First, let’s understand what hybrid encryption is and why it’s useful. Hybrid encryption combines symmetric key cryptography (which is faster for bulk data) with asymmetric key cryptography (which provides secure distribution of keys). This allows us to have the best of both worlds fast decryption times and secure transmission of keys over insecure channels.

In our hybrid scheme, we will be using ECC for generating a public-private key pair and AES-GCM for symmetric encryption/decryption. The reason why we chose BrainpoolP256r1 curve is that it provides high security with relatively small key sizes (compared to other curves).

To implement this scheme, follow these steps:

Step 1: Generate a public-private key pair using the ECC library in Pycryptodome. This can be done by calling the generate() function on an instance of the EllipticCurve object with the desired curve name (in our case, ‘brainpoolP256r1’).

Step 2: Export the public key as a DER-encoded byte string using the export_key() method. This can be done by calling the private_bytes() function on an instance of the EccPrivateKey object with the format parameter set to ‘DER’.

Step 3: Encrypt the data using AES-GCM and generate a nonce (a random number used for encryption) using the Random library in Python. This can be done by calling the encrypt() function on an instance of the Cipher object with the desired key, mode, and IV parameters set accordingly.

Step 4: Combine the public key, ciphertext, nonce, and authentication tag (generated during encryption) into a single byte string using the DER encoding format. This can be done by calling the dump() function on an instance of the PKCS12 object with the appropriate parameters set accordingly.

Step 5: Send this byte string over an insecure channel to the recipient, who will then import it as a PKCS#8-encoded private key using the import_key() method. This can be done by calling the load_pkcs12() function on an instance of the FileIO object with the appropriate parameters set accordingly.

Step 6: Decrypt the data using AES-GCM and verify the authentication tag to ensure that it hasn’t been tampered with during transmission. This can be done by calling the decrypt_and_verify() function on an instance of the Cipher object with the appropriate parameters set accordingly.

Step 7: Export the private key as a DER-encoded byte string using the export_key() method, which can then be used for generating new public keys and encrypting data in future sessions.