A Capacity-Achieving T -PIR Scheme Based On MDS Array Codes

Transcription

1 A Capacty-Achevng T -PIR Scheme Based On MDS Array Codes Jnge Xu, Yaqan Zhang, Zhfang Zhang KLMM, Academy of Mathematcs and Systems Scence, Chnese Academy of Scences, Bejng , Chna School of Mathematcal Scences, Unversty of Chnese Academy of Scences, Bejng , Chna Emals: xujnge14@mals.ucas.edu.cn, zhangyaqan15@mals.ucas.ac.cn, zfz@amss.ac.cn arxv: v1 [cs.it] 17 Jan 2019 Abstract Suppose a database contanng M records s replcated n each of N servers, and a user wants to prvately retreve one record by accessng the servers such that dentty of the retreved record s secret aganst any up to T servers. A scheme desgned for ths purpose s called a T -prvate nformaton retreval (T -PIR) scheme. In ths paper we focus on the feld sze of T -PIR schemes. We desgn a general capacty-achevng T -PIR scheme whose queres are generated by usng some MDS array codes. It only requres feld sze q l N, where l = mn{t M 2,(n t) M 2 }, t = T/gcd(N,T), n = N/gcd(N,T) and has the optmal sub-pacetzaton Nn M 2. Comparng wth exstng capacty-achevng T -PIR schemes, our scheme has the followng advantage, that s, ts feld sze monotoncally decreases as the number of records M grows. In partcular, the bnary feld s suffcent for buldng a capacty-achevng T-PIR scheme as long as M 2+ log µ log 2 N, where µ = mn{t,n t} > 1. I. INTRODUCTION Prvate nformaton retreval (PIR) s a canoncal problem n the study of prvacy ssues that arse from the retreval of nformaton from publc databases. Typcally, a PIR model nvolves a database contanng M records stored across N servers and a user who wants to prvately retreve one record by accessng the servers. Specfcally, the prvacy requrement means any colludng subset contanng no more than T servers nows nothng about dentty of the retreved record. Snce t s closely related to cryptography [1] and codng theory [2], PIR has become a central research topc n the computer scence lterature snce t was frst ntroduced by Chor et al. [3] n The effcency of PIR scheme s characterzed by ts rate. Specfcally, the rate of a PIR scheme s measured as the rato between the retreved data sze and the downloaded sze, and the capacty s defned as the supremum of the rate over all PIR schemes. Recently, much wor has been done on determnng the capacty of PIR n varous cases. Sun and Jafar derved that the capacty for the non-colludng servers (.e., T = 1) s 1 1/N n [4] and further proved that the capacty for the 1 (1/N) M colludng servers (.e., T > 1) s the latter s called T -PIR. They also determned the capacty of PIR wth symmetrc prvacy n [6]. The capacty of PIR wth MDS coded non-colludng servers s determned n [7]. It remans an open problem to determne the capacty of PIR wth MDS coded colludng servers. For non-mds coded 1 T/N n [5]. Moreover, 1 (T/N) M storage, PIR schemes wth colludng or non-colludng servers are presented n [9], [10]. In general, the capacty of PIR s acheved by dvdng each record nto multple sub-pacets and queryng from each server specally desgned combnatons of these subpacets. Therefore, both the number of sub-pacets and the sze of each sub-pacet are mportant metrcs for measurng the mplementaton complexty of a PIR scheme. As to the former, we call the number of sub-pacets contaned n each record as sub-pacetzaton. The optmal sub-pacetzaton for capacty-achevng PIR schemes has been determned n some cases [8], [12], [13]. As to the latter, snce all exstng PIR schemes are lnear schemes over some fnte felds, t s actually about the