Resilience against Xxxxx Attacks Sample Clauses

Resilience against Xxxxx Attacks. The Xxxxx attack cannot be mounted as in Xxxx’x scheme. Consider that m private keys KC1M1 , · · · , KCmM1 and the corresponding public keys VC1 , · · · , VCm associated with one of the master keys M1 have been obtained. The attacker chooses a public key IDX seed sX1 and constructs the public key VX1 as a Vandermonde column vector, such that,

Resilience against Xxxxx Attacks. The Xxxxx attack cannot be mounted as in Xxxx’x scheme. Consider that m private keys KC1M1 , · · · , KCmM1 and the corresponding public keys VC1 , · · · , VCm associated with one of the master keys M1 have been obtained. The attacker chooses a public key IDX seed sX1 and constructs the public key VX1 as a Vandermonde column vector, such that, VX1 = α11 VC1 + · · · + α1m VCm (mod q) (10) The coefficients α11 , · · · , α1m can be obtained and used to construct the private key associated with M1 and VX1 , 1M 1 X1 C1 Cm KX = VT M1 = (α1 VT + · · · + α1 VT )M1 Here, KC1M = α11 KC1M1 + · · · + α1m KCmM1 (mod p) (11) is the private key associated with the master key M1 and public key VC1 . The difficulty is identifying which of the Nη private keys in the node is this particular one, and similarly for KC2M 1 , etc. Each private key is a row vector with elements, which are sums and products of random numbers, and is indistinguishable from the others. The order of storage in memory is also random and unrelated to the order in which they were computed. An adversary cannot derive any information about the private-public-master-key associations (PPMka) from examining the keys or its storage location. follows. From each node, there are (Nη)! ways to select the η private keys associated with M1 and the (Nη−η)! public keys VC1 , · · · , VCη . To select all of the private keys in the captured nodes associated with M1 and the corresponding public keys for use in Equation (11), we have Φ1 possible ways, given by, Σ (Nη)! Σ m (Nη − η)! To complete the Xxxxx attack, all of the public and private keys are similarly constructed for each of the master keys and used together. The total number of possible solutions for all of the PPMka’s is, NΣ−1 Σ (Nη − iη)! Σ m (Nη − iη − η)! Table 2. Number of Solutions Φ. η N Master Key Size m 7 1.64×1019 5.67×1028 2.07×1038 2.91×1057 8 9.45×1019 7.46×1028 6.30×1039 4.79×1059 7 6 1.97×1022 2.55×1033 3.43×1044 4.65×1055 7 2.07×1023 8.37×1034 3.55×1046 1.53×1058 8 1.57×1024 1.68×1036 1.90×1048 2.20×1060 8 6 2.41×1026 2.41×1026 3.55×1039 5.37×1052 7 3.52×1027 3.52×1027 1.91×1041 1.08×1055 8 3.54×1028 3.54×1028 5.88×1042 1.03×1057 As an example, with N = 7, η = 6 and m = 16, Φ = 5.67 × 1028 possible solutions. Hence, without knowing the PPMka, the Xxxxx attack requires an unfeasibly large number of trials. Table 2 gives the possible number of solutions for various keying parameters.