Question

Suppose N people want to communicate with each of N – 1 other people using symmetric key encryption. All communication between any two people, i and j, is visible to all other people in this group of N, and no other person in this group should be able to decode their communication. How many keys are required in the system as a whole? Now suppose that public key encryption is used. How many keys are required in this case?

Short answer

For symmetric key encryption, \( \frac{N(N-1)}{2} \) keys are needed; for public key encryption, 2N keys are needed.

Step-by-step solution

Understand Symmetric Key Encryption

In symmetric key encryption, each pair of users (person i and person j) needs a unique key for their communication. Since communication is private, the key cannot be shared with others.

Calculate Key Requirements for Symmetric Key

For N people, each person needs to communicate securely with N-1 others. Thus, the number of unique pairs of people is given by the combination formula \( \binom{N}{2} = \frac{N(N-1)}{2} \), representing the number of unique keys needed.

Understand Public Key Encryption

In public key encryption, each person has a pair of keys: a public key (shared with everyone) and a private key (kept secret). This allows all others to send encrypted messages using the public key, which only the private key holder can decrypt.

Calculate Key Requirements for Public Key

Since each of the N people has one public and one private key, a total of 2N keys (N public + N private) are required for the system.

Key concepts

Understanding Symmetric Key Encryption

Symmetric key encryption is like having a secret handshake between friends. Each pair of friends (or people) uses a unique key, like a special handshake, to exchange messages that only they can understand. To maintain this secrecy, the same key must be used for both encryption and decryption.
In this system, if you have a group of N people and everyone wants to communicate privately with everyone else, each pair needs its own key. This is because the key used between one pair cannot be shared with another, as that would compromise the privacy.
To figure out how many unique keys we need, we use combinatorial math. The formula for the number of unique pairs is \( \binom{N}{2} = \frac{N(N-1)}{2} \). This means for 5 people, you'd need 10 unique keys. The rule of thumb here is the more people there are, the more keys are needed exponentially.

Exploring Public Key Encryption

Public key encryption is like handing out a contact card that lets anyone send you a message, but only you can read it. Each person has two keys: a public key (the contact card) and a private key (your personal decoder). While the public key is shared with everyone, the private key is kept secret.
This method allows others to send you secure messages using your public key, which only your private key can decrypt. Because of this two-key system, the communication can remain private without the headache of managing a different key for every pair of people.
If there are N people, each person needs their own two keys (one public, one private), so the total number of keys in the system is 2N. This structure simplifies the management of keys, especially as the number of people increases.

Calculating Encryption Keys

The calculation of keys is essential to ensure secure communication, and it varies based on the type of encryption used.
With symmetric key encryption, the number of keys needed grows quickly because each communication pair requires its own key. The formula \( \frac{N(N-1)}{2} \) efficiently calculates this need. For instance, for the case where 5 people communicate, this results in 10 unique keys.
On the other hand, public key encryption simplifies this by assigning two keys to each participant—one public, one private, which totals to 2N keys for the system. This system becomes especially beneficial as N gets large, providing a clear systematic approach for secure communications among participants.