Multicast Delivery Scheme for Threshold Secret Shared Content

Nagao Ogino and Hidetoshi Yokota KDDI R&D Laboratories Inc.

Saitama 356-8502 JAPAN {ogino, yokota}@kddilabs.jp

Abstract—A threshold secret sharing scheme can deliver

important content reliably using redundant delivery routes via a network. Furthermore, simultaneous delivery of the threshold secret shared content to multiple receivers can achieve efficient resource utilization thanks to multicast and network coding techniques. Nevertheless, the network coding technique results in a tradeoff between reliability and efficiency. This paper proposes a multicast delivery scheme for threshold secret shared content that can control the tradeoff between reliability and efficiency. In the proposed scheme, all the pieces obtained from the original content are grouped in advance, and network coding is only applied to the pieces included in the same group. This paper also proposes two kinds of heuristic multicast delivery route computation methods to optimize the proposed multicast delivery scheme in networks on a practical scale. The evaluation results show that the proposed scheme adopting an optimized number of groups can actually minimize the total link bandwidth required while satisfying the given requirement on content loss probability.

Keywords—threshold secret sharing; multicast content delivery; network coding; tradeoff between reliability and efficiency; route optimization

I. INTRODUCTION Secret sharing schemes have been proposed to preserve

important content securely by using multiple content servers [1]. In particular, a (k, n) threshold secret sharing scheme divides content into n pieces and only requires k (≤ n) pieces to reconstruct the original content [2]. The coding and decoding of the content can be simplified in the threshold secret sharing scheme. The threshold secret sharing scheme can also achieve reliable delivery of important content via the network [3, 4]. In this case, n pieces obtained from the threshold secret shared content are delivered to a receiver using redundant n routes. Even if n - k delivery routes incur damage due to a multiple-link failure in the network, the receiver can still reconstruct the original content from k pieces delivered using the remaining normal k routes. A heuristic method has been proposed to compute the sub-optimum redundant n routes and minimize the content loss probability due to the multiple-link failures [5].

The above existing studies have only considered unicast delivery of the threshold secret shared content. In contrast, this paper investigates multicast delivery of threshold secret shared content. Although a time lag may occur from content delivery requests to actual content delivery, network resources can be utilized efficiently when the content is simultaneously delivered to multiple receivers. In particular, a network coding technique can be applied to multiple pieces obtained from the threshold secret shared content, which can attain efficient utilization of link bandwidth [6-9]. Nevertheless, damage to

one piece due to link failure may preclude network decoding of multiple pieces in the receiver. This means that a certain tradeoff exists between reliability and efficiency when the threshold secret shared content is multicast using the network coding technique.

This paper proposes a multicast delivery scheme for threshold secret shared content. In the proposed scheme, all the pieces obtained from the original content are grouped in advance, and the network coding is only applied to the pieces involved in the same group. The proposed multicast delivery scheme can control the tradeoff between reliability and efficiency by adjusting the number of groups. Furthermore, this paper proposes two kinds of heuristic delivery route computation methods to optimize the proposed multicast delivery scheme. One method computes the delivery routes for minimizing the content loss probability that each receiver cannot reconstruct the original content. The other method computes the delivery routes that minimize the total utilized link bandwidth. From the evaluation results, the proposed scheme can minimize the required link bandwidth while satisfying the given requirement on the content loss probability.

Section II proposes a multicast delivery scheme for the threshold secret shared content. Section III explains two kinds of heuristic route computation methods to obtain suboptimum delivery routes, and the proposed scheme is evaluated in section IV. Finally, section V concludes this paper.

