Vulnerability of MRD-Code-Based Universal Secure Error-Correcting Network Codes under Time-Varying Jamming Links

Transcription

1 Vulnerability of MRD-Code-Based Universal Secure Error-Correcting Network Codes under Tie-Varying Jaing Links Jun Kurihara KDDI R&D Laboratories, Inc 2 5 Ohara, Fujiino, Saitaa, Japan Eail: kurihara@kddilabsjp Toohiko Uyeatsu Tokyo Institute of Technology 2 2 Ookayaa, Meguro, Tokyo, Japan Eail: uyeatsu@ieeeorg Abstract In order to provide reliable and secure counication against eavesdroppers and jaers over networks, Universal Secure Error-Correcting Network Codes (USECNC) based on Maxiu-Rank-Distance (MRD) codes have been introduced This code can be applied to any underlying network codes However, Shioji et al introduced a reasonable network odel against the code In their odel, an attacker eavesdrops inforation sybols fro soe links, where the set of eavesdropping links is re-selected during one packet transission The MRD-code-based USECNC cannot guarantee the security against eavesdroppers under this odel Inspired by Shioji et al s result, this paper considers the odel such that the set of links that jaing (error) sybols are injected into is re-selected for each tie slot We show that the MRDcode-based USECNC cannot guarantee the error-correcting capability under the odel of tie-varying jaing links, even if the nuber of jaing links is liited to only one Furtherore, by introducing a restriction on the field of local coding vectors in the network coding, we propose a siple solution to the proble of tie-varying jaing links for MRD-code-based USECNC Keywords-Network Coding; Secure Network Coding; Network Error-Correction; Jaing I INTRODUCTION Network coding [] has been attracting uch attentions since it can achieve better perforance than ordinary networks with routing ethods in ters of throughput, energy consuption, etc [2][3] In reality, network coding ay suffer fro two kinds of adversaries: eavesdropping and jaing Silva et al s presented Universal Secure Error-Correcting Network Codes (USECNC) [4][5] based on Maxiu-Rank-Distance (MRD) codes [6][7] to provide secure and reliable counication against eavesdroppers and jaers over the network Their code can be applied to any underlying network codes and hence it is called universal In the construction of Silva et al s USECNC, packets are split into segents and transitted to sink nodes through the network over tie slots It has been assued that packets are tapped by eavesdroppers and corrupted by jaers on several links that are fixed during one transission of packets, ie tie slots Shioji et al introduced the tie-varying eavesdroppinglink odel in which attacker can re-select the set of eavesdropping links at each tie slot of one packet transission [8] They showed that Silva et al s USECNC is insecure under this odel, that is, inforation is leaked to eavesdroppers [8] When the network coding is ipleented on an overlay network of the Internet, the assuption of tie-varying eavesdropping links is reasonable Although a packet is not split fro the perspective of the overlay network and they are transitted through fixed logical paths, a packet is split into ultiple fragents and they are routed over different physical paths at an interediate router Hence, fro the point of view of the overlay network, the set of tapped logical links ay be re-selected if physical links are eavesdropped Such attackers against the network ight be active, ie jaers who inject jaing packets into links to corrupt the network Thus, this paper considers the odel such that the attacker injects jaing (error) sybols into soe links (called jaing links ) in the network, and the set of jaing links is re-selected during one transission of packets We show that the MRD-code-based USECNC cannot guarantee the error-correcting capability under the odel of tie-varying jaing links, even if the nuber of jaing links is liited to only one Furtherore, by introducing a restriction on the field of local coding vectors (LCV s) in the network coding, we propose a siple solution to