Fifty members want every pair of them to be able to communicate with encryption without the other members learning the contents. Which combination gives the number of keys needed with symmetric-key cryptography and with public-key cryptography (counting public and private keys together)?

Answer: A. Symmetric-key: 1225 keys; public-key: 100 keys

With symmetric-key cryptography each pair needs its own key, because if three or more people shared a key, other members could read the contents too. The number of pairs that can be chosen from n people is n(n−1)/2, so 50 people need 50 × 49 / 2 = 1225 keys. With public-key cryptography, if each person has one public key and one private key, anyone can encrypt to anyone else with the recipient's public key, so 2n = 2 × 50 = 100 keys are enough. As the number of people grows, the number of symmetric keys grows roughly with the square of the number of people, whereas public-key cryptography grows only in proportion to it. This is one reason public-key cryptography makes key management easier.

• A. Correct. Symmetric-key needs 50 × 49 / 2 = 1225 keys, and public-key needs 2 × 50 = 100 keys.
• B. 50 × 49 = 2450 counts the pair from A to B and the pair from B to A separately. The same two people need to share only one key.
• C. This swaps the two counts. It is symmetric-key cryptography whose key count grows roughly with the square of the number of people.