II. MULTICAST OF THRESHOLD SECRET SHARED CONTENT

A. Motivation of Proposed Scheme The threshold secret sharing scheme can achieve reliable

content delivery using redundant routes for the pieces that are obtained from the division of important content. Furthermore, multicast delivery of the threshold secret shared content can efficiently utilize network resources, although a time lag may occur from the content delivery requests to actual content delivery. Figure 1 illustrates the multicast delivery of the threshold secret shared content via the backbone network. All the pieces obtained from the original content are preserved on multiple content servers and are simultaneously delivered to multiple content receivers via the backbone network equipped with a centralized route computation server [10, 11]. Each transit node in the backbone network has both multicast and network coding functions for multiple pieces originating from identical content. Although each transit node independently executes network coding, a feasible code assignment is always guaranteed for each receiver node [7]. This means that each receiver node can successfully decode all the pieces when all delivery routes to the receiver node incur no damage.

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Fig. 1. Multicast delivery of threshold secret shared content

The route computation server accepts multiple requests for identical content from several content receivers. Subsequently, it computes the multicast delivery routes from the server nodes accommodating the requested content servers to the receiver nodes accommodating the content receivers that have requested the content. The route computation server notifies the content servers of the computed delivery routes, and the pieces are multicast from the content servers along the reported delivery routes. Each piece cannot be rerouted dynamically even if it happens to meet with a link failure on the delivery route. Each receiver node executes network decoding for the pieces and reconstructs the original content from the decoded k pieces.

Figure 2 shows an example of efficient link bandwidth utilization in the multicast delivery of the threshold secret shared content. The threshold secret shared content is divided into four pieces P1 through P4. Two pieces P1 and P2 are stored on a content server that server node s1 accommodates, while two pieces P3 and P4 are stored on a content server that server node s2 accommodates. The threshold secret shared content is multicast to the content receivers that receiver nodes d1 and d2 accommodate. Multiple identical pieces delivered to the different receiver nodes can share the link bandwidth thanks to the multicast technique. In Fig. 2, two identical pieces delivered to receiver nodes d1 and d2 can share the link bandwidth on link1 and link2. Furthermore, multiple different pieces delivered to different receiver nodes can share the link bandwidth thanks to the network coding technique [6]. For example, piece P1 delivered to receiver node d2 and piece P3 delivered to receiver node d1 can share the link bandwidth on link7. Likewise, two pieces P2 and P4 delivered to receiver nodes d2 and d1 can also share the link bandwidth on link7.

Fig. 2. An example of link bandwidth utilization in the multicast delivery

Although the link bandwidth can be utilized efficiently thanks to the network coding technique, damage to one piece due to a link failure may preclude network decoding of multiple pieces in the receiver node. For example, damage to two pieces P1 and P2 due to a failure on link3 causes the loss of all four pieces P1 through P4 in receiver node d1. Likewise, damage to two pieces P3 and P4 due to a failure on link4 induces loss of all four pieces in receiver node d2. This means that a tradeoff exists between reliability and efficiency when the threshold secret shared content is multicast using the network coding technique.

B. Proposed Multicast Delivery Scheme This paper proposes a multicast delivery scheme for

threshold secret shared content. In the proposed scheme, all the pieces obtained from the original content are classified into multiple groups in advance, and the network coding technique is only applied to the pieces involved in the same group. Although the proposed scheme weakens the efficient utilization of link bandwidth procured by network coding, damage to one piece due to a link failure only precludes network decoding of the pieces included in the same group. This means that no piece classified into the different groups is lost while all the pieces in the same group may be lost in the worst case. The proposed scheme can control the tradeoff between reliability and efficiency by adjusting the number of groups. Minimizing the total link bandwidth required can be expected while satisfying the given requirement on the content loss probability thanks to adoption of a suitable number of groups.

