Suppose that everyone in a group of \(𝑁\) people wants to communicate…
2018
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
- A.
\(N(N - 1)\) - B.
\(N(N-1)/2\) - C.
\(2N\) - D.
\((N-1)^2\)
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Correct answer: B
Answer: N(N-1)/2
Explanation: A single symmetric key can be shared between two people to secure their communication in both directions, so each unordered pair of people needs exactly one unique key.
Count ordered pairs: choose the first person in N ways and the second in N−1 ways, giving N(N−1).
Adjust for double counting: each unordered pair was counted twice in the ordered count, so divide by 2 to obtain N(N−1)/2.
Alternate viewpoint: this is the number of 2-person combinations from N people, i.e., C(N,2) = N(N−1)/2.
Example: For N = 4, required keys = 4×3/2 = 6 (pairs: person1-person2, person1-person3, person1-person4, person2-person3, person2-person4, person3-person4).
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