(Which additional input element can be used to implement integrity in combination with symmetric ciphers?

Symmetric encryption alone typically provides confidentiality, but it does not automatically provide integrity. Many encryption modes (especially older ones like CBC without authentication) are malleable, meaning an attacker may be able to modify ciphertext and cause predictable changes in plaintext after decryption. To add integrity, systems commonly combine symmetric encryption with a cryptographic hash-based integrity mechanism, such as a hash function used in an HMAC (Hash-based Message Authentication Code) or a dedicated authenticated-encryption mode like GCM that internally uses authentication tags. Among the given options, a hash function is the fundamental additional element that enables integrity checks: it allows construction of a MAC (e.g., HMAC-SHA-256) that the receiver verifies to detect any tampering. An initialization vector and a nonce value are used to ensure uniqueness and randomness properties for encryption but do not, by themselves, guarantee integrity. An encoding algorithm changes representation, not security. Therefore, the correct additional input element for implementing integrity alongside symmetric encryption is a hash function, typically as part of an HMAC or similar MAC construction.