Figure 3 shows an example of the proposed multicast delivery scheme. Although four pieces P1 through P4 are multicast as shown in Fig. 2, three pieces P1 through P3 are classified into group G1, and one piece P4 is classified into another group G2. In Fig. 3, two pieces P1 and P3 share the bandwidth on link7, since these two pieces are classified into the same group. Nevertheless, a piece P2 delivered to receiver node d2 and piece P4 delivered to receiver node d1 cannot share the bandwidth on link7, since these two pieces are classified into the different groups. In contrast, damage to piece P2 in group G1 due to a failure on link3 does not preclude network decoding of piece P4 in group G2 in receiver node d1. Likewise, damage to piece P4 in group G2 due to a failure on link4 does not induce loss of piece P2 in group G1 in receiver node d2. This means that damage to two pieces due to a failure on link3 only causes the loss of three pieces in receiver node d1. Likewise, damage to two pieces due to a failure on link4 only induces the loss of three pieces in receiver node d2.

Fig. 3. An example of the proposed multicast delivery scheme

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When piece P3 is also classified into group G2 in Fig. 3, network coding is not applied to two pieces P1 and P3. This means that damage to two pieces due to a failure on link3 does not induce loss of the other two pieces in receiver node d1. Likewise, damage to two pieces due to a failure on link4 permits normal reception of the other two pieces in receiver node d2. In contrast, network coding is applied to two pairs of pieces as shown in Fig. 2 if piece P2 is also classified into group G2 in Fig. 3. In this case, damage to two pieces due to a failure on link3 precludes network decoding of the other two pieces in receiver node d1 and damage to two pieces due to a failure on link4 precludes network decoding of the other two pieces in receiver node d2. Thus, a method for grouping the pieces also enables control of the tradeoff between reliability and efficiency in the proposed scheme.

The multicast technique enables link bandwidth sharing between delivery routes for multiple identical pieces to the different receiver nodes. Furthermore, the network coding technique enables link bandwidth sharing between the delivery routes for multiple different pieces to different receiver nodes [6]. Thus, the required link bandwidth is given by the maximum value of the number of delivery routes to each receiver node if the link bandwidth for each delivery route is 1.0. In the proposed scheme, the link bandwidth required for each group is given by the maximum value of the number of delivery routes belonging to the group and directed to each receiver node. The total utilized link bandwidth is identical to the sum of the link bandwidth required for each group.

Figure 4 illustrates an example of the bandwidth utilized in a specific link. In Fig. 4, the existence of each square box means that a delivery route carrying the piece indicated by the vertical axis and directed to the receiver node indicated by the horizontal axis traverses the considered link. Two delivery routes for identical piece #1 can share the link bandwidth thanks to the multicast technique. Likewise, three delivery routes for piece #2 and two delivery routes for piece #(m+2) can also share the bandwidth, respectively. Meanwhile, two delivery routes for two pieces #(m+1) and #(m+n) in the same group G2 can share the link bandwidth thanks to the network coding technique. If the pieces are not classified into two groups, three delivery routes for three pieces #m, #(m+1), and #(m+n) can share the bandwidth thanks to the network coding technique. When m = n = 3, the link bandwidth required for group G1 and group G2 are 3.0 and 2.0, respectively. Thus, the total utilized link bandwidth is 5.0 in the proposed scheme. In contrast, the total utilized link bandwidth is reduced to 4.0 when the pieces are not classified into the two groups.

Fig. 4. An example of link bandwidth utilization in the proposed scheme

III. OPTIMIZATION OF MULTICAST DELIVERY SCHEME

A. Optimization of Proposed Scheme The proposed multicast delivery scheme can minimize the

total utilized link bandwidth while satisfying the given requirement on the content loss probability that each receiver cannot reconstruct the original content. The optimum number of groups and optimum delivery routes for the pieces need to be selected in the proposed scheme. Figure 5 shows the optimization procedure for the proposed scheme. Since the content loss probability is reduced due to the increase in the number of groups, the number of groups (Ng) is increased starting from one until the delivery routes satisfying the requirement on the content loss probability can be obtained. The delivery routes satisfying the requirement are computed by solving the delivery route optimization problem when the number of groups is given. The number of groups and the delivery routes that satisfy the requirement for the first time indicate the optimum number of groups and the optimum delivery routes in the multicast delivery. If the delivery routes satisfying the constraint on the content loss probability cannot be obtained even when the number of groups (Ng) is equal to the number of pieces (n), the proposed scheme cannot realize the multicast delivery that satisfies the given requirement.

