Explicit combinatorial design

Explicit com binatorial design Xiongfeng Ma, 1, 2 , ∗ Zhen Zhang, 1 , † and Xiao qing T an 3, 2 , ‡ 1 Center for Quantum Information, Institute for Inter disciplinary Information Scien c es, Tsinghua University, Beijing, China 2 Center for Quantum Inf o rmation and Quantum C ontr ol, Dep artment of Physics and Dep artment of Ele ctric al & Co mputer Engin e ering, University of T or onto, T or onto, Ontario, Canada 3 Dep artment of Mathematics, Co l le ge of Information Scienc e and T e chnolo gy, Jinan Univers i ty, Guangzho u, Guangdong, P. R. China Abstract A com b inatorial design is a family of sets that are almost disj oin t, which is ap p lied in pseud o random n umb er generat ions and randomness extrac tions. Th e parameter, ρ , quan tifying the o ver- lap b et ween the sets within the family , is directly related to the length of a random seed needed and the efficiency of an extracto r . Nisan and Wigderson prop osed an explicit construction of designs in 1994. Later in 200 3, Hartman and Raz pro ve d a b ound of ρ ≤ e 2 for the Nisan-Wigderson construction in a limited p arameter regime. In this work, w e prov e a tigh ter b oun d of ρ < e w ith the entire parameter range b y sligh tly refining the Nisan-Wigderson construction. F ollo wing the blo c k idea used by Raz, Reingold, and V adhan, we presen t an exp licit weak design w ith ρ = 1. ∗ xma@tsinghua.edu.cn † zzhang12@ mails.tsinghua.edu.cn ‡ ttanxq@jnu.edu.cn 1 I. INTR ODUCT ION Com binatorial designs play an imp orta nt ro le in pseudo random n umber generations [1] and randomness extractions [2]. Nisan and Wigderson prop ose a simple construction of designs (Nisan-Wigderson design) fo r pseudo random num b er generators [1], whic h is later applied to construct randomness extractors b y T revisan [2]. A com binato rial design is a family of subsets, drawn from t he set, whic h ha ve a same size, q , and a re almost disjoin t. F o r a family of disjoin t subsets, the size of the set, l , grow s linearly with t he n um b er of subsets, n . La t er, we will see that with a design, the size of the set only grow s as pol y (log n ). One k ey parameter of a design, ρ , is used to quan tify the ov erlap b etw een subsets in the f a mily . Generally speaking, the smaller ρ is, the more disjoin t the subsets are. This parameter is link ed to the seed length and approx imat ely indicates the r a tio of randomness that can b e extracted by T revisan’s extractor [2 , 3]. In the application of extractors, ρ is normally required to b e close to 1. F ur t hermore, the size of the set, l , is linked to the initial randomness input (as seed) required for T revisan’s extractor. In g eneral, the size ( l ) should b e small compared to the num b er of subsets ( n ). Hartman and R az prov ed a b ound of ρ ≤ e 2 ( e as the Euler’s n um b er) for the Nisan- Wigderson design [4] when n is a p o we r o f a prime p ow er n umber, q (subset size). By sligh tly refining the Nisan-Wig derson design, w e pro ve a b etter bo und ρ < e for t he en tire range of n ≤ q q . F urthermore, we f o llo w the blo ck idea used by Raz, Reingold, and V adhan to construct a n explicit design with ρ = 1 and l = O (log 3 n ). In Section I I, w e review the definitions of com binat o rial designs, the Nisan-Wigderson design and the Hartman-R a z b o und. In Section I I I, w e refine the Nisan-Wigderson design and show a b etter b ound of ρ . In Section IV, w e construct an explicit