If N people use secret-key encryption for privacy and every pair must be able…
2016
If N people use secret-key encryption for privacy and every pair must be able to communicate privately, which expression gives the number of distinct secret keys required?
Answer: C. N(N − 1)/2 — ConceptWhen one unique resource is assigned to each unordered pair chosen from N distinct entities, the total is the combination C(N, 2). Counting ordered…
- A.
N
- B.
N − 1
- C.
N(N − 1)/2
- D.
N(N + 1)/2
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Correct answer: C
Concept
When one unique resource is assigned to each unordered pair chosen from N distinct entities, the total is the combination C(N, 2). Counting ordered selections gives N(N − 1), but each unordered pair appears twice, so C(N, 2) = N(N − 1)/2.
Application
Each person must share one secret key with every other person, so begin with N choices for the first person and N − 1 choices for the partner.
The product N(N − 1) counts every pair twice: once as person A with person B and once as person B with person A.
Divide by 2 to remove this double counting, giving N(N − 1)/2 distinct shared keys.
Cross-check
For N = 4, the pairs are AB, AC, AD, BC, BD, and CD: 6 pairs. The formula gives 4(4 − 1)/2 = 6, so the count agrees with direct enumeration.
Therefore, the required number of secret keys is N(N − 1)/2.