GATE CS 2015 Set 1 Q.31 (CS Q.21): Application Layer: DNS, HTTP, SMTP, FTP

Suppose that everyone in a group of $N$ people wants to communicate secretly with the $N-1$ others 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:

$2N$

$N(N-1)$

$N(N-1)/2$

$N^2$

Official answer and solution

Official answer: C
From: Official GATE 2015 answer key

Solution

Calculate the number of pairwise communication channels among N individuals.

In a symmetric key cryptosystem, each pair of communicating parties must share a unique secret key that no other party possesses.

Among a group of $N$ people, the number of distinct two-person pairs is given by the combination formula:

\binom{N}{2} = \frac{N(N-1)}{2}.

Thus, the total number of symmetric keys required across the entire system is $\frac{N(N-1)}{2}$ (option C).

Final answer

Common trap

Confusing symmetric key requirements ( $\binom{N}{2}$ ) with asymmetric key pairs ( $2N$ ).'}

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 1
Question: 31 of 65, printed as CS Q.21
Section: Computer Science and Information Technology

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