ρ = 1 design. W e finally conclude with discussions in Section V. I I. PRELIMINARIES A. Notations and Definitions Notations: [ l ] = { 0 , 1 , 2 , . . . , l − 1 } ; log is base 2; ln is the natural logarithm; and e is the base of the natura l log arithm or t he Euler’s num b er. 2 Define a Galois (finite) field, GF ( q ) = [ q ] where q = p r , r is a p ositiv e in teger, and p is a prime. Here, we represen t an elemen t, j ∈ [ q ], b y a p - nary string. D efine F q to b e the ring of p olynomials ov er the field GF ( q ). F or a p olynomial φ ( x ) ∈ F q , denote λ ( φ ) to b e its n um b er of ro ots ov er GF ( q ). F or the sak e of simplicit y , we use p = 2 in the following. W e remark that our results a pply to the case of a general prime p with minor mo difications. Denote M q d +1 = { φ 0 , φ 1 , . . . , φ q d +1 − 1 } ⊆ F q to b e the set of all p o lynomials ov er GF ( q ) with the highest o rder no g reater than d ∈ [ q ], and hence, |M q d +1 | = q d +1 . W e further divide the set M q d +1 ev enly in to q disjoint subsets, N d,j with j ∈ GF ( q ), N d,j , { j x d + φ ( x ) | φ ( x ) ∈ M q d } . (1) That is, the co efficien t of x d of eac h p o lynomial in N d,j is j . It is not hard to see that M q d +1 = q − 1 [ j =0 N d,j , N d, 0 = M q d (2) and hence for ev ery j ∈ [ q ], |N d,j | = q d . (3) F or a p olynomial set, M , define a f unction, Λ( M ) , X φ ∈M 2 λ ( φ ) (4) In the summation on the r ig h t side, w e assume that the num b er of ro ots of the trivial p olynomial φ = 0 is zero. That is, for ev ery constant function φ , λ ( φ ≡ const) = 0 . (5) B. Designs A com binato rial design is a family (collection) of nearly disjoin t subsets of a set [ l ]. Here are the tw o definitions of designs used in the lit erature. Definition I I.1. (Standar d Design ) A family o f sets S 0 , S 1 , . . . , S n − 1 ⊆ [ l ] is a standar d ( n, q , l, ρ ) -desig n if 1. F or al l i ∈ [ n ] , | S i | = q . 3 2. F or al l i 6 = j ∈ [ n ] , | S i ∩ S j | ≤ log ρ. (6) Definition I I.2. (We ak des i g n) A family of sets S 0 , S 1 , . . . , S n − 1 ⊆ [ l ] is a we ak ( n, q , l , ρ ) - design if 1. F or al l i ∈ [ n ] , | S i | = q . 2. F or al l i ∈ [ n ] , X j | j ∈ GF ( q ) } (10) where < j, φ i ( j ) > presen ts an elemen t in [ l ]. The fo llo wing fa cts can b e easily verified [1]: 1. The size of each set is exactly q , | S i | = q for ev ery i ∈ [ q ]. 2. An y tw o sets in tersect in at most d p oints . 3. There a r e at least q d +1 p ossible sets (the n um b er of p o lynomials o n GF ( q ) of degree at most d ). In the original prop osal of the Nisan-Wigderson design, the p olynomials (with a degree at most d ) are c hosen in an arbitrary manner. A natural w ay to c ho ose these p olynomials is to go from low o rder p olynomials to higher ones, which results the highest order of p olynomials to b e d = ⌈ log n/ log q − 1 ⌉ ≤ log n . According to Definition I I.1, it is straigh tfor ward to see that ρ ≤ log n as sho wn by Nisan and Wigderson [1 ]. D. Hartman-Raz b ound Hartman and Ra z pro ve d that the Nisan-Wigderson design is an explicit mo dified w eak ( n, q , l, ρ )-design with l = q 2 and ρ ≤ e 2 in Theorem 1 of ref. [4]. W e remark that Hartman and Ra z’s result is only pro ven to for t he case when n is a p ow er o f q . 