sze of the feld on whch the PIR scheme can be bult. The man concern of ths wor s to reduce the feld sze for T -PIR schemes whle mantanng the rate achevng the capacty and the optmal sub-pacetzaton. In [5], t requres a feld of sze q = Ω(N 2 T M 2 ) for the capacty-achevng T -PIR scheme. The feld sze s reduced to q = Ω(Nt M 2 ) for the capacty-achevngt -PIR scheme wth optmal sub-pacetzaton n [12], where t = T/gcd(N,T The best nown result of feld sze for capacty-achevng T - PIR scheme s q = Ω(N) n [14]. But the feld sze s stll unfrendly wth the growth of the number of servers. The man contrbuton of ths wor conssts of desgnng a T -PIR scheme that smultaneously acheves the capacty and the optmal sub-pacetzatonnn M 2 over a fnte feld F q for all possble parameters (N,T,M), and t requres the feld sze q l N, where l = mn{t M 2,(n t) M 2 },n = N/gcd(N,T),t = T/gcd(N,T When l = 1, the constrant of the feld sze n our scheme degenerates nto q N, whch s the same wth that of the capacty-achevng T -PIR scheme n [14]. When l > 1, the bnary feld s suffcent for buldng a capacty-achevng T -PIR scheme provded M 2+ log µ log 2 N, where µ = mn{t,n t}. Reference Feld sze(q) Sun et al. [5] q max{n 2 T M 2,N 2 (N T) M 2 } Zhang et al. [11] q max{nnt M 2,Nn(n t) M 2 } Zhang et al. [12] q max{nt M 2,N(n t) M 2 } Xu et al. [14] q N Ths paper q l N,l = mn{t M 2,(n t) M 2 } Table 1: A lst of all exstng capacty-achevng T -PIR schemes wth T 2. And N n = gcd(n,t),t = T gcd(n,t).

2 Comparng wth all exstng capacty-achevng T -PIR schemes wth T 2 n [5], [11], [12], [14], as dsplayed n Table 1, the man dfference n our scheme s to employ MDS array codes to generate queres, whch s a ey dea for reducng the feld sze. Moreover, an advantage of our scheme s that ts feld sze monotoncally decreases as the number of records M grows. The rest of ths paper s organzed as follows. Frst, the T - PIR model s formally ntroduced and the MDS array code s defned n Secton II. Then n Secton III an example of the T -PIR scheme s presented to explan the desgn dea. The recovery property of MDS array codes s proved and the general descrptons of our scheme are gven n Secton IV. Fnally, Secton V concludes the paper. II. PRELIMINARIES A. Notatons and the T -PIR model For an nteger n N, we denote by [n] the set {1,..., n}. For a vector u = (u 1,...,u n ) and a subset Γ = { 1,..., m } [n], denote u Γ = (u 1,...,u m Most vectors n ths paper are row vectors and they are denoted by the bold lowercase letters (eg. a,b For a bloc matrx A = (A (1),A (2),...,A (N) ) and Γ = { 1,..., m } [N], denote A Γ = (A (1),...,A (m) Suppose there are M records W 1,...,W M and N servers Serv (1),...,Serv (N), each server stores all the M records. Moreover, the records are ndependent and each can be seen as an L-length vector over F q. Then suppose a user wants to prvately retreve W θ for some θ [M]. Formally, a T -PIR scheme conssts of two phases: Query phase. Gven θ [M], the user generates the query Que(θ,S) = (Q (1) θ,...,q(n) θ ), and sends Q (j) θ to Serv (j) for1 j N, wheres are some random resources prvately chosen by the user. Note that Que(, ) s the query functon defned by the scheme. Response phase. After recevng Q (j) θ, the Serv(j) computes the answers Ans (j) (Q (j) θ,w [M]) = A (j) θ for 1 j N, and sends t bac to the user, where Ans (j) (, ) s the answer functon defned by the scheme. Moreover the functons Que(, ) and Ans (j) (, ),1 j N must satsfy the followng two condtons: (1) Correctness: The user can reconstruct W θ after collectng all answers