As mentioned below, the Bitcoin network relies on many cryptographic primitives (hashing, signatures, random number generation, etc.) however it does not rely on encryption specifically to function. That being said, encryption (hiding information) is often used in conjunction with Bitcoin.
Some examples of where encryption is used to secure Bitcoin communications and private keys:
- Bitcoin Core can encrypt the wallet.dat file which holds your private keys, thus preventing a casual attacker from stealing your bitcoins if they get that file but do not have the password.
- BIP-38 Allows users to encrypt a so called “paper wallet” so that you can keep your private keys offline and isolated from computer-based attacks. Standard (unencrypted) paper wallets allow anyone holding the piece of paper to spend the bitcoins. BIP-38 requires both the piece of paper and a password to access the private key.
- BIP-75 Uses encryption to hide payment invoice details and ensure only the parties involved in a transaction can see the details.
- The new BIP-151 (not implemented yet) establishes a system to encrypt communication between nodes on the Bitcoin network to enhance transaction privacy for all.
- Any situation where you are using https, TLS or SSL — you are using encryption. Examples: all web-wallets like Blockchain.info, payments through a payment processors like Coinbase or BitPay and most “light” wallets such as Mycelium, Electrum and Copay use TLS to communicate with a backend server.
So, technically, Bitcoin does not require encryption to function, but it would be very unsafe to use without encryption. Many bitcoins were stolen before encryption became commonly used in Bitcoin wallet software.