5 I I I. NEW BOUND In tuitiv ely , the more sets the design has, the harder to make sets disjoin t. Th us, one migh t conjecture that the pa r a meter ρ , defined in Eq. (7), grows with n . Mathematically , this is not necessarily true, b ecause the o verlap is normalized by n , as sho wn in Eq. ( 7). In fact, one can find coun ter examples to this conjecture for Nisan-Wigderson design. In the follow ing, w e presen t a new design construction b y sligh tly refining the original Nisan-Wigderson design. W e show that for an y n ≤ q q , one can obtain the upp er b ound ρ < (1 + q − 1 ) q , whic h show s that the refined Nisan-Wigderson design is an explicit weak ( n, q , l , ρ )-design with ρ < e (see, Theorem I I I.5). A. Refined Nisan-Wigderson design Here, we refine the Nisan-Wigderson design b y c ho osing the i th p olynomial fo r Eq. (10) in the following manner: φ i ( x ) = d X k =0 ( ⌊ i/q k ⌋ mo d q ) x k , (11) where i ∈ [ n ], d = ⌈ log n/ log q − 1 ⌉ (then, q d < n ≤ q d +1 ), and the co efficien ts calculated by the mo dulo function ( ⌊ i/q k ⌋ mod q ) are elemen ts of GF ( q ). These p olynomials form a set M n = { φ 0 , φ 1 , . . . , φ n − 1 } , (12) and by the definition of Eq. (1) , N d, 0 = { φ 0 , φ 1 , . . . , φ q d − 1 } ⊂ M n . (13) Eac h p olynomial, φ i , in M n corresp onds to a set S i in the design in the fo r m of Eq. (10). B. Ev aluation of ρ In the fo llowing discussion, we ev aluate the par a meter ρ in Eq. (8) fo r the design giv en b y Eq. (11). The num b er of interse ction elemen ts | S i ∩ S j | equals to the n um b er of ro ots of φ i = φ j or | S i ∩ S j | = λ ( φ i − φ j ) . (14) 6 Then, the left hand side of Eq. ( 8) can b e written as X j 0, k ∗ ≤ k ≤ d and 0 ≤ c k < a k , define a set A k ,c k = { a d x d + · · · + a k +1 x k +1 + c k x k + φ ( x ) | φ ( x ) ∈ N k , 0 } . (24) It is not hard to see that the p olynomial sets, A k ,c k , are disjoint for differen t v alues o f k and c k , and the M i +1 defined in Eq. (12) can b e partitioned b y M i +1 = d [ k = k ∗ a k − 1 [ c k =0 A k ,c k , (25) 8 where we use the fact tha t A k ,c k = ∅ when a k = 0. F or the last pa rtition, where k = k ∗ and c k ∗ = a k ∗ − 1, o ne can see that X φ j ∈A k ∗ ,c k ∗ 2 λ ( φ i − φ j ) = X φ j ′ ∈N k, 0 2 λ ( φ j ′ ) = Λ( N k , 0 ) , (26) F or an y other partitions, X φ j ∈A k,c k 2 λ ( φ i − φ j ) = X φ j ′ ∈N k, 1 2 λ ( φ j ′ ) = Λ( N k , 1 ) (27) where the first equalities in Eq. (26 ) and Eq. (27) come from the fact t ha t the co efficien ts of the highest d − k orders in φ i are the same as the ones in ev ery p olynomial φ j in A k ∗ ,c k ∗ or A k ,c k . Now with Eq. (25), (26 ), and (27), w e can ev aluate the left hand side of Eq. (1 9 ), i X j =0 2 λ ( φ i − φ j ) = X φ j ∈ M i 2 λ ( φ i − φ j ) = X φ j ∈ S A k,c k 2 λ ( φ i − φ j ) = X φ j ∈A k ∗ ,a k ∗ − 1 2 λ ( φ i − φ j ) + X φ j ∈ S A k,c k / A k ∗ ,a k ∗ − 1 2 λ ( φ i − φ j ) = Λ( N k ∗ , 0 ) + X ( k, c k ) 6 =( k ∗ ,a k ∗ − 1) Λ( N k , 1 ) = d X k = k ∗ a k Λ( N k , 1 ) − Λ( N k ∗ , 1 ) + Λ( N k ∗ , 0 ) . (28) C. Main result Theorem I I I.5. F or a prime p ower numb er q and every p osi tive inte ger n ≤ q q , ther e exists an explicit we ak ( n, q , l , ρ ) -design with l = q 2 and ρ < ( 1 + q − 1 ) q < e . Pr o of. W e pro v e this theorem by show ing that the design constructed by Eq. (11) is a w eak ( n, q , l, ρ )-design with ρ < (1 + q − 1 ) q . F rom the definition of Eq. (8) and (15), o ne can see that ρ = P j 2 for the refined Nisan-Wigderson design (as constructed b y Eq. (11 )) in a reasonable regime of n and q , e.g., q ≥ 16 a nd n > q 2 . Th us, our b ound in Theorem I I I.5 is relative ly tigh t. In the applicatio n of