Xxxxx Attacks Sample Clauses

Xxxxx Attacks. In this attack, the attacker would fabricate new public and private keys by combining captured keys and use them to masquerade legitimate nodes. Consider that n nodes and their public and private keys have been obtained. The attacker can fabricate a new public key VX by linear combination of captured public keys as follow: VX = α1V1 + · · · + αnVn (mod p) (7) The corresponding private key KX would also be a similar linear combination of the captured private keys, KX = VT M = (α1VT + · · · + αnVT )M‌ = α1VT M + · · · + αnVT M = α1K1 + · · · + αnKn (mod p) (8) By choosing various combinations of α1, · · · , αn, the attacker is able to fabricate any public key and the corresponding private key at will. To defeat this attack, three conditions must be met: (1) the public keys must conform to a prescribed structure, (2) the public keys are linearly independent, and (3) no more than (m − 1) nodes are captured, i.e., n < m. The first condition ensures that a key formed from arbitrary linear combinations of captured keys would not be accepted. If all of the public keys are of a prescribed structure, such as those of the column of the Vandermonde matrix, arbitrary public keys would simply be discarded. If all of the public key vectors are linearly independent and n < m, then by definition, the solution of Equation (7) is trivial, i.e., α1, · · · , αn = 0. On the other hand, if n ≥ m, then, as there are at most m linearly independent (m× 1) vectors, any m public keys can be combined to obtain a non-trivial solution in Equation (7) and obtain the corresponding private key using Equation (8).