Symmetric Encryption vs Public Key Encryption

Suppose you have N people who want to communicate with each other using symmetric keys. All communication between any two people, i and j, is visible to group N. Only person i and person j can decrypt each others messages.

How many keys would Symmetric Encryption require to protect group N?

I solved this with the following python function:

def count_symmetric_keys( N=2 ):
    """Provide the number of entities in group N.
    return the number of symmetric keys needed for this group"""
    keys = 0
    for i in range( 0, N ): keys += i
    return keys

If group N had 10 members, it would need to generate and maintain 45 Symmetric Keys.

If group N had 50 members, it would need to generate and maintain 1125 Symmetric Keys.

Symmetric keys are also susceptible to man-in-the-middle attacks. This attack occurs when an entity poses as a trusted entity. Let i and j be trusted entities. Let k be an un-trusted attacker. If k determined the Symmetric key it could send or recieve messages posing as i or j.

How many keys would Public-key Encryption require to protect group N?

Public Key Encryption requires 2n keys or two keys per person in group N. Public key encryption also does not require ‘pre sharing’ the secret key before communication may start. Each member would need 1 public key and 1 private key.

If group N had 10 members, it would need to generate and maintain 20 Public/Private Keys.

If group N had 50 members, it would need to generate and maintain 100 Public/Private Keys.