the proble of tie-varying jaing links for MRDcode-based USECNC The rest of this paper is organized as follows: Section II gives several definitions, the basic odel of network coding and a brief review of MRD-code-based USECNC presented by Silva et al In Section III, we introduce the tie-varying jaing-link odel, and show the vulnerability of MRDcode-based USECNC under this odel In Section IV, we propose a siple countereasure against this odel Finally, we conclude this paper in Section V II PRELIMINARIES In this section, we give definitions about the representation of finite field extensions, the basic odel of network coding Copyright (c) IARIA, 20 ISBN:

2 and a brief review of MRD-code-based USECNC presented by Silva et al A Field Extensions Let F q be a finite field containing q eleents, and let F q be an -degree field extension of a base field F q Then, F q can be viewed as a vector space over F q When the basis of the space is fixed, ie, an irreducible polynoial generating the field extension is deterined, an eleent of F q can be represented by an -diensional vector over F q We suppose that the vector representation of x F q is written by [, x (2),, x () ] F q B Network Coding Let G = (E, V) be a delay-free acyclic directed network, where E and V denote a set of links (edge, channel) and a set of nodes, respectively Let s V and R V respectively denote a source node and a set of sink nodes, where s R In this network odel, we suppose that each link can carry an eleent of F q per one tie slot The source node wishes to ulticast the sequence x = [x, x 2,, x n ] T F n q to all sink nodes at rate n, where the rate is defined as the nuber of eleents in F q transitted fro s per one tie slot Suppose that n in{axflow(s, r) : r R} holds, where axflow(s, r) (i, j V) denotes the ax-flow fro i to j Then, there exists a network coding ethod in which s can ulticast x to all nodes in R at rate equal to n [] We assue that linear network coding [9] is eployed over G, ie, the type of data processing perfored on the packets at each node is liited to linear cobination This iplies that the data flow on any link over G can be represented as an F q -linear cobination of the sequence x, x 2,, x n Thus, the inforation flow on link e E can be denoted as y e = b T e x using a global coding vector (GCV), b e = [b, b 2,, b n ] T F n q When one has access to, say, l links e, e 2,, e l, then the inforation obtained fro these links is denoted as M x F l q, where M = [ b e, b e2,, b en ] T Constructing a network code is equivalent to deterining the GCV of each link by setting the coefficients of the linear cobination perfored at each node C The MRD-Code-Based USECNC Here we introduce the fixed jaing-link odel and the construction of Silva et al s USECNC eployed over this odel [4][5] ) Fixed Jaing-Link Model: Suppose that one packet is coposed of an eleent of F q that is an -degree field extension of F q, and is represented by an -diensional vector over F q We define n packets transitted fro the source node s by X = [X, X 2,, X n ] T F n q (= F n q ) Then, the duration of one transission of X is coposed of tie slots The source node s splits X and sends the over tie slots using a linear network code though G, ie, transits [X (i), X(i) 2,, X(i) n ] T F n q at each tie slot i =,, Here we suppose that there exist t(< n) jaing links in E and they inject error packets into G Silva et al [4][5] assued that these links are fixed during one transission, ie, tie slots At a specific sink node, the received packets though G with injection of jaing (error) packets is represented by Y = AX + DZ F n q, () where A F n n q is a transition atrix corresponding to GCV s A linear network code eployed over G is called feasible [0] if rank A = n for all sink nodes, otherwise it is rank-deficient The rank deficiency is defined as ρ = n rank A On the other hand, Z F t q denotes t jaing (error) packets D F n t q is a transition atrix corresponding to jaing links D and Z are unknown rando variables with unknown distributions Then rank D t holds since t < n ust hold Throughout this paper, we focus our attention only on injection of jaing packets, and oit eavesdropping on links (cf, Shioji et al s analysis in [8]) 2) Silva et al s Schee: For the construction against the jaing(error)-packet injection, Silva et al s Universal Secure Error-Correcting Network Codes (USECNC) [4][5] is equivalent to [n, k] Maxiu Rank Distance (MRD) code C F n q satisfying n and 0 < k n 2t ρ, where ρ (= n rank A) is the rank deficiency of A Naely, the transitted packet X in Eq() is a codeword of C MRD code is a class of linear codes over F q, which is optial in the rank-distance sense For the [n, k] MRD code C, we have d R (C) 2t + ρ +, where d R (C) denotes the iniu rank-distance [6][7] between all pairs of distinct codewords of C We also have d R (AX, Y ) = rank (Y AX) = rank DZ rank Z t, (2) where d R (P, Q) denotes the rank-distance between atrices P, Q F n q Hence, the iniu rank-distance decoder of C can correct t jaing packets (and ρ rank deficiency of A) under the fixed jaing-link odel (n k 2t + ρ) [0] III TIME-VARYING JAMMING LINKS OVER THE NETWORK In this section, we introduce the odel of tie-varying jaing links, and show that MRD-code-based USECNC cannot guarantee the error-correcting capability under this odel Copyright (c) IARIA, 20 ISBN:

3 A The Model Suppose that the source node s ulticast n packets, ie, n vectors in F q, over tie slots i =, 2,, We consider a odel such that the jaer(s) can re-select the set of t jaing links at each tie slot i Denote the t error packets injected into G by a t atrix Z = [ z, z 2,, z ] F t q, where z i F t q (i =, 2,, ) is a t-diensional colun vector chosen according to arbitrary distribution At a specific sink node, let D i Fq n t (i =, 2,, ) be a n t atrix representing the linear cobination of jaing-packet segents at tie i Jaer(s) specify the set of jaing-links arbitrarily, and hence we assue that D i is chosen arbitrarily by jaer(s) The received atrix Y F n q at the specific sink node is written as Y = AX + V F n q, (3) where the atrix V denotes jaing packets conveyed over the tie-varying jaing links, given by V = [D z, D 2 z 2,, D z ] F n q When D = D 2 = = D, Eq(3) is equivalent to Silva et al s fixed jaing-link odel [4][5] B MRD-Code-Based USECNC under the Tie-Varying Jaing Link Model On the tie-varying jaing-link odel presented in the previous subsection, we first have the following lea Lea Suppose that the jaer(s) can arbitrarily select non-zero atrices D, D 2,, D fro F n t q and can generate non-zero vectors z, z 2,, z F t q Then the axiu possible rank of V is n Proof: It is only necessary to show an exaple having rank V = n Choose D i for i =, 2,, n (n ) as follows: D = [ u, 0,, 0 ], D 2 = [ u 2, 0,, 0 ], D n = [ u n, 0,, 0 ], where u j denotes the j-th colun vector of an n n identity atrix and 0 is a colun zero vector Let z i be represented as z i = [ i, 0,, 0] T F t q We assue that i is chosen fro F q = F q \{0} for i =,, n Then, we denote V by V = [D z, D 2 z 2,, D n z n D n+ z n+,, D + z + ] = [V V 2 ], where V can be represented as V = [D z, D 2 z 2,, D n z n ] 0 = 0 z n () We thus have rank V = n and hence rank V = n holds This copletes the lea Denote rank V = v We then have v n fro Lea Let X in Eq(3) be a codeword of Silva et al s USECNC, that is, [n, k] MRD code C with n and 0 < k n 2t ρ Fro Lea, jaers ay select D i and z i to satisfy v > t even if rank D i t holds for all i =,, We then have t < v = rank V = rank (Y AX) = d R (Y, AX) Thus, the decoder of C cannot correct t jaing (error) packets under the tie-varying jaing-link odel Moreover, we give the following lea that shows the nuber of jaing packets is independent of rank V Lea 2 Suppose that jaer(s) can arbitrarily choose D i s in Eq(3) fro F n t q Then, the axiu possible rank of V is n, which is independent of the value t > 0 Proof: Since the supposition n is required in Silva et al s USECNC, the exaple presented in the proof of Lea always satisfy rank V = n Moreover, in the exaple in the proof of Lea, we have assued rank D i = for i =, 2,, n