extractors, such as [3], the v alue o f ρ roughly indicates the ra tio of randomness t ha t can b e extracted. Th us, we need to ac hiev e a ρ that is close t o 1. Then, w e ha v e to go b ey ond the Nisan-Wigderson design. In order to reduce the par ameter ρ , one can extend the size of the set, from [ l ] to [ l ′ ]. Raz et a l. prop osed a blo c k design idea to reduce ρ [3, 4]. The basic idea is break the set [ l ′ ] into b blo cks (smaller sets), eac h o f which has a size of l (hence, l ′ = l b ). That is, the i th subset is { il + 1 , il + 2 , . . . , ( i + 1) l } and i ∈ [ b ]. The design sets are subsets of o ne of subsets. Ob viously , the sets from differen t subsets are disjoin t. Hartman a nd Raz sho w that with this tec hnique (Lemma 17 of ref. [3 ]) , ρ can b e reduced to 1 exp onentially fa st with the n umber of subsets grows. With this techniq ue, w e 10 can reduce ρ dow n to 1 with a finite num b er, O ( ρ log ( nρ )), of blo c ks b y digg ing into details of the design constructed by Eq. (11). Corollary IV.1. Given the explicit we ak ( n, q , l, ρ ) -design c o n structe d by Eq. (11) with l = q 2 and 1 < ρ < e , ther e exists an explicit we ak ( n ′ , q , l ′ , 1) - d esign with n ′ = nρ , l ′ = q 2 b and b =  log n + log ρ − log q log ρ − log ( ρ − 1)  = O (log n ) (33) as the numb er of blo cks. Pr o of. Denote the n um b er of subsets from i th subset to b e n i . W e construct the design in suc h a wa y that n i = (1 − ρ − 1 ) i n n b = nρ − b − 1 X i =0 n (1 − ρ − 1 ) i = nρ (1 − ρ − 1 ) b (34) where the first equation holds fo r i ∈ [ b ]. It is not har d to v erify that P b i =0 n i = nρ and n b ≤ q with Eq. (33). Now, we can v erify the conditions in Definition I I.2. Condition 1 is ob viously satisfied. F or a set S j in blo c k i ∈ [ b ], X j ′ q q = 4. The k ey p oint is that one do es no t need to pic k only one elemen t from one blo c k, as used in Eq. (10). In general, one might expect n = O (  l q  ) or l = O ( lo g n ). If one can find suc h a design with a reasonable ρ , one can apply the blo c k design idea as sho wn in Eq. ( 3 3) so that the seed length for the T revisan extractor is O ( log 2 n ) . Ac kno wledgments W e thank H.-K. Lo, B. Qi, C. Ro ck off, F . Xu, and H. Xu for enligh tening discus- sions. Financial supp orts from the National Basic Researc h Program of China Gran ts No. 2011CBA00300 and No. 2011CBA0030 1, National Natural Science F oundatio n of China Gran ts No. 61073 174, No. 61033001, No. 61061130 540, and No . 61003258, the 1000 Y outh F ellow ship pro gram in China, CFI, CIPI, the CR C prog r a m, CIF AR, MIT A CS, NSER C, OIT, Quan tumW or ks, and Sp ecial F unds for W ork Safety of Guangdong Prov ince of 2010 from Administration o f W ork Safety of G uangdong Provinc e of China are gratefully ac- kno wledged. X. Q. T an esp ecially thanks H.-K. Lo for the ho spitalit y during her sta y a t the 12 Univ ersit y o f T oronto. [1] N. Nisan and A. Wigderson, J. Comput. S yst. Sci., 49 , 149 (1994), ISSN 0022-000 0. [2] L. T revisan, Journal of the A CM, 48 , 2001 (1999). [3] R. Raz, O. Reingold, and S. V adhan, J ournal of Compu ter and S ystem Sciences, 65 , 97 (2002), ISSN 0022-000 0. [4] T. Hartman and R. Raz, Random S tructures & Algorithms, 23 , 235 (2003) . [5] In the original definition of w eak design, ( n − 1) ρ in stead of nρ is used on the r igh t side of Eq. (7). Here w e follo w the definition in [4]. [6] V. L eont’ev, Mathematical Notes, 80 , 300 (2006). 13