from the N servers,.e., H(W θ A [N] θ,q[n] θ,s,θ)=0, where H( ) s the condtonal entropy. (2) Prvacy: For any Γ [N] wth Γ =T, the serves n Γ can t obtan any nformaton on θ even f they collude wth each other,.e., I(θ;Q Γ θ,a Γ θ,w [M] ) = 0, where I( ; ) denotes the mutual nformaton. Defne the rate R of a T -PIR scheme by R = H(W θ ) N =1 H(A() θ ), that s, R characterzes the amount of retreved nformaton per unt of downloaded data. Furthermore, the capacty of T -PIR s defned by the largest rate over all achevable T - PIR schemes, denoted by C T -PIR. By [5], t has that C T -PIR = 1 T/N 1 (T/N) M. B. MDS Array Codes In ths secton we ntroduce MDS array code used n ths paper and then gve a method to construct such code over F q. Suppose N > T 1 and N,T are two postve ntegers. For a lnear [N,T] code C over F q l, a codeword c = (c 1,c 2,...,c N ) can be seen as an Nl-length vector c = (c 1,c 2,...,c N ) over F q,.e., for [N], the code bloc c = (c,1,c,2,...,c,l ) F l q denotes the l-length vector correspondng to the symbol c F q l. So we call the code C a lnear array code overf q, and refer to the code as an(n,t;l) q lnear array code. Equvalently, an (N,T;l) q lnear array code can be defned by a Tl Nl full ran matrx G over F q as follows, C = {c = (m 1,m 2,...,m T )G : (m 1,m 2,...,m T ) F Tl q }. The matrx G s called a generator matrx of the array code C. Then the generator matrx G can be vewed as a bloc matrx G = (G (1) For [N], the Tl l sub-matrx G () s represented as the thc column assocated wth the th code bloc n the codewords of C. Defnton 1. (MDS Array Codes) A lnear array code C over F q s called an (N,T;l) MDS array code f ts generator matrx G = (G (1) ) Fq Tl Nl has the followng MDS property: Γ [N] wth Γ = T,ran(G Γ ) = Tl (1) where G () F Tl l q for [N] and T < N. By the defnton of (N,T;l) MDS array code C, t degenerates nto a MDS code over F q for l = 1. Next we gve a method to construct an (N,T;l) MDS array code. Suppose α s a prmtve element of F q l, then F q l = {α j : 0 j q l 2} {0}. Suppose m(x) s the mnmal polynomal of α over F q. Let C F l l q be the companon matrx of m(x) and F = {C j : j Z} {0}. Then F s a fnte feld of sze q l and the map whch s defned by ϕ(α j ) = C j and ϕ(0) = 0 s a feld somorphsm from F q l to F by [15]. Let G = (α,j) F T N be a generator matrx of an q l [N,T] MDS code over F q l. Note that each symbol of F q l can be represented as an l l matrx n F over F q by usng the feld somorphsm ϕ( ), then the matrx G can be seen as an T N bloc matrx G,.e., G = (ϕ(α,j )) [T],j [N], and each thc column s an Tl l matrx over F q. It s easy to verfy that for any Γ [N] wth Γ = T, det((ϕ(α,j )) [T],j Γ ) = ϕ(det((α,j ) [T],j Γ )) 0. Hence the lnear array code whch s defned by the generator matrx G over F q s an (N,T;l) MDS array code. Then we can drectly obtan the followng theorem.

3 Theorem 2. Supposeq s a power of a prme and N,T,l N wth N > T 2. If q l N, then there exsts an (N,T;l) MDS array code over F q. Recall that for exstng capacty-achevng T -PIR schemes n [5], [11], [12], some [Nl,Tl ] MDS codes over F q are used to construct the query. And all Nl symbols of each codeword are equally dvded nto N blocs. Then ths MDS code can be seen as an (N,T;l ) MDS array code over F q. Based on ths observaton, we fnd a drecton to reduce the feld sze. That s, we generate the query by usng some MDS array codes over a smaller fnte feld rather than MDS codes. Moreover, the MDS array codes need to satsfy some specal property that s determned by the correctness