Common use of Strength of Keys against Brute Force Attacks Clause in Contracts

Strength of Keys against Brute Force Attacks. ‌ The master keys and private keys are random and large. For example, with values of N = 7, m = 16, η = 6 and p = 31, there are 2634 possible master keys and 2208 private keys. A brute force attack is not feasible. Pairwise Key One limitation in the original Xxxx’x scheme is that the pairwise key is only the same size as the data size of the master key elements. In our BYka scheme, the pairwise key size can be up to pNη2 integers ∈ [0, p − 1]. The BYka scheme can be viewed as a mechanism for two nodes to derive a common secret pairwise key set R consisting of Nη2 integers from which to construct their pairwise key. The number of possible keys, the “key space”, is limited by the number of possible combinations of the Nη2 integers. To determine the key space size, we consider the following partitioning problem. Given a row of Nη2 items, we wish to partition them into p groups. This is illustrated in Figure 1 for the case of partitioning eight items into four groups. To create the partitions, we first insert (p − 1) items into the row, so that there are now (Nη2 + p − 1) items. If any (p − 1) items are now removed, (p − 1) Nη2 + p − 1 gaps would be created, separating the remaining items into p groups as desired. Let group g0 contain the integer zero, g1 contain one, g2 contain two, etc. The total number of integers is always Nη2. The number of ways to remove (p − 1) items from (Nη2 + p − 1) gives the key space size as follows, Ksp = = log2 p − 1 2 Nη + p − 1 p − 1 bits

Strength of Keys against Brute Force Attacks. ‌ The master keys and private keys are random and large. For example, with values of N = 7, m = 16, η = 6 and p = 31, there are 2634 possible master keys and 2208 private keys. A brute force attack is not feasible. Pairwise Key One limitation in the original Xxxx’x scheme is that the pairwise key is only the same size as the data size of the master key elements. In our BYka scheme, the pairwise key size can be up to pNη2 integers ∈ [0, p − 1]. The BYka scheme can be viewed as a mechanism for two nodes to derive a common secret pairwise key set R consisting of Nη2 integers from which to construct their pairwise key. The number of possible keys, the “key space”, is limited by the number of possible combinations of the Nη2 integers. To determine the key space size, we consider the following partitioning problem. Given a row of Nη2 items, we wish to partition them into p groups. This is illustrated in Figure 1 for the case of partitioning eight items into four groups. To create the partitions, we first insert (p − 1) items into the row, so that there are now (Nη2 + p − 1) items. If any (p − 1) items are now removed, (p − 1) Nη2 .Nη2 + p − 1 1Σ gaps would be created, separating the remaining items into p groups as desired. Let group g0 contain the integer zero, g1 contain one, g2 contain two, etc. The total number of integers is always Nη2. The number of ways to remove (p − 1) items from (Nη2 + p − 1) gives the key space size as follows, Ksp = = log2 p − 1 2 Σ. ΣΣ2 Nη + p − 1 p − 1 bits