Question:

How is the broadcast address of a subnet calculated from the IP address and the prefix length?

For example, given the IP address 85.75.75.180 and the prefix length /9, the network address is 85.0.0.0. However, the broadcast address is not 85.255.255.255, as one might expect, but 85.127.255.255. What is the logic behind this? How does the second octet change from 0 to 127?

Answer:

How to calculate the broadcast address of a subnet

The broadcast address of a subnet is the IP address that can be used to send a message to all the hosts in that subnet. It is usually the last address in the range of valid addresses for the subnet. To calculate the broadcast address, we need to know two things: the IP address of any host in the subnet, and the prefix length of the subnet.

The prefix length of a subnet is the number of bits that are fixed in the IP address, from left to right. For example, a prefix length of /9 means that the first 9 bits of the IP address are fixed, and the remaining 23 bits can vary. The prefix length can also be expressed as a subnet mask, which is a 32-bit binary number that has 1s in the fixed bits and 0s in the variable bits. For example, a prefix length of /9 corresponds to a subnet mask of 11111111 10000000 00000000 00000000, or 255.128.0.0 in decimal notation.

The network address of a subnet is the IP address that has all the variable bits set to 0. It is usually the first address in the range of valid addresses for the subnet. To calculate the network address, we need to perform a bitwise AND operation between the IP address of any host in the subnet and the subnet mask. For example, given the IP address 85.75.75.180 and the prefix length /9, the network address is:

85.75.75.180 AND 255.128.0.0 = 85.0.0.0

The broadcast address of a subnet is the IP address that has all the variable bits set to 1. It is usually the last address in the range of valid addresses for the subnet. To calculate the broadcast address, we need to perform a bitwise OR operation between the network address and the inverse of the subnet mask. The inverse of the subnet mask is a 32-bit binary number that has 0s in the fixed bits and 1s in the variable bits. For example, the inverse of the subnet mask 255.128.0.0 is 00000000 01111111 11111111 11111111, or 0.127.255.255 in decimal notation. Therefore, given the network address 85.0.0.0 and the prefix length /9, the broadcast address is:

85.0.0.0 OR 0.127.255.255 = 85.127.255.255

This explains why the broadcast address is not 85.255.255.255, as one might expect, but 85.127.255.255. The second octet changes from 0 to 127 because the prefix length /9 only fixes the first 9 bits of the IP address, which are 01010101 in binary. The remaining 7 bits of the second octet can vary, and they are all set to 1 in the broadcast address, which is 01111111 in binary, or 127 in decimal.