condton. To formally llustrate ths dea, we wll gve an example n the next secton. III. EXAMPLE FOR N = 5,T = 3,M = 3 Before constructng our schemes, we frst gve an example by usng the method descrbed n [12]. And then, we explan how to reduce the feld sze by modfyng ths scheme. Example 1. Suppose M = 3, N = 5 and T = 3. The feld sze q 15 s enough and the sub-pacetzaton of ths case s L = N M 1 = 25, so each record can be seen as a 25- dmensonal vector over F q,.e., W 1,W 2,W 3 F 25 q. WLOG, suppose W 1 s the desred record,.e., θ = 1. Let S 1,S 2,S 3 be three matrces chosen by the user ndependently and unformly from all nvertble matrces over F q. Actually, S 1,S 2 and S 3 are the random resources prvately held by the user. Then, defne (a 1,a 2,...,a 25 ) = W 1 S 1 (b 1,b 2,...,b 25 ) = W 2 S 2 [:,(1 : 15)]G (c 1,c 2,...,c 25 ) = W 3 S 3 [:,(1 : 15)]G where S [:,(1 : 15)] denotes the 25 ( 15 matrx ) formed by G1 0 the frst 15 columns of S and G = 0 G. Moreover, 2 G 1 F 9 15 q s a generator matrx of an [15,9] MDS code and G 2 Fq 6 10 s a generator matrx of an [10,6] MDS code over F q. It can be seen that the answers are all sums of the symbols a,b,c n Fg.1. For each sum x n Fg.1, we defne ts support as a subset of [3] and ths subset s composed of the label of all terms n the sum x, denoted by supp(x For example, supp(a ) = {1},supp(b + c j ) = {2,3}. For any Λ [3], a sum x n Fg.1 s called an Λ-type sum f Λ = supp(x For Λ [M] {θ}, defne Λ = Λ {θ} and call Λ-type sums as nterference. Let γ () be the number of Λ-type sums n Serv () for each -subset Λ [3] and [5]. Now we show the scheme satsfes the correctness condton and the prvacy condton. Recall the suffcent condtons for the correctness (s1) requrement n [12], that s, for any Λ [M] {θ}, the nterference parts of all Λ-type sums can be lnearly expressed by the Λ-type sums whch appears n all servers. For Λ [M] {θ}, we collect all Λ-type sums and the nterference parts of all Λ-type sums to form a matrx and (2) Serv (1) Serv (2) Serv (3) Serv (4) Serv (5) a 1 a 4 a 7 a 10,a 11,a 12 a 13,a 14,a 15 b 1 b 4 b 7 b 10,b 11,b 12 b 13,b 14,b 15 c 1 c 4 c 7 c 10,c 11,c 12 c 13,c 14,c 15 a 2 +b 2 a 5 +b 5 a 8 +b 8 a 3 +b 3 a 6 +b 6 a 9 +b 9 a 16 +c 2 a 18 +c 5 a 20 +c 14 a 17 +c 3 a 19 +c 6 a 21 +c 15 b 16 +c 16 b 18 +c 18 b 20 +c 20 b 17 +c 17 b 19 +c 19 b 21 +c 21 a 22 +b 22 +c 22 a 23 +b 23 +c 23 a 24 +b 24 +c 24 a 25 +b 25 +c 25 Fg. 1: Answers of the (N = 5,T = 3,M = 3) PIR scheme for retrevng W 1. call ths matrx as the dstrbuton matrx of Λ-type sums. For example, for {2}-type sums {b }, ts dstrbuton matrx has the followng form, b 1 b 4 b 7 b 10 b 13 b 2 b 5 b 8 b 11 b 14. (3) b 3 b 6 b 9 b 12 b 15 where the bold symbols are all {2}-type sums and the rest are the nterference parts of all {2, 3}-type sums. Smlarly, the dstrbuton matrx of {2,3}-type sums {b +c j } s ( ) b16 +c 16 b 18 +c 18 b 20 +c 20 b 22 +c 22 b 24 +c 24 b 17 +c 17 b 19 +c 19 b 21 +c 21 b 23 +c 23 b 25 +c 25. (4) Then by the MDS property of G 1 and G 2, the coordnates labeled by the bold symbols n (3) and (4) form an nformaton set of G 1 and G 2, respectvely. That s, the rest symbols can be recovered by the bold symbols n (3) and (4), respectvely. Note that for any -subset Λ [M] {θ}, the dstrbuton matrx of Λ-type sums n Fg.1 has a smlar form, as the matrx (3) or (4), so the nterference parts of all Λ-type sums can be recovered by all Λ-type sums appeared n 5 columns. Hence ths scheme satsfes the condton (s1) n [12],.e, the correctness condton s guaranteed. As to the