The number of symmetric keys required for a group of nnn people, where each person wants to communicate secretly with every other person, and the communication between any two persons should not be decodable by others, is given by the formula:
n(n−1)2\frac{n(n-1)}{2}2n(n−1)
This is because each pair of people needs a unique shared key, and there are (n2)=n(n−1)2\binom{n}{2}=\frac{n(n-1)}{2}(2n)=2n(n−1) such pairs in a group of nnn individuals
Explanation:
- Each person needs a distinct key to communicate securely with each of the other n−1n-1n−1 people.
- However, the key between person A and person B is the same as the key between person B and person A (symmetric key).
- Hence, the total number of unique keys is the number of unique pairs, which is n(n−1)2\frac{n(n-1)}{2}2n(n−1).
For example:
- For 6 people, the number of keys required is 6×52=15\frac{6\times 5}{2}=1526×5=15.
- For 100 people, the number of keys required is 100×992=4950\frac{100\times 99}{2}=49502100×99=4950.
This ensures that communication between any two people remains confidential from the others in the group.
