On the minimum distance of elliptic curve codes

Transcription

1 On the minimum istance of elliptic curve coes Jiyou Li Department of Mathematics Shanghai Jiao Tong University Shanghai PRChina Daqing Wan Department of Mathematics University of California Irvine CA USA Jun Zhang School of Mathematical Sciences Capital Normal University Beijing PRChina Abstract Computing the minimum istance of a linear coe is one of the funamental problems in algorithmic coing theory Vary [] showe that it is an NP-har problem for general linear coes In practice one often uses coes with aitional mathematical structure such as cyclic coes an algebraic geometry (AG coes etc In this paper we stuy the minimum istance of a family of AG coes For AG coes of genus 0 (generalize Ree-Solomon coes the minimum istance has a simple explicit formula An interesting result of Cheng [] says that the minimum istance problem is alreay NP-har (uner RP-reuction for general elliptic curve coes (ECAG coes or AG coes of genus In this paper we show that the minimum istance of ECAG coes also has a simple explicit formula if the evaluation set is suitably large (at least / of the group orer Our metho is purely combinatorial an base on a new sieving technique from Li-Wan [] I INTRODUCTION Let F n q be the n-imensional vector space over the finite fiel F q with q elements For any vector x (x x x n F n q the Hamming weight Wt(x of x is efine to be the number of non-zero coorinates ie Wt(x {i i n x i 0} A linear [n ] coe C is a -imensional linear subspace of F n q The minimum istance (C of C is the minimum Hamming weight of all non-zero vectors in C ie (C min{wt(c c C \ {0}} A linear [n ] coe C F n q is calle a [n ] linear coe if C has minimum istance A well-nown trae-off between the parameters of a linear [n ] coe is the Singleton boun which states that n An [n ] coe is calle a maximum istance separable (MDS coe if the equality above hols ie n The minimum istance of a linear coe etermines the ability of etecting an correcting of the coe Computing the minimum istance of a linear coe is one of the most important problems in algorithmic coing theory It was prove to be NPhar for general linear coes in [] The gap version of the problem was also shown to be NP-har in [4] An the same paper showe that approximating the minimum istance of a linear coe cannot be achieve in ranomize polynomial time to the factor log ɛ n unless NP RTIME( polylog(n In [5] Cheng an the secon author eranomize the reuction an showe there is no eterministic polynomial time algorithm to approximate the minimum istance to any constant factor unless NP P An they prove that approximating the minimum istance of a linear coe cannot be achieve in eterministic polynomial time to the factor log ɛ n unless NP RTIME( polylog(n Despite the above complexity results it is more interesting to compute the minimum istance of linear coes that are use in practical applications An important class of such coes is algebraic geometry (AG coes with parameters [n ] as efine in Section 4 The minimum istance of such AG coes from algebraic curves of genus g is nown to satisfy the inequality n g n In the simplest case g 0 ie generalize Ree-Solomon coes the minimum istance has the simple formula n In the next simplest case g either n or n an Cheng [] showe that etermining the minimum istance of ECAG coes between the two options is NP-har uner RP-reuction For genus g there is no such complexity result so far But it is believe to be an NP-har problem as well We are intereste in positive results for etermining the minimum istance of ECAG coes It was shown in [] an also in [6] from a ifferent aspect that computing the minimum istance of an ECAG coe is equivalent to a subset sum problem (SSP in the group of rational points on the elliptic curve We now mae this more precise Let E be an elliptic curve over the finite fiel F q Let G be the group of F q -rational points on the elliptic curve E The Hasse boun shows that (q q Let D G be a nonempty subset of carinality n which will be our evaluation set for ECAG coe For a positive integer n < an element b G let N( b D be the number of - subsets T D such that x T x b The counting version of the -subset sum problem for the pair (G D is to compute N( b D The minimum istance of the ECAG [n ]-coe is equal to n if an only if the number N( b D is positive This -subset sum problem is in general NP-har if the evaluation set D is small On the other han the ynamic programming metho implies that there is a polynomial time algorithm to compute N( b D if n D is large say n δ for some constant δ > 0 In this paper we obtain an asymptotic formula for N( b D if n D is suitably large say D > ( ɛ As an application we show that if the carinality n of the evaluation set is suitably large (at least / of the group size