independently fro the value t > 0 Thus the lea is copleted The above analysis proves the following result Theore Suppose that there are t jaing links in E under the tie-varying jaing-link odel Let the transitted packet over the network is a codeword of Silva et al s USECNC, ie, [n, k] MRD code C with 0 < k n 2t ρ and n Then, for any t > 0, the decoder of C cannot correct t jaing (error) packets injected fro tievarying jaing links This theore iplies that, under the tie-varying jaing-link odel, Silva et al s USECNC cannot guarantee the error-correction capability even if t = C An Exaple Assue q = 2, = n = 3 and t = We do not consider the rank deficiency and the existence of eavesdroppers, and set ρ = µ = 0 Fro these paraeters, the USECNC is defined as a [3, ] MRD code C F A codeword of C is defined as a 3 3 atrix X = [X, X 2, X 3 ] T C, where X i F 3 2 Copyright (c) IARIA, 20 ISBN:

4 Under the tie-varying jaing-link odel, a specific sink node receives the following packet Y = AX + V = A X X 2 + [D z, D 2 z 2, D 3 z 3 ], X 3 where A F 3 3 2, D i F 3 2 and z i F 2 for i =, 2, 3 Here we assue rank A = 3 and rank D i t = for i =, 2, 3 For exaple, we suppose A is a 3 3 identity atrix I and D i s are D = [, 0, 0] T, D 2 = [0,, 0] T, D 3 = [0, 0, ] T We note that rank D i = t = for each of i =, 2, 3 siilar to the fixed jaing-link odel We also represent X i = [X () i, X (2) i, X (3) i ] F 3 2 and z i = [z i ] F 2 for i =, 2, 3 We then receive X () X (2) X (3) z 0 0 Y = X () 2 X (2) 2 X (3) z 2 0 X () 3 X (2) 3 X (3) 0 0 z 3 3 In this case, we have rank V = 3 if all z, z 2, z 3 are nonzero and hence X cannot be uniquely reconstructed This can be viewed as the worst case of Silva et al s USECNC under the fixed jaing-link odel, that is, 3(> t) error packets are injected into the network IV A SIMPLE WAY TO PREVENT THE JAMMERS The siplest way to prevent tie-varying jaing links is to construct the network in which no packet is split into segents Naely, packets are transitted not as - diensional vectors using tie slots but as eleents of an -degree field extension itself at one tie slot in Silva et al s USECNC We thus consider how to transit codewords of [n, k] MRD code at one tie slot over a linear network code and how to decode the received codeword by the iniu rank-distance decoder Let p be a prie and the nuber of eleents in a base field F p And let p ( q) be the nuber of eleents in an extended field F p We redefine that each link can carry an eleent of F p per one tie slot The source node s wishes to transit x = [x, x 2,, x n ] T F n p to all sink nodes in R at one tie slot using a linear network code Then, the received packet at a specific sink node is represented as y = [y, y 2,, y n ] T = A x + D z F n p, (4) where A F n n p is a transition atrix according to a linear network code over F p z F t p denotes t jaing packets and D F t t p to jaing links is a transition atrix corresponding Assue x F n p(= Fn p ) in Eq(4) is a codeword of [n, k] MRD code defined over F p, where n and 0 < k n 2t + ρ Let M( ) denote the atrix representation of a vector in F n p to easure the rank distance between vectors For exaple, the atrix representation of x is M( x) = x (2) 2 x (2) x () 2 x () 2 Fn n x (2) n x () n p, where [ i, x (2) i,, x () i ] F p is the vector representation of x i F p (i =,, n) Then, according to Eq(4), we have d R (M(A x), M( y)) = rank (M( y) M(A x)) = rank (M( y A x)) = rank M(D z), (5) fro the properties of the additive operation over a field extension F p However, unlike Eq(2), rank M(D z) t does not always hold fro the property of the ultiplicative operation over F p Hence, the iniu rank-distance decoder of the [n, k] MRD code cannot be siply applied to Eq(4) Hence, we add the following condition to the construction of the underlying network coding such that rank M(D z) t always holds Condition For all nodes, eleents of local coding vectors (LCV s) are chosen fro the subfield (base field) F p of the field extension F p Note that eleents of LCV s are coefficients of linear cobination of incoing sybols at each node and they define eleents of GCV s [2] Thus, this condition also yields that the transition atrices A and D are defined over the subfield, ie, A F n n p F n n p and D Ft t p F t t p For any a F p and b F p, their vector representations are [0,, 0, a] F p and [b (), b (2),, b () ] F p, respectively Then, the ultiplication ab over F p is given by ab := [ab (), ab (2),, ab () ], fro the fact of finite fields That is, all operations can be executed over the base field F p Thus, under Condition, Copyright (c) IARIA, 20 ISBN:

