Each pair needs its own shared key, so the total is n·(n-1)/2 — quadratic in n. Worse, every new user has to securely exchange a key with every existing user before they can talk.

Two compounding problems:

1. Storage: every user must store n−1 keys. Add a million users and each device holds nearly a million keys.
2. Distribution: how do you securely deliver each fresh pairwise key in the first place? A secure channel doesn't exist yet — that's the whole point of crypto.

The fix — public-key (asymmetric) cryptography. Each user has one keypair (public + private). Anyone who wants to talk to Alice uses Alice's public key; only she can decrypt. Number of keys grows linearly with n, not quadratically — and the public key can be shared on a billboard.

Tip: Modern systems are hybrid: asymmetric crypto only bootstraps a per-session symmetric key, then the actual data is encrypted symmetrically (1000× faster). TLS, SSH, Signal all work this way.