2 then the minimum istance of an ECAG coe [n ] is always n We conjecture that the conition D > ( ɛ in our results can be improve to D > ( ɛ Our main technical tool is the sieve metho of Li-Wan [7] Let Ĝ be the group of aitive characters of G Note that Ĝ is isomorphic to G Denote the amplitue by Φ(D max χ(a χ Ĝχ is non-trivial a D Our main technical result is the following asymptotic formula for N( b D Theorem : Notations as above We have N( b D ( n S ( Φ(D < exp(g φ( ( nφ(d ( nφ(d where S is the set of characters in Ĝ which have orer greater than an exp(g is the exponent of G We apply this theorem to etermine the minimum istance of ECAG coes (for etails see Section IV an obtain Theorem : Suppose that n ( ɛq an q > 4 ɛ where ɛ is positive There is a positive constant C ɛ such that if C ɛ ln q < < n C ɛ ln q then ECAG coes [n ] have the eterministic minimum istance n In other wors if an elliptic curve coe [n ] over F q is an MDS coe then the length n shoul not excee ( ɛq Munuera [8] got an upper boun q q for the length of MDS elliptic curve coes For fixe when q is sufficiently large Munuera s boun tens to about q But for large saying cq Munuera s boun becomes looser an looser when c becomes larger an larger an it turns to the MDS conjecture when c is close to However compare with Munuera s boun our result hols for almost all in the range [ n] Especially our boun performances better than Munuera s boun when is large If we allow the length of the coes to be larger we then have a better boun on Theorem : If n q then for q > 64 an < < q then ECAG [n ] coes have the eterministic minimum istance n Note that one can chec the cases q 64 by a computer search we have a complete result for the minimum istance of the ECAG coe [n ] if n q This gives a new proof of MDS conjecture on ECAG coes in a purely combinatoric metho For recent progress on MDS conjecture on general coes an relate problems please see [9] This paper is organize as follows Section recalls the sieve metho of Li-Wan Section uses the sieve metho to get an estimate of counting subset sum problems on any large subset of the rational point group of an elliptic curve An Section 4 escribes the relation between minimum istance of ECAG coes an subset sum problems on the evaluation set of the ECAG coe The main result of this paper then follows II A DISTINCT COORDINATE SIEVING FORMULA For the purpose of the proof we introuce a sieving formula iscovere by Li-Wan [] which significantly improves the classical inclusion-exclusion sieve in many interesting cases We cite it here without any proof For etails an relate applications we refer to [] [7] Before we present the sieving formula we introuce some notations vali for the whole paper Let D be an alphabet set X a finite set of vectors of length over D Denote X {(x x x X x i x j i j} the pairwise istinct component subset Let S be the symmetric group on { } For τ S the sign function is efine to be sign(τ ( l(τ where l(τ is the number of cycles of τ incluing the trivial cycles which have length Let τ (i i i a (j j j a (l l l as with a i i s be any permutation enote the τ-symmetric subset X τ { (x x X x i x ia x l x las } ( Let f(x x x be a complex value function efine on X Denote the istinct sum an the τ-symmetric sum F τ F x X f(x x x x X τ f(x x x We now state our sieve formula We remar that there are many other interesting corollaries of this formula For intereste reaer we refer to [] Theorem : Let F an F τ be efine as above Then F τ S sign(τf τ ( For τ S enote by τ the conjugacy class etermine by τ whose elements are permutations conjugate to τ Conversely for given conjugacy class τ C enote by τ a representative permutation of this class For convenience we usually ientify these two symbols Since two permutations in S are conjugate if an only if they have the same type of cycle structure (up to the orer C is exactly the set of all partitions of The symmetric group S acts on D naturally by permuting coorinates That is for τ S an x (x x x D τ x (x τ( x τ( x τ( A subset X in D is sai to be symmetric if for any x X an any τ S τ x X In particular if X is symmetric an f is a symmetric function uner the action of S we then have the following simpler formula for ( Proposition : Let C be the set of conjugacy classes of S If X is symmetric an f is symmetric then F τ C sign(τc(τf τ ( where C(τ is the number of permutations conjugate to τ

3 For the purpose of evaluating the above summation we nee several combinatorial formulas A permutation τ S is sai to be of type (c c c if τ has exactly c i cycles of length i Note that i ic i Let N(c c c be the number of permutations in S of type (c c c an it is well-nown that N(c c c! c c! c c! c c! Lemma : Define the generating function C (t t t N(c c c t c tc tc ici If t t t q then we have C (q q q N(c ici c c q c q c q c (q In another case if t i q for i an t i s for i then we have {}}{{}}{ C ( s s q s s q N(c ici c c q c q c s c q c! / ( q s i i0 III q s! ( s(q s/ ( s i s SUBSET SUM PROBLEM IN A SUBSET OF THE RATIONAL POINT GROUP Lemma (Hasse-Weil Boun: Let E be an elliptic curve over the finite fiel F q Then the number of rational points on E has the following estimate #E(F q q q It is well-nown that the rational points E(F q form a finite group an more precisely it has the structure G E(F q Z/n Z/n for some n n By Lemma G has orer q c q with c Denote by exp(g the exponent of G Let D G be a nonempty subset of carinality n Let Ĝ be the group of aitive characters of G with the trivial character χ 0 Note that Ĝ is isomorphic to G Denote the partial character sum s χ (D a D χ(a an the amplitue Φ(D max χ Ĝχ χ 0 s χ (D Let N( b D be the number of -subsets T D such that x S x b In the following theorem we will give an asymptotic boun for N( b D which ensures N( b D > 0 when G D is not too large compare with G Theorem : Let N( b D be efine as above ( N( b D n S ( Φ(D ( nφ(d < φ( ( nφ(d exp(g (4 where S is the set of characters which has orer greater than Proof: Let X D D D be the Cartesian prouct of copies of D Let X { (x x x D x i x j i j} } It is clear that X n an X (n We have!n( b D (x x x X χ Ĝ (n (n χ(x x x b χ(x χ(x χ (b χ χ 0 (x x x X χ χ 0 χ (b (x x x X i χ(xi Denote f χ (x f χ (x x x i χ(x i For τ S let F τ (χ f χ(x χ(x i x X τ i x X τ where X τ is the τ-symmetric subset which is efine as in ( Obviously X is symmetric an f χ (x x x is normal on X Applying ( in Corollary we get!n( b D (n χ (b sign(τc(τf τ (χ χ χ 0 τ C where C is the set of conjugacy classes of S C(τ is the number of permutations conjugate to τ If τ is of type (c c c then F τ (χ x X τ i c x X τ i c i i( a D χ(x i χ(x i c i χ (x c i χ (x c c i χ i (a c i n c i m i (χ s χ(d c i ( m i (χ where m i (χ if χ i an otherwise m i (χ 0 Now suppose or(χ with n n Note that C(τ N(c c c In the case s χ (D is an integer Applying Lemma we have sign(τc(τf τ (χ τ C ( C(τ( n ci m i (χ ( s c χ(d i ( m i (χ τ C (!!! / / i0 ( nφ(d ( ( nsχ(d i s χ(d i i i (s χ(d i i i0 ( n sχ(d The last inequality in the case s χ (D > 0 is from the ientity i0 ( ( a b i i ( a b

4 In the case s χ (D < 0 since the summation has alternative signs the inequality follows from a simple combinatorial argument In the case since s χ (D Φ(D we have sign(τc(τf τ (χ τ C C(τn cim i(χ Φ(D c i( m i(χ τ C! ( nφ(d Similarly if or(χ is greater than then ( Φ(D sign(τc(τf τ (χ! τ C Let S be the set of characters which have orer greater than Summing over all nontrivial characters we obtain ( N( b D n S ( Φ(D ( nφ(d < exp(g φ( ( nφ(d where φ( is the number of characters in Ĝ of orer This completes the proof Corollary : We have ( n N( b D where M is efine as {( Φ(D M max ( M ( ( nφ(d nφ(d } an is the smallest nontrivial ivisor of that is not equal to Corollary 4: Let q 64 an n q For 6 < q we have N( b D > 0 for every b G Proof: By symmetry it is sufficient to consier the case n/ To ensure N( b D > 0 by (4 it suffices to have ( n > S ( ( Φ(D nφ(d ( nφ(d φ( < exp(g For a nontrivial character χ g G χ(g 0 an it follows that Φ(D Φ(G D < D q Since G is the prouct of at most two cyclic groups by the efinition of φ( we have φ( For simplicity set K For the case q / it is sufficient to have (q (q ( q q K ( q q ( q q K > 0 When one has 5/6q 79/6q 5/ 589/8q 59/7q / 49/q 67/q / > 0 Similarly when 6 one has q 64 This is one by first taing K 40 we solve that q 97 But notice that now K shoul be 7 Then taing K 7 we solve q 79 Iteratively we can get q 64 finally One checs that when q / this function is unimoal on For q / < < (q q/6 it then suffices to have ( q > (q ( q q q an for (q q/6 (q / ( q > (q ( q q q It follows from a simple asymptotic analysis an the proof is complete A similar argument gives Corollary 5: Suppose that n ( ɛq an q > 4 ɛ where ɛ is positive Then there is a positive constant C ɛ such that N( b D > 0 for every b G provie C ɛ ln q < < n C ɛ ln q Proof: Similarly as above we only consier the case n/ To ensure N( b D > 0 by (4 it suffices to have ( ( ( n Φ(D nφ(d > S ( nφ(d φ( < exp(g For a nontrivial character χ g G χ(g 0 an it follows that Φ(D Φ(G D < D ( ɛq q For small q/6 it suffices to have ( q (q q ( q q > 0 which hols when > C ln q for some constant C For q/6 < n/ ( ɛ q it suffices to have ( ( ɛq > (q ( ( q ɛ q q which hols when q > 4 ɛ an > C ɛ ln q for some constant C ɛ So the proof is complete From the proof of the above corollary if follows that Corollary 6: Suppose n ( ɛq where ɛ is positive an ɛ / When q is large enough (in application we nee to use long length coes so it is reasonable to assume q is large then there is a positive constant C (inepenent of ɛ an q such that N( b D > 0 for every b G provie C ln q < < n C ln q