prvacy, recall the suffcent condtons for the prvacy (s2) requrement n [12], t s suffcent to ensure that for any Λ [M] {θ}, there are the same number of ndependent symbols contaned n any 3 columns of Λ-type sums dstrbuton matrx (.e., (3) or (4) Actually ths s guaranteed by the MDS property of the lnear code whch s used to generate such type nterference. Thus the prvacy condton s guaranteed. Moreover the desred record conssts of 25 symbols whle the answers totally contan 49 symbols, so the scheme has rate attanng the capacty for ths case The feld sze reles on the maxmum length of the MDS codes used n ths scheme, so t requresq 15 n Example 1. Note that for any Λ-type nterference, f ts dstrbuton matrx s a codeword of some MDS array code, then there are also the same number of ndependent symbols contaned n any 3 columns of ts dstrbuton matrx. For example, suppose Λ = {2}, the matrx (3) can be vewed as a codeword of an

4 (5, 3; 3) MDS array code. Smlarly, the [10, 6] MDS code also can be vewed as an (5,3;2) MDS array code. So f we adopt (5,3;3) and (5,3;2) MDS array codes rather than [15,9] and [10, 6] MDS codes, then the new obtaned scheme also satsfes the T -prvacy condton. However, there s a problem that how to guarantee the correctness condton. So the MDS array codes have to satsfy some property determned by the correctness condton. More precsely, for any -subset Λ of [M] {θ} and MDS array code correspondng to the Λ-type sums, denoted by (5,3;l ), ts generator matrx needs to have the followng recovery property: (a1) for th thc column, there are γ () columns whch are (a2) used to generate the Λ-type sums, and the rest γ () +1 columns are used to generate Λ-type sums, that s, l = γ () +γ() +1. All these N =1 γ() = Tl. columns have full column ran, that s, N =1 γ() Now we gve two admssble matrces G 1 F , G 2 F That s, G 1 = , G 2 = Then one can verfy that the columns labeled by {1,4,7,10,11,12,13,14,15} n G 1 have full column ran and the columns labeled by {1,2,3,4,5,6} n G 2 also have full column ran. Hence G 1 and G 2 satsfy the recovery property. Actually, G1 s obtaned by applyng the method n Theorem 2 to a generator matrx of an [5, 3] Generalzed Reed-Solomon code over F 2 3 and rearrangng the order of columns n each thc column by multplyng some 3 3 permutaton matrx. Smlarly, G2 s obtaned by usng the same method to a generator matrx of a[5, 3] doubly-extended Generalzed Reed- Solomon code overf 2 2. Then the MDS property of G 1 and G 2 s also satsfed. Therefore, the new scheme obtaned by usng G to replace G n (2) s a capacty-achevng T -PIR scheme wth optmal sub-pacetzaton, where G = ) ( G1 0. Note 0 G 2 that the feld sze s reduced to 2. As dsplayed n the example, the man desgn dea behnd our scheme s to mae each (N,T;l ) MDS array code correspondng to Λ-type nterference n the scheme satsfy the recovery property for any -subset Λ of [M] {θ} and some fxed γ (1),...,γ(N). Fortunately, we prove that every MDS array code trvally satsfes the recovery property by Lemma 3 n Secton IV-A. IV. THE GENERAL T -PIR SCHEME BASED ON MDS ARRAY CODES In ths secton we frst characterze the recovery property of MDS array codes and then descrbe our general capactyachevng T -PIR scheme based on MDS array codes. A. The Recovery property of MDS array code Lemma 3. Suppose G = (G (1) ) Fq Tl Nl s a generator matrx of an (N,T;l) MDS array code over F q, where G () = (g () 1,...,g() l ) FTl l q s a Tl-length column vector. Then for any (m 1,...,m N ) {0,1,...,l} N and g () j wth N =1 m = Tl, there exst N