o ( n ) and that operations can be performed with o ( n ) complexity , by using a generalized merge . intersection query . let i = s 1 ∩ s 2 ∩ . . . ∩ s t ={ y 1 , y 2 , . . . , y δ }. we express the correctness of the set intersection operation by means of the following two conditions : ( s 1 − i )∩( s 2 − i )∩ . . . ∩( s t − i )= ø . ( 7 ) the completeness condition in equation 7 is necessary since set i must contain all the common elements . given an intersection i , and for every set s j , 1 ≦ i ≦ t , we defin the degree - n j polynomial the following result is based on the extended euclidean algorithm over polynomials and provides our core verificatio test for checking the correctness of set intersection . lemma 5 set 1 is the intersection of sets s 1 , s 2 , . . . , s t if and only if there exist polynomials q 1 ( s ), q 2 ( s ), . . . , q t ( s ) such that q 1 ( s ) p 1 ( s )+ q 2 ( s ) p 2 ( s )+ . . . + q t ( s ) p t ( s )= 1 , where p j ( s ), j = 1 , . . . , t , are defined in equation 8 . moreover the polynomials q 1 ( s ), q 2 ( s ), . . . , q t ( s ) can be computed with o ( n log 2 n log log n ) complexity . using lemmas 2 and 5 we next construct efficien proofs for both conditions in equations 6 and 7 . in turn , the proofs are directly used to defin the algorithms query and verify of our ads scheme for intersection queries . proof of subset condition . for each set s j , 1 ≦ j ≦ t , the subset witnesses w i , j = g p j ( s ) = g øχεs j - 1 ( z + s ) are computed , each with o ( n j log n j ) complexity , by lemma 2 . ( recall , w i , j serves as a proof that i is a subset of set s j ) thus , the total complexity for computing all t required subset witnesses is o ( n log n ), where n = σ i = 1 t n i . 11 11 this is because σn j log n j ≦ log nσn j = n log n . proof of completeness condition . for each q j ( s ), 1 ≦ j ≦ t , as in lemma 5 satisfying q 1 ( s ) p 1 ( s )+ q 2 ( s ) p 2 ( s )+ . . . + q t ( s ) p t ( s )= 1 , the completeness witnesses f i , j = g q j ( s ) are computed , by lemma 5 with o ( n log 2 n log log n ) complexity . algorithm { π ( q ), α ( q )}← query ( q , d h , auth ( d h ), pk ) ( intersection ): query q consists of t indices { 1 , 2 , . . . , t }, asking for the intersection i of s 1 , s 2 , . . . , s t . let i ={ y 1 , y 2 , . . . , y δ }. then α ( q )= i , and the proof π ( q ) consists of the following parts . 1 . coefficients b δ , b δ - 1 , . . . , b 0 of polynomial ( y 1 + s )( y 2 + s ) . . . ( y δ + s ) that is associated with the intersection i ={ y 1 , y 2 , . . . , y δ }. these are computed with o ( δ log δ ) complexity ( lemma 2 ) and they have o ( δ ) group complexity . 2 . accumulation values acc ( s j ), j = 1 , . . . , t , which are associated with sets s j , along with their respective correctness proofs π j . these are computed by calling algorithm querytree ( j , d h , auth ( d h ), pk ), for j = 1 , . . . , t , with o ( tm ε log m ) total complexity and they have o ( t ) total group complexity ( lemma 4 ). 3 . subset witnesses w 1 , j , j = 1 , . . . , t , which are associated with sets s j and intersection i ( see proof of subset condition ). these are computed with o ( n log n ) complexity and have o ( t ) total group complexity ( lemma 2 ). 4 . completeness witnesses f i , j , j = 1 , . . . , t , which are associated with polynomials q j ( s ) of lemma 5 ( see proof of completeness condition ). these are computed with o ( n log 2 n log log n ) complexity and have o ( t ) group complexity ( lemma 5 ). algorithm { accept , reject }← verify ( q , α , π , d h , pk ) ( intersection ): verifying the result of an intersection query includes the following steps . 1 . first , the algorithm uses the coefficient b =[ b δ , b δ - 1 , . . . , b 0 ] and the answer α ( q )={ y 1 , y 2 , . . . , y δ } as an input to algorithm certify ( b , α ( q ), pk ), in order to certify the validity of b δ , b δ - 1 , . . . , b 0 . if certify outputs reject , the algorithm also outputs reject . 12 this step has o ( δ ) complexity ( lemma 3 ). 12 algorithm certify is used to achieve optimal verificatio and avoid an o ( δ log δ ) fft computation from scratch . 