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Suppose that everyone in a group of $N$ people wants to communicate secretly with $(N-1)$ other people using symmetric key cryptographic system. The communication between any two persons should not be decodable by the others in the group. The number of keys required in the system as a whole to satisfy the confidentiality requirement is

1. $N(N-1)$
2. $N(N-1)/2$
3. $2N$
4. $(N-1)^2$

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The problem is actually the structure of an undirected complete graph with N vertices. The solution is the number of edges, which is equal to n(n-1)/2
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Symmetric Key Cryptographic system uses only one key for encryption and decryption, i.e. only one key will be needed between two nodes acting as sender and receiver.

Science there are N number of nodes and everyone wants to communicate with each other, there will be N * N-1 ports.

This simply means (N * N-1)/2 connections, hence only (N * N-1)/2 keys are required.