subsets,..., of [l] wth Γ = m such that ran(g (1) ) = Tl. Proof. For any fxed (m 1,...,m N ) wth N =1 m = Tl, there exst at least T nonzero numbers of them. Wthout loss of generalty, we may assume that m 0,1 N. Because that f m = 0, the new matrx obtaned by deletng the thc bloc columng () s also a generator matrx of an(n 1,T;l) MDS array code. Let b(m 1,m 2,...,m N) = max ran(g (1) Γ,..., [l], 1 Γ =m,1 N. Then there exst N subsets,..., of [l] wth Γ = m such that b = ran(g (1) Choose a maxmum lnearly ndependent subset of the vectors {G (1) }, denoted by {G () : [N]}, then b = ran(g (1) ), where Γ for 1 2 N [N] and N =1 Γ(1) = b. Then t s suffcent to show that b = Tl. On the contrary, we assume that b < T l. Let V = Cspan(G (1) ), where Cspan( ) denotes the lnear space spanned by all columns of the matrx over F q. Then dmv = b < Tl and {G () : [N]} s a base of the vector space V. To derve a contradcton, we assume the followng clam has been proved. Clam : f b < Tl, then for 1 f T, there exst f dsjont nonempty subsets X 1,...,X f of [N] such that u f =1 X, j [l],g (u) Partcularly, let f = T. Then t follows from the Clam that there exst T dsjont nonempty subsets X 1,...,X T of [N] such that u T =1 X, j [l],g (u) Hence ran(g (u) : u T =1 X ) dmv = b. On the other hand, note that T =1 X = T =1 X T. Combnng wth the MDS property of G, then ran(g (u) : u T =1 X ) = Tl. So one can obtan that Tl b < Tl, a contradcton. To complete the proof, t remans to prove the Clam. Now we prove t by nducton on f. For f = 1, let X 1 = { [N] : Γ }. Snce b < Tl, then X 1 1. For any u X 1, t s suffcent to show that

5 for j [l] Γ u, g (u) j V. Then choosng a m u -subset Γ u of [l] such that u {j} Γ u, one can obtan that V Cspan(G (1) Γ,G(u+1) u Γ u+1 By the defnton of b(m 1,m 2,...,m N ), t holds that ran(g (1) Γ,G(u+1) u Γ u+1 ) b = dmv, whch mples that V = Cspan(G (1) Γ,G(u+1) u Γ u+1 Hence, g (u) Suppose that there exst f 1 dsjont nonempty subsets X 1,...,X f 1 of [N] such that f 1 u X, j [l],g (u) =1 Consder the case f, note that ran(g (u) : u f 1 =1 X ) dmv = b < Tl. By the MDS property (1) of G, then the f 1 =1 X < T, whch mples that the vectors {g (u) j : u f 1 =1 X,j [l]} are lnearly ndependent over F q, so are the vectors {g (u) j : u f 1 =1 X,j Γ u }. Therefore the vectors can extend to be a base of V. Then there exst Γ (f) Γ for 1 N and Γ (f) u = Γ u for u f 1 =1 X such that {G () : [N]} s a base of the vector space V. Let X Γ (f) f = { [N] : Γ (f) Γ }. Then X f, otherwse dmv = N =1 Γ = Tl. By the defnton of X f, one can obtan that X f ( f 1 =1 X) =, that s, such f subsets X, [f] are dsjont. Smlarly, by usng the same way n the case f = 1, one can obtan that u X f, j [l],g (u) Remar 1. Usng the notatons ntroduced above, we may assume that for any fxed (m 1,m 2,...,m N ) wth N =1 m = Tl, Γ = {1,2,...,m }, [N] n a generator matrx of the (N,T;l) MDS array code. Ths s because that we can rearrange the order of l columns n each thc column by multplyng some l l permutaton matrx. B. Formal Descrpton of the general scheme Our scheme can be obtaned by modfyng the capactyachevng T -PIR schemes n [12]. As n Example 1, we replace M 1 MDS codes wth some M 1 MDS array codes. Next we gve these M 1 desred MDS array codes. Specally, for 1 M 1, the th MDS code defned by the generator matrx G n [12] has the parameters [ N T (Tα + (N T)β ),Tα + (N T)β ] over F q, where α,β are defned as n the denttes (35),(36) n [12]. Note that Tα + (N T)β = T(n t) 1 t M 1, and defne l = (n t) 1 t M 1, where t = T/gcd(N,T), n = N/gcd(N,T Then for 1 M 1, the th MDS code can be vewed as an (N,T;l ) MDS array code. By Lemma 3, one can choose a generator matrx G of an (N,T;l ) MDS array code whch has the recovery property for (γ (1),...,γ(N) ), where γ () = α for 1 T and γ () = β for T +1 N. Then these M 1 matrces G are desred. One can verfy that the new scheme satsfes the correctness condton and T -prvacy condton, whch are guaranteed by the recovery property and MDS property of all M 1 MDS array codes, respectvely. Moreover, the new scheme doesn t change the sub-pacetzaton of records and download sze. Therefore the new scheme has the hghest rate and the optmal sub-pacetzaton. Note that there are M 1 MDS array codes used n our scheme over F q, by Theorem 2 t only needs to requres that for 1 M 1, q l N. That s, q l N, where l = mn{t M 2,(n t) M 2 }. V. CONCLUSION In ths paper we buld a general capacty-achevng T -PIR scheme based on MDS array codes overf q, that s, the queres are generated by usng M 1 MDS array codes rather than MDS codes. It requres the feld sze q l N and has optmal sub-pacetzaton. In partcular, the bnary feld s enough to buld our scheme as long as M 2+ log µ log 2 N, where µ = mn{t,n t} > 1. REFERENCES [1] A. Bemel, Y. Isha, E. Kushlevtz, and I. Orlov, Share converson and prvate nformaton retreval, n Proc. 27th Annu. Conf. Comput. Complex., pp , Jun [2] S. Yehann, Locally Decodable Codes and Prvate Informaton Retreval Schemes, Ph.D. dssertaton, Massachusetts Insttute of Technology, [3] B. Chor, E. Kushlevtz, O. Goldrech, M. Sudan, Prvate nformaton retreval, Proc. 36-th IEEE Symposum on Foundatons of Computer Scence, pp.41 50, [4] H. Sun, S. A. Jafar, The capacty of prvate nformaton retreval, IEEE Trans. on Inf. Theory, vol.63, no.7, pp , Jul [5] H. Sun, S. A. Jafar, The capacty of prvate nformaton retreval wth colludng databases, IEEE Global Conference on Sgnal and Informaton Processng (GlobalSIP), pp , [6] H. Sun, S. A. Jafar, The Capacty of Symmetrc Prvate Informaton Retreval, IEEE Globecom Worshops (GC Wshps), pp.1 5, [7] K. Banawan and S. Uluus, The capacty of prvate nformaton retreval from coded databases, IEEE Trans Inf Theory, vol.64 no.3, pp , Mar [8] H. Sun, S. A. Jafar, Optmal Download Cost of Prvate Informaton Retreval for Arbtrary Message Length, IEEE Transactons on Informaton Forenscs and Securty, vol.12, no.12, pp ,2017 [9] H.-Y. Ln, S. Kumar, E. Rosnes, and A. Graell Amat, An MDS- PIR Capacty-Achevng Protocol for Dstrbuted Storage Usng Non- MDS Lnear Codes, Proceedngs of IEEE Internatonal Symposum on Informaton Theory (ISIT), July 2018, pp [10] R. Frej-Hollant, O. W. Gnle, C.Hollant, et al. t-prvate Informaton Retreval Schemes Usng Transtve Codes, IEEE Transactons on Informaton Theorey, DOI /TIT [11] Ywe Zhang and Gennan Ge, A general prvate nformaton retreval scheme for MDS coded databases wth colludng servers. arxv: [12] Zhfang Zhang and Jnge Xu, The Optmal Sub-Pacetzaton of Lnear Capacty-Achevng PIR Schemes wth Colludng Servers, IEEE Transacton on Informaton Theory, DOI /TIT [13] Jnge Xu and Zhfang Zhang, On sub-pacetzaton and access number of capacty-achevng PIR schemes for MDS coded noncolludng servers, Scence Chna Informaton Scence, 2018, Vol. 61 (10), pp [14] Jnge Xu and Zhfang Zhang, Buldng Capacty-Achevng PIR Schemes wth Optmal Sub-Pacetzaton over Small Felds, Proceedngs of IEEE Internatonal Symposum on Informaton Theory (ISIT), July 2018, pp [15] R. Ldl and H. Nederreter, Introducton to fnte felds and ther applcatons, Cambrdge Unversty Press, Cambrdge, UK (1994