Fig. 5. Optimization procedure for the proposed multicast delivery scheme

B. Delivery Route Optimization Problem The delivery route optimization problem needs to be solved

in Step 2 in the above procedure for optimizing the proposed multicast delivery scheme. The delivery route optimization problem can be formulated using an integer linear programming (ILP) model [8, 9]. Table I defines notations used in the ILP model. The constraints in the ILP model are given as follows. First, the following constraints hold as the route preservation rule:

D~d 1=∀ np pdX pdX

psOutl l

psInl l ~1 ;1),(;0),(

))(())((　　 =∀=∑=∑

∈∈

np pdX pdXdOutl

l dInl

l ~1 ;0),(;1),()()(

　　 =∀=∑=∑∈∈

npd-ps-Ndnd

pdpdndOutl

l ndInl

l XX

~1 ,)(

;1),(),()()(

　　　 =∀∈∀

≤∑∑∈∈

= (1)

In the proposed multicast delivery scheme, damage to one piece in a group may preclude network decoding of multiple pieces in the same group. The detailed delivery routes determine which of the pieces in the same group are lost in each receiver node. Nevertheless, the detailed delivery routes are unknown prior to solving the ILP model. Thus, this paper assumes that damage to one piece in a group precludes network decoding of all the pieces in the same group. This assumption

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overestimates the content loss probability and gives a measure related to the reliability on the safety side.

Since the value of Y c (d, gr) becomes 1 when at least one piece in the group gr incurs damage due to a failure on link combination c, the following constraint holds:

Cc D,d ∈∀=∀ ~1 　clPpgrd Ypd X grc l ∈∀∈∀≤ , );,(),( (2)

Since the value of Z c (d) becomes 1 only when more than n – k pieces are lost in receiver node d due to a failure on link combination c, the following constraint holds:

Cc D,d ∈∀=∀ ~1

;)(

)()),((1))(1(1

dZA

kngrdY|P|dZA

c

N

grc grc

g

×≤

−−∑ ×≤+−×−=

(3)

In the above constraint, the notation |Pgr| indicates the number of pieces in group gr.

The content loss probability that each receiver node cannot reconstruct the original content is given by the sum of the probabilities of multiple-link failures that induce the loss of more than n – k pieces in the receiver node. Since the average content loss probability must be less than or equal to the allowed value specified as the reliability requirement, the following constraint holds:

;1

)( ReqDd ZpbD

dcl

clCc≤

×∏∑ ∑

=∈∈

(4)

In the above constraint, it is assumed that the probability of a multiple-link failure can be expressed by the product of the probabilities of single-link failures on the component links.

TABLE I. NOTATIONS IN THE ILP MODEL

As explained in subsection II-B, the link bandwidth required for each group is given by the maximum value of the number of delivery routes belonging to the group and directed to each receiver node when the link bandwidth for each delivery route is 1.0. Thus, the following constraint holds:

gN~gr L,l 1=∀∈∀

Ddgr BWpd X l l Pp gr

~1 );(),( =∀≤∑∈

(5)

The total bandwidth required for each link is the sum of the link bandwidth utilized by each group. Since the delivery routes are optimized to attain the highest efficiency, the objective function to be minimized is the total link bandwidth required for each receiver node and is given as follows:

;1

)( DgrBWObj Ng

gr l

Ll∑∑=∈

= (6)

The final values of Xl (d, p) indicate the optimum delivery routes. The number of binary variables involved in the above ILP model becomes O(nD|L| fm) when the notation |L| denotes the total number of links, and notation fm indicates the maximum number of links involved in a combination of links on which a multiple-link failure is considered. This means that the number of binary variables increases rapidly as the number of links for the multiple-link failure increases. Thus, solving the globally optimum delivery routes from the ILP model requires large amounts of computational resources.