2 . subsequently , the algorithm uses the proof π j to verify the correctness of acc ( s j ), by running algorithm verifytree ( j , acc ( s j ), π j , d h , pk ) for j = 1 , . . . , t . if , for some j , verifytree running on acc ( s j ) outputs reject , the algorithm also outputs reject . this step has o ( t ) complexity ( lemma4 ). 3 . next , the algorithm checks the subset condition : 13 13 group element π i = 0 δ g s i b i = g ( y 1 + s )( y 2 + s ) . . . ( y δ + s ) is computed once with o ( δ ) complexity . if , for some j , the above check on subset witness w i , j fails , the algorithm outputs reject . this step has o ( t + δ ) complexity . if the above check on the completeness witnesses f i , j = 1 ≦ j ≦ t , fails , the algorithm outputs reject . or , if this relation holds , the algorithm outputs accept , i . e ., it accepts α ( q ) as the correct intersection . this step has o ( t ) complexity . note that for equation 10 , it holds π j = 1 t e ( w i , j , f i , j )= e ( g , g ) σ j = 1 r q j ( s ) p j ( s ) = e ( g , g ) when all the subset witnesses w i , j , all the completeness witnesses f i , j and all the sets accumulation values acc ( s j ) have been computed honestly , since q 1 ( s ) p 1 ( s )+ q 2 ( s ) p 2 ( s )+ . . . + q t ( s ) p t ( s )= 1 . this is a required condition for proving the correctness of our ads scheme , as define in definitio 2 . we continue with the description of algorithms query and verify for the union query . union query . let u = s 1 ∪ s 2 ∪ . . . ∪ s t ={ y 1 , y 2 , . . . , y δ }. we express the correctness of the set union operation by means of the following two conditions : the superset condition in equation 12 is necessary since set u must exclude none of the elements in sets s 1 , s 2 , . . . , s t . we formally describe algorithms query and verify of our ads scheme for union queries . algorithm { π ( q ), α ( q )}← query ( q , d h , auth ( d h ), pk ) ( union ): query q asks for the union ∪ of t sets s 1 , s 2 , . . . , s t . let ∪={ y 1 , y 2 , . . . , y δ }. then α ( q )=∪ and the proof π ( q ) consists of the following parts . ( 1 ) coefficients b δ , b δ - 1 , . . . , b 0 of polynomial ( y 1 + s )( y 2 + s ) . . . ( y δ + s ) that is associated with the union ∪={ y 1 , y 2 , . . . , y δ }. ( 2 ) accumulation values acc ( s j ), j = 1 , . . . , t , which are associated with sets s j , along with their respective correctness proofs π j , both output of algorithm querytree ( j , d h , auth ( d h ), pk ). ( 3 ) membership witnesses w y i , s k of y i , i = 1 , . . . , δ ( see equation 1 ), which prove that y i belongs to some set s k , 1 ≦ k ≦ t , and which are computed with o ( n log n ) total complexity and have o ( δ ) total group complexity ( lemma 2 ). ( 4 ) subset witnesses w s j ,∪ , j = 1 , . . . , t , which are associated with sets s i and union ∪ and prove that ∪ is a superset of s j , 1 ≦ k ≦ t , and which are computed with o ( n log n ) total is complexity and have o ( t ) total group complexity ( lemma2 ). algorithm { accept , reject }← verify ( q , α , π , d h , pk ): ( union ): verifying the result of a union query includes the following steps . ( 1 ) first , the algorithm uses b =[ b δ , b δ - 1 , . . . , b 0 ] and the answer ∪= α ( q )={ y 1 , y 2 , . . . , y δ } as an input to algorithm certify ( b , α ( q ), pk ), in order to certify the validity of b δ , b δ - 1 , . . . , b 0 . ( 2 ) subsequently , the algorithm uses the proofs ø j to verify the correctness of acc ( s j ), by using algorithm verifytree ( j , acc ( s j ), π j , d h , pk ) for j = 1 , . . . , t . if the verificatio fails for at least one of acc ( s j ), the algorithm outputs reject . ( 3 ) next , the algorithm verifie that each element y , i = 1 , . . . , δ , of the reported union belongs to some set s k , for some 1 ≦ k ≦ t ( o ( δ ) complexity ). this is done by checking that relation e ( w y i , s k , g y i g s )= e ( acc ( s k ), g ) holds for all i = 1 , . . . , δ ; otherwise the algorithm outputs reject . ( 4 ) finally , the algorithm verifie that all sets specifie by the query are