5 IV MINIMUM DISTANCE OF ELLIPTIC CODES AND SSP In this section we iscuss the relationship between the minimum istance of ECAG coe an SSP on the group of rational points of the elliptic curve Using the results in the previous section our main theorems in Introuction follow automatically Let E/F q be an elliptic curve over the finite fiel F q with function fiel F q (E Let E(F q be the set of all F q -rational points on E Suppose D {P P P n } is a proper subset of rational points E(F q an G is a ivisor of egree (g < < n with Supp(G D Without any confusion we also write D P P P n Denote by L (G the F q -vector space of all rational functions f F q (E with the principal ivisor iv(f G together with the zero function (cf [0] The functional AG coe C L (D G is efine to be the image of the following evaluation map: ev : L (G F n q ; f (f(p f(p f(p n It is well-nown that C L (D G has parameters [n ] where the minimum istance has two choices: n or n Suppose O is one of the F q -rational points on E The set of rational points E(F q forms an abelian group with zero element O (for the efinition for the sum of any two points we refer to [] an it is isomorphic to the Picar group iv o (E/Prin(F q (E where Prin(F q (E is the subgroup consisting of all principal ivisors Denote by an the aitive an minus operator in the group E(F q respectively Proposition 4 ([] [6]: Let E be an elliptic curve over F q D {P P P n } a subset of E(F q such that rational points (not necessarily istinct O P / D an let G ( O P (0 < < n Enow E(F q a group structure with the zero element O Then the AG coe C L (D G has minimum istance n if an only if N( P D 0 An the minimum istance n if an only if N( P D > 0 Proof: We have alreay seen that the minimum istance of C L (D G has two choices: n n So C L (D G is not MDS ie n if an only if there is a function f L (G such that the evaluation ev(f has weight n This is equivalent to that f has zeros in D say P i P i That is iv(f ( O P (P i P i which is equivalent to iv(f ( O P (P i P i The existence of such an f is equivalent to saying P i P i P Namely N( P D > 0 It follows that the AG coe C L (D G has minimum istance n if an only if N( P D 0 Proposition 4 establishes the relation between minimum istance of ECAG coe an SSP on the rational point group of the elliptic curve Together with Corollaries 4 an 5 we obtain the main result of this paper Theorem ACKNOWLEDGMENT This paper was written when the first author was visiting the Department of Mathematics University of Delaware an the Department of Mathematics University of California Irvine The first author warmly thans both epartments for their hospitality an Professor Qing Xiang for his help The thir author woul lie to than Maosheng Xiong for inviting him to visit Hong Kong University of Science an Technology uring ISIT 05 The authors woul lie to than the anonymous reviewers for remining the references [8] an [9] The wor of Jiyou Li is supporte by the Ky an Yu- Fen Fan Fun Travel Grant from the AMS the National Science Founation of China (0070 an the National Science Founation of Shanghai Municipal (ZR4500 The research of Daqing Wan is partially supporte by NSF This research of Jun Zhang is supporte by the National Key Basic Research Program of China (0CB8404 the National Natural Science Founation of China ( an REFERENCES [] A Vary The intractability of computing the minimum istance of a coe IEEE Transactions on Information Theory vol 4 no 6 pp [] Q Cheng Har problems of algebraic geometry coes IEEE Transactions on Information Theory vol 54 pp [] J Li an D Wan A new sieve for istinct coorinate counting Science China-mathematics vol 5 pp 00 [4] I Dumer D Micciancio an M Suan Harness of approximating the minimum istance of a linear coe IEEE Transactions on Information Theory vol 49 no pp 7 00 [5] Q Cheng an D Wan A eterministic reuction for the gap minimum istance problem:[extene abstract] in Proceeings of the 4st annual ACM symposium on Theory of computing ACM 009 pp 8 [6] J Zhang F-W Fu an D Wan Stopping sets of algebraic geometry coes IEEE Transactions on Information Theory vol 60 no pp March 04 [7] J Li an D Wan Counting subset sums of finite abelian groups Journal of Combinatorial Theory Series A vol 9 no pp [8] C Munuera On the main conjecture on geometric MDS coes IEEE Transactions on Information Theory vol 8 no 5 pp Sep 99 [9] S Ball On sets of vectors of a finite vector space in which every subset of basis size is a basis Journal of the European Mathematical Society vol 4 no pp [0] H Stichtenoth Algebraic function fiels an coes n e ser Grauate Texts in Mathematics Berlin: Springer-Verlag 009 vol 54 [] J Silverman The arithmetic of elliptic curves n e ser Grauate Texts in Mathematics Dorrecht: Springer 009 vol 06