C. Heuristic Delivery Route Computation Methods Two kinds of heuristic delivery route computation methods

are proposed to optimize the proposed multicast delivery scheme in networks on a practical scale. Figure 6 shows the overall procedure in the proposed heuristic methods. The notations used in Fig. 6 are identical to those shown in Table I. Method 1 computes the sub-optimum delivery routes for minimizing the content loss probability in each receiver node. In Method 1, the given constraint on reliability may be satisfied immediately by a smaller value of Ng, and this may result in delivery routes with higher efficiency thanks to the small value of Ng. Method 2 computes the sub-optimum delivery routes that minimize the total link bandwidth required for each receiver node. In Method 2, the computed delivery routes with higher efficiency may just satisfy the reliability requirement. Both of the methods are based on the greedy approach and compute each of the delivery routes successively by using the conventional shortest route algorithm [12]. The methods repeatedly adjust the distance of the links and compute the shortest route between each pair of server and receiver nodes. The proposed methods can compute all delivery routes quickly since they only execute the shortest route algorithm n|D| times.

The adjustment of link distance is only different from each other in Method 1 and Method 2. Method 1 aims for the highest reliability and thus adjusts the link distance based on the given probability of a failure on each link [5]. The distance of a link is adjusted to the increase in the probability that the receiver node cannot reconstruct the original content if the new computed delivery route only traverses the considered link whose distance is adjusted. When the first through (n – k)-th delivery routes directed to each receiver node are computed, the distance of the link is adjusted to the probability that all existing and new delivery routes directed to the receiver node

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incur damage if the new delivery route only traverses the considered link. When the (n – k + 1)-th through n-th delivery routes directed to each receiver node are computed, the distance of the link is adjusted to the probability that just n - k existing delivery routes and the new delivery route directed to the receiver node incur damage. It is assumed that damage to one piece in a group precludes the network decoding of all pieces in the same group when the probability that each delivery route will incur damage is computed. Each of the shortest routes obtained from the adjusted link distance gives the delivery route that minimizes the increase in the probability that the receiver node cannot reconstruct the original content.

(a) Procedure in Method 1

(b) Procedure in Method 2

Fig. 6. Procedure in the proposed delivery route computation methods

Method 2 aims for the highest efficiency of the multicast delivery and thus adjusts the link distance on the basis of the utilized bandwidth on the considered link. The distance of the link reflects the increase in the utilized bandwidth when the new delivery route traverses the considered link whose distance is adjusted. When an existing delivery route for the identical piece already traverses the considered link, the link distance can be adjusted to a sufficiently small value thanks to the multicast technique. When an existing delivery route carrying a different piece and directed to a different receiver node already passes through the considered link, the link distance can also be adjusted to a sufficiently small value thanks to the network coding technique. Each of the shortest routes computed from the adjusted link distance gives the delivery route that minimizes the increase in the total utilized link bandwidth.

IV. EVALUATION OF MULTICAST DELIVERY SCHEME

A. Setting for Evaluation This section verifies that the proposed multicast delivery

scheme can control the tradeoff between reliability and

efficiency. As a measure of reliability, the content loss probability (Loss) that each receiver node cannot reconstruct the original content is evaluated. In the evaluation, the probability of failure on each link is assumed to be identical and sufficiently small. This means that the probability of a multiple-link failure on f + 1 links can be neglected in comparison with the probability of a multiple-link failure on f links. Thus, the minimum number of links fm whose failures induce damage to more than n - k delivery routes to each receiver node and the number of risky link combinations |C| composed of such fm links represent the approximate content loss probability. When the probability of a multiple-link failure can be expressed by the product of the probabilities of failures on the component links, the content loss probability can be expressed as follows:

|| CpbLoss mf ×= (7) In the above expression, the probability of failure on each link is denoted by notation pb. The value of pb is set at 0.001 as a typical value to clarify the performance of the proposed scheme.

As a measure of efficiency, the required link bandwidth (BW) in each receiver node is also evaluated. The required link bandwidth is given by dividing the total link bandwidth required for multicast delivery by the number of receiver nodes. The link bandwidth of each delivery route is assumed to be 1.0. The content loss probability (Loss) is derived from the computed delivery routes assuming that network coding vectors for the pieces directed to each receiver node are always independent of each other. From the above assumption, each receiver node cannot decode received coded pieces only when the number of received coded pieces is less than the number of native pieces embedded in the received coded pieces. The required link bandwidth (BW) is also obtained from the computed delivery routes. The multicast delivery routes are computed using a CPU with a clock speed of 3.16 GHz and RAM with a capacity of 4.0 GB.

B. Evaluation in Small-Scale Network The proposed multicast delivery scheme is evaluated for a

small-scale network. Figure 7 shows the evaluated small-scale network with the NSFNET topology composed of 14 nodes and 21 bi-directional links [13]. The number of pieces n and the threshold value k for the threshold secret sharing scheme are assumed to be nine and six, respectively. The server nodes are nodes A, B, and C, and each of the server nodes delivers three pieces. The receiver nodes are nodes L, M, and N.

Fig. 7. Evaluated small-scale network

Table II shows the evaluation results when the number of receiver nodes (D) to which the content is delivered simultaneously varies. In the case where D = 1, the evaluation results indicate the average value when the content is

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individually delivered to each of nodes L, M, and N. The rows referred to as the “Proposed Method 1” and “Proposed Method 2” indicate the results obtained from the delivery routes computed by Method 1 and Method 2 in subsection III-C, respectively. As shown in Table II, Method 1 can achieve less of a content loss probability while Method 2 can reduce the required link bandwidth. When the number of groups (Ng) for the pieces is nine, only the multicast technique is effective. As shown in Table II, the required link bandwidth for multicast delivery can be significantly reduced thanks to the multicast technique and slightly reduced thanks to the network coding technique when compared with that for unicast content delivery. In contrast, the content loss probability increases due to the network coding technique although it is almost identical when the multicast technique is only effective.

TABLE II. EVALUATION RESULTS WHEN THE NUMBER OF RECEIVER NODES VARIES – CASE OF SMALL-SCALE NETWORK

Figure 8 shows the evaluation results when the number of

groups for network coding (Ng) varies. Fig. 8 also shows the results obtained from the optimum delivery routes that are solved using the ILP model formulated in subsection III-B. The optimum Method 1 indicates the results derived from the ILP model with the objective function given by the left side of inequality (4). Meanwhile, the optimum Method 2 indicates the results derived from the ILP model with the objective function given by expression (6). In both ILP models, reliability constraint (4) is not taken into account. Optimum Method 1 can achieve the lowest content loss probability while optimum Method 2 can minimize the required link bandwidth. Nevertheless, optimum Method 1 becomes intractable in the utilized computational environment when the value of fm is more than one and the content loss probability is reduced to less than 0.001. In Fig. 8, two methods for grouping the pieces are considered. In grouping 1, a set of pieces classified into the same group are delivered from an identical server node. In contrast, a set of pieces classified into the same group are delivered from different server nodes in grouping 2.

As shown in Fig. 8(a), the proposed Method 1 has sufficient accuracy in comparison with optimum Method 1 that strictly minimizes the content loss probability. Likewise, the proposed Method 2 has sufficient accuracy in comparison with optimum Method 2 that strictly minimizes the required link bandwidth as shown in Fig. 8(b). The content loss probability in the proposed scheme can be reduced due to the increase in the number of groups, although the required link bandwidth increases slightly. As shown in Fig. 8(a), the content loss probability in Method 1 decreases for grouping 1 because pieces in different groups are delivered from different server nodes using separated routes and seldom incur damage simultaneously. This means that the number of pieces whose

network decoding may be precluded decreases in grouping 1. As shown in Fig. 8(b), the required link bandwidth in Method 2 decreases for grouping 2 since the delivery routes for pieces in an identical group often intersect on the way from different server nodes to different receiver nodes. This means that the number of nodes where the network coding technique can be applied increases in grouping 2.

(a) Content loss probability (b) Required link bandwidth

Fig. 8. Evaluation results when the number of groups varies – Case of small-scale network

C. Evaluation in Networks on a Practical Scale The proposed multicast delivery scheme is evaluated for

ten random networks of a practical scale [14]. All the evaluated networks are composed of 100 nodes and 200 bi-directional links. The number of pieces n and the threshold value k for the threshold secret sharing scheme are assumed to be 16 and 8, respectively. Four server nodes and four receiver nodes are randomly selected in each network. However, the maximum number of link-disjoint routes from the four server nodes to each receiver node is always three. Each server node delivers four pieces.

TABLE III. EVALUATION RESULTS WHEN THE NUMBER OF RECEIVER NODES VARIES – CASE OF PRACTICAL-SCALE NETWORKS

Table III shows the evaluation results when the number of

receiver nodes (D) to which the content is delivered simultaneously varies. In the case where D = 1, the evaluation results indicate the average value when the content is individually delivered to each of the receiver nodes. In the case where D = 2, the evaluation results indicate the average value when the content is individually delivered to each pair of the receiver nodes. When the number of groups (Ng) for the pieces

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is 16, only the multicast technique is effective. As shown in Table III, the required link bandwidth can be significantly reduced thanks to the multicast technique and slightly reduced thanks to the network coding technique, when the number of receiver nodes increases. In contrast, the content loss probability increases due to the network coding technique as the number of receiver nodes increases.

Figure 9 shows the evaluation results when the number of groups for the network coding (Ng) varies. Figure 9 only shows the results in the proposed Method 1 and Method 2, since the optimum Method 1 and Method 2 become intractable in the utilized computational environment. In Fig. 9, both grouping 1 and grouping 2 are considered as the methods for grouping the pieces. In Fig. 9, the average values obtained from the results in the ten random networks are plotted.

(a) Content loss probability (b) Required link bandwidth

Fig. 9. Evaluation results when the number of groups varies – Case of practical-scale networks

Figure 9 shows similar characteristics as in Fig. 8. When the number of groups increases, the content loss probability can be reduced while the required link bandwidth increases slightly. This means that the proposed multicast delivery scheme can control the tradeoff between reliability and efficiency. For the same reason as in Fig. 8, the content loss probability in Method 1 decreases for grouping 1 and the required link bandwidth in Method 2 decreases for grouping 2. The proposed scheme can minimize the utilized link bandwidth while satisfying the given requirement on reliability. For example, the proposed scheme that adopts Method 1 can minimize the required link bandwidth by adjusting the number of groups to four and classifying the pieces with grouping 1 when the required content loss probability is less than 0.0001.

V. CONCLUSIONS This paper proposed a multicast delivery scheme for the

threshold secret shared content, which can control the tradeoff between reliability and efficiency. This paper also proposed

two kinds of heuristic delivery route computation methods to optimize the proposed multicast delivery scheme. The content loss probability and required link bandwidth were evaluated as two measures of reliability and efficiency, respectively. Although the required link bandwidth can be reduced thanks to multicast delivery, the content loss probability increases due to network coding between multiple pieces obtained from the original content. The evaluation results revealed that the proposed multicast delivery scheme, where the appropriate method classifies the pieces into the optimized number of groups, can actually minimize the total link bandwidth required while satisfying the given requirement on content loss probability.

ACKNOWLEDGMENT The authors would like to thank Dr. Nakajima, president &

CEO, and Dr. Ano, executive director, of KDDI R&D Laboratories Inc. for their encouragement throughout the study.

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