subsets of the union , by checking the following conditions : if any of the above checks fails , the algorithm outputs reject , otherwise , it outputs accept , i . e ., ∪ is accepted as the correct union . subset and set difference query . for a subset query ( positive or negative ), we use the property s i ⊃ s j ∀ y εs i , yεs j . for a set difference query we use the property the above conditions can both be checked in an operation - sensitive manner using the techniques we have presented before . we now give the main result in our work . theorem 1 consider a collection of m sets s 1 , . . . , s m and let m = σ i = 1 m | s i | and 0 ≦ ε ≦ 1 . for a query operation involving t sets , let n be the sum of the sizes of the involved sets , and δ be the answer size . then there exists an ads scheme ={ genkey , setup , update , refresh , query , verify } for a sets collection data structure d with the following properties : ( 1 ) is correct and secure according to definitions 2 and 3 and based on the bilinear q - strong diffie - hellman assumption ; ( 2 ) the access complexity of algorithm ( i ) genkey is o ( 1 ); ( ii ) setup is o ( m + m ); ( iii ) update is o ( 1 ) outputting information upd of o ( 1 ) group complexity ; ( iv ) refresh is o ( 1 ); ( 3 ) for all queries q ( intersection / union / subset / difference ), constructing the proof with algorithm query has o ( n log 2 n log log n + tm ε log m ) access complexity , algorithm verify has o ( t + δ ) access complexity and the proof π ( q ) has o ( t + δ ) group complexity ; ( 4 ) the group complexity of the authenticated data structure auth ( d ) is o ( m + m ). in this section we give an overview of the security analysis of our ads scheme , describe how it can be employed to provide verificatio protocols in the three - party [ 36 ]( fig2 ) and two - party [ 30 ] authentication models ( fig1 ), and finall discuss some concrete applications . security proof sketch . we provide some key elements of the security of our verificatio protocols focusing on set intersection queries . the security proofs of the other set operations share similar ideas . let d 0 be a sets collection data structure consisting of m sets s 1 , s 2 , . . . , s m , 14 and consider our ads scheme ={ genkey , setup , update , refresh , query , verify }. let k be the security parameter and let { sk , pk }← genkey ( 1 k ). the adversary is given the public key pk , namely { h (•), p , e , g , g s , . . . g s q }, and unlimited access to all the algorithms of , except for setup and update to which he only has oracle access . the adversary initially outputs the authenticated data structure auth ( d 0 ) and the digest d 0 , through an oracle call to algorithm setup . then the adversary picks a polynomial number of updates μ t ( e . g ., insertion of an element χ into a set s r ) and outputs the data structure d i , the authenticated data structure auth ( d i ) and the digest d i through oracle access to update . then he picks a set of indices q ={ 1 , 2 , . . . , t }( wlog ), all between 1 and m and outputs a proof π ( q ) and an answer ≠ s 1 ∩ s 2 ∩ . . . ∩ s t which is rejected by check as incorrect . suppose the answer α ( q ) contains d elements . the proof π ( q ) contains ( i ) some coefficient b 0 , b 1 , . . . , b d ; ( ii ) some accumulation values acc j with some respective correctness proofs π j , for j = 1 , . . . , t ; ( iii ) some subset witnesses w j with some completeness witnesses f j , for j = 1 , . . . , t ( this is , what algorithm verify expects for input ). 