5 we have d, d,2 d,t z d 2, d 2,2 d 2,t M(D z) = M z 2 d t, d t,2 d t,t z t w= d,wz w t w= = M 2,wz w w= d t,wz w w= [d,wz w (), d,w z w (2),, d,w z w () ] = w= [d 2,wz w (), d 2,w z w (2),, d 2,w z () w ] w= [d t,wz w (), d t,w z w (2),, d t,w z w () ] z () = D t z () t = DM( z), where d u,v F p (u, v =,, t) and [z w (), z w (2),, z w () ] F p is the vector representation of z w F p (w =,, t) Since rank M( z) t, we have rank M(D z) = rank DM( Z) t Cobining Eq(5) and this, the iniu rank-distance decoder of the [n, k] MRD code can decode the received packets y as with the Silva et al s USECNC for the fixed jaing-link odel However, this solution contains the following probles; a) The schee is not universal since the condition of LCV s is introduced b) Since the field for LCV s is changed fro F q to F p with p < q, the probability of rank deficiency becoes high when rando network coding [] is eployed, or the size of F p ight be insufficient for the deterinistic construction of network coding depending on the network structure [2] We believe the copletely different technique fro MRD codes ust be required to realize universal secure error-correcting codes against the tie-varying jaing links V CONCLUSION This paper considered the reasonable network odel such that the set of links that jaing (error) sybols are injected into is re-selected for each tie slot We revealed that the MRD-code-based USECNC cannot guarantee the errorcorrecting capability under the odel of tie-varying jaing links, even if the nuber of jaing links is liited to only one Further, by introducing a severe restriction to the field of LCV s, we presented a siple way to avoid jaing with the tie-varying jaing links for MRD-code-based USECNC REFERENCES [] R Ahlswede, N Cai, S-Y R Li, and R W Yeung, Network inforation flow, IEEE Transactions on Inforation Theory, vol 46, no 4, pp , 2000 [2] C Fragouli and E Soljanin, Network Coding Fundaentals Now Publishers, 2007 [3] C Fragouli and E Soljanin, Network Coding Applications Now Publishers, 2007 [4] D Silva and F R Kschischang, Universal secure errorcorrecting schees for network coding arxiv:003387v, Jan 200 Appeared in IEEE International Syposiu on Inforation Theory (ISIT) 200 Available at abs/003387v [5] D Silva and F R Kschischang, Universal secure network coding via rank-etric codes arxiv: v2, Apr 200 Available at [6] E M Gabidulin, Theory of codes with axiu rank distance, Probles of Inforation Transission, vol 2, no, pp 2, 985 [7] R M Roth, Maxiu-rank array codes and their application to crisscross error correction, IEEE Transactions on Inforation Theory, vol 37, no 2, pp , 99 [8] E Shioji, R Matsuoto, and T Uyeatsu, Vulnerability of MRD-code-based universal secure network coding against stronger eavesdroppers, IEICE Transactions on Fundaentals, vol E93-A, no, pp , 200 [9] S-Y R Li, R W Yeung, and N Cai, Linear network coding, IEEE Transactions on Inforation Theory, vol 49, no 2, pp 37 38, 2003 [0] D Silva and F R Kschischang, On etrics for error correction in network coding, IEEE Transactions on Inforation Theory, vol 55, no 2, pp , 2009 [] T Ho, M Médard, R Koetter, D R Karger, M Effros, J Shi, and B Leong, A rando linear network coding approach to ulticast, IEEE Transactions on Inforation Theory, vol 52, no 0, pp , 2006 [2] S Jaggi, P Sanders, P A Chou, M Effros, S Egner, K Jain, and L M G M Tolhuizen, Polynoial tie algoriths for ulticast network code construction, IEEE Transactions on Inforation Theory, vol 5, no 6, pp , 2005 Copyright (c) IARIA, 20 ISBN: