A secure group key management scheme for sensor networks?

Answer 1

Secure group communication in mobile ad hoc networks is often dynamic and impromptu, and thus re-

quires efficient and automated secure group management and seamless combination of secure groups with

distributed applications running upon them. CRTDH sceme by Balachandran et. al. [75] has been proposed

in 2005. This scheme has been used in later access control frameworks like in [76]. The scheme considers the

Chinese Remainder Theorem and Diffie-Hellman Key exchange to achieve distributed key-management for

SGC in wireless ad hoc networks. Assuming a group of n users, U1;U2; :::Un has to be formed, the following

operations are performed by each member Ui (where i = 1; 2; :::n) to obtain the shared group key.

Ui selects the Diffie-Hellman private share xi and broadcasts the public share yi = gximodp. Ui then

receives the public shares from all other members in the group and computes the DH key shared with each

of them. as mij = yxi

i mod p, where j = 1; :::; i 1; i + 1; :::n and j = i. Least Common Multiple (LCM)

of all the DH keys calculated is noted as lcmi. Ui then randomly selects ki, such that ki < min(mij ; 8j),

which will be its share of the group key. Ui also selects an arbitrary number D such that D = ki and another

number Dp such that the gcd(Dp; lcmi) = 1. Each member then solves the CRT

crti = ki mod lcmi

crti = D mod Dp

and broadcasts it to the group. After Ui receives the crt values from all the other members in the group and

calculate kj = crtj mod mij , for all j = i and compute the group key GK = k1 k2 :::kn As can be seen

from the above steps, the Chinese Remainder Theorem is used to send each member's key share (disguised)

to all the other members in the group. The Difie-Hellman key exchange is performed to derive the modulo

value in the CRT calculation. To understand the details of the scheme, let us consider a member U1 in a

group of 4 members. The first two steps of the protocol involve the generation and distribution of the DH

public share by each member in the group. U1 selects a DH private share x1 and computes its DH public share y1 = gx1 mod p. U1 then broadcasts the DH public share y1 to all the other members in the group. In

Step 3 of the protocol, all the mij values are generated, which are nothing but the DH keys shared between

U1 and the other members. U1 calculates three m values m12;m13;m14 which are equal to yx12; yx13; yx14

respectively. y2; y3; y4 are the DH public shares of members U2;U3;U4 broadcasted in Step 2. The three DH

keys (m12;m13;m14) generated by U1 are equal to m21;m31;m41 generated by U2;U3;U4 respectively. U1

then calculates the LCM of the DH keys m12;m13andm14. This LCM value will be later used for the CRT

calculation in Step 6. Step 5 of the protocol involves the generation of a random key share k1 by U1. This

k1 share has to be less than all DH keys m1;m2andm3 and the lcm1 value since we want the other members

to obtain k1 and not k1(modmij) or k1(modlcm1) respectively. In the next step, U1 generates an arbitrary

number D and Dp which will be used in solving the CRT. The Dp value should be selected such that Dp

and lcmi are co-primes, in order to solve the CRT. Also, the number D should not be equal to k1, since if

they are, then the solution to the CRT will be equal to the group key, k1. After solving the CRT in Step 6,

the solution is broadcasted to the group in Step 7. U1 solves the CRT to obtain crt1 and broadcasts it to

the the group. U1 also receives the CRT values crt2; crt3; crt4 from the other members in the group. U1 can

obtain k2; k3; k4 by performing the following operations.

k2 = crt2( mod m12)

k3 = crt3( mod m13)

k4 = crt4( mod m14).

The individual ki shares are then XOR-ed to obtain the group key GK. Similarly all the members in the group arrive at the same group key, since the following holds kj = crtj mod LCMj = crtj mod mij. Any member,

such as Ui, receives the (broadcast) values crt1 from U1; :::; crti 1 from Ui 1; crti + 1from Ui + 1; :::;and

crtn from Un. Ui can then compute k1; :::; ki 1; ki+1; :::and kn using m(i; 1); :::;m(i; i 1);m(i; i+1); ::: and

m(i; n) respectively. Along with its own ki;Ui has all the elements for computing the group key. As a result,

all the members will compute the same key.