14 note here that since the sets are picked by the adversary , we have to make sure that no element in any set is equal to s , the trapdoor of the scheme ( see definitio of the bilinear - map accumulator domain ). however , this event occurs with negligible probability since the sizes of the sets are polynomially - bounded and s is chosen at random from a domain of exponential size . suppose verify accepts . then : ( i ) by lemma 3 , b 0 , b 1 , . . . , b d are indeed the coefficient of the polynomial π χεχ ( χ + s ) , except with negligible probability ; ( ii ) by lemma 4 , values acc j are indeed the accumulation values of sets s j , except with negligible probability ; ( iii ) by lemma 1 , values w j are indeed the subset witnesses for set ( with reference to s j ), i . e ., w j = g p j ( s ) percent with negligible probability ; ( iv ) however , p 1 ( s ), p 2 ( s ), . . . , p t ( s ) are not coprime since is incorrect and therefore cannot contain all the elements of the intersection . thus the polynomials p 1 ( s ), p 2 ( s ), . . . , p t ( s ) ( equation 8 ) have at least one common factor , say ( r + s ) and it holds that p j ( s )=( r + s ) q j ( s ) for some polynomials q j ( s ) ( computable in polynomial time ), for all j = 1 , . . . , t . by the verificatio of equation 10 ( completeness condition ), we have therefore we can derive an ( r + s )- th root of e ( g , g ) as this means that if the intersection is incorrect and all the verificatio tests are satisfied we can derive a polynomial - time algorithm that outputs a bilinear q - strong diffie - hellma challenge ( r , e ( g , g ) 1 /( r + s ) ) for an element r that is a common factor of the polynomials p 1 ( s ), p 2 ( s ), . . . , p t ( s ), which by assumption 1 happens with probability neg ( k ). this concludes an outline of the proof strategy for the case of intersection . protocols . as mentioned in the introduction , our ads scheme can be used by a verificatio protocol in the three - party model [ 36 ]( see fig2 ). here , referring to fig5 , a trusted entity , called source , owns a sets collection data structure d h , but desires to outsource query answering , in a trustworthy ( verifiable way . as shown in fig6 , the source runs genkey and setup and outputs the authenticated data structure auth ( d h ) along with the digest d h . the source subsequently signs the digest d h , and it outsources auth ( d h ), d h , the digest d h and its signature to some untrusted entities , called servers as shown in fig7 . on input a data structure query q ( e . g ., an intersection query ) sent by clients , the servers use auth ( d h ) and d h to compute proofs π ( q ), by running algorithm query , and they return to the clients π ( q ) and the signature on d h along with the answer α ( q ) to q ( see fig8 and 9 ). clients can verify these proofs π ( q ) by running algorithm verify ( since they have access to the signature of d h , they can verify that d h is authentic ). when there is is an update in the data structure ( issued by the source ), the source uses algorithm update to produce the new digest d ′ h to be used in next verifications while the servers update the authenticated data structure through refresh . additionally , our ads scheme can also be used by a non - interactive verificatio protocol in the two - party model [ 30 ] as shown in fig2 . in this case , the source and the client coincide , i . e ., the client issues both the updates and the queries , and it is required to keep only constant state , i . e ., the digest of the authenticated data structure . whenever there is an update by the client , the client retrieves a verifiable constant - size portion of the authenticated data structure that is used for locally performing the update and for computing the new local state , i . e ., the new digest . a non - interactive two - party protocol that uses an ads scheme for a data structure d is directly comparable with the recent protocols for verifiabl computing [ 1 , 12 , 16 ] for the functionalities offered by the data structure d , e . g ., computation of intersection , union , etc . due to space limitations , we defer the detailed description of these protocols to the full version of the paper . furthermore , our ads scheme can also be used by a non - interactive verificatio protocol in the multi - party model [ 30 ] as shown in fig3 . that is , where there is more than one trusted source . in this instance , the multiple sources must synchronize together whenever there is an update by one of them to the server in order that they maintain a consistent collection of sets and can produce a digest that is verifiabl across all sets . applications . first