ISRO CS 2025 Q.8: solution and answer

ISRO 2025 · CSQ.8 · 1 mark · MCQ

How many keys are needed for encrypted communication among 16 persons using AES encryption assuming unique key is used between any two persons:

See the question in the paper

Answer and solution

ANSWER

120 keys, option (A).

AES is a symmetric cipher: the same secret key is used to encrypt and to decrypt, so one key serves both directions between two persons.
Each pair of persons needs its own unique key, and the pair (X, Y) is the same pair as (Y, X). So we count unordered pairs, that is, 2-element subsets of the 16 persons.
Number of pairs $= (2 16) = \frac{16 \times 15}{2}$ .
$\frac{16 \times 15}{2} = \frac{240}{2} = 120$ keys.
Other options: $256 = 1 6^{2}$ counts ordered pairs including a person paired with themselves. $32 = 2 \times 16$ grows only linearly and gives each person just two keys. $2^{16}$ counts all subsets of the 16 persons, not pairs.

KEY IDEA

A symmetric cipher needs one shared key per unordered pair of users, so $n$ persons need $(2 n) = \frac{n ( n - 1 )}{2}$ keys.

WATCH OUT FOR

Counting ordered pairs, giving $n^{2} = 256$ (or $n (n - 1) = 240$ ), when the key between X and Y is the same as the key between Y and X.