of all , our scheme can be used to verify keyword - search queries implemented by the inverted index data structure [ 4 ]: each term in the dictionary corresponds to a set in our sets collection data structure which contains all the documents that include this term . a usual text query for terms m 1 and m 2 returns those documents that are included in both the sets that are represented by m 1 and m 2 , i . e ., their intersection . moreover , the derived authenticated inverted index can be efficientl updated as well . however , sometimes in keyword searches ( e . g ., keyword searches in the email inbox ) it is desirable to introduce a “ second ” dimension : for example , a query could be “ return emails that contain terms m 1 and m 2 and which were received between time t 1 and t 2 ”, where t 1 & lt ; t 2 . we call this variant a timestamped keyword - search , which is shown in fig4 . one solution for verifying such queries could be to embed a timestamp in the documents ( e . g ., each email message ) and have the client do the filterin locally , after he has verified — usin our scheme — the intersection of the sets that correspond to terms m 1 and m 2 . however , this approach is not operation - sensitive : the intersection can be bigger than the set output after the local filtering making this solution inefficient to overcome this inefficien y , we can use a segment - tree data structure [ 35 ], verifying in this way timestamped keyword - search queries efficientl with o ( t log r + δ ) complexity , where r is the total number of timestamps we are supporting . this involves building a binary tree t on top of sets of messages sent at certain timestamps and requiring each internal node of t be the union of messages stored in its children . finally , our method can be used for verifying equi - join queries over relational tables , which boil down to set intersections . in this paper , we presented an authenticated data structure for the optimal verificatio of set operations . the achieved efficien y is mainly due to new , extended security properties of accumulators based on pairing - based cryptography . our solution provides two important properties , namely public verifiability and dynamic updates , as opposed to existing protocols in the verifiabl computing model that provide generality and secrecy , but verifiability in a static , secret - key setting only . a natural question to ask is whether outsourced verifiabl computations with secrecy and efficient dynamic updates are feasible . analogously , it is interesting to explore whether other specifi functionalities ( beyond set operations ) can be optimally and publicly verified finally , according to a recently proposed definitio of optimality [ 33 ], our construction is nearly optimal : verificatio and updates are optimal , but not queries . it is interesting to explore whether an optimal authenticated sets collection data structure exists , i . e ., one that asymptotically matches the bounds of the plain sets collection data structure , reducing the query time from o ( n log 2 n ) to o ( n ). it would be appreciated by those skilled in the art that various changes and modification can be made to the illustrated embodiments without departing from the spirit of the present invention . all such modification and changes are intended to be within the scope of the present invention except as limited by the scope of the appended claims . b . applebaum , y . ishai , and e . kushilevitz . from secrecy to soundness : efficien verificatio via secure computation . in int . colloquium on automata , languages and programming ( icalp ), pp . 152 - 163 , 2010 . m . j . atallah , y . cho , and a . kundu . efficien data authentication in an environment of untrusted third - party distributors . in int . conference on data engineering ( icde ), pp . 696 - 704 , 2008 . m . h . au , p . p . tsang , w . susilo , and y . mu . dynamic universal accumulators for ddh groups and their application to attribute - based anonymous credential systems . in rsa , cryptographers &# 39 ; 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