Optimizing Encrypted Email Overlap Calculation with Jaxite

[bllmo]

I'm using jaxite to encrypt SHA-256 hashes of email addresses and trying to calculate the overlap between two lists of emails, company_emails and partner_emails. My current approach involves comparing each bit of the encrypted hash arrays, which seems inefficient given that I have to compare 256 * 881 elements to determine similarity.

Here's a snippet of the code for clarity:

company_emails = [
    "fabrizio@example.com",
    "filippo@example.com",
]
partner_emails = [
    "fabrizio@example.com",
    "fabio@example.com",
]

company_emails_hashes = [email_to_sha256_bytes(email) for email in company_emails]
partner_emails_hashes = [email_to_sha256_bytes(email) for email in partner_emails]

def encrypt_bits(byte_array, client_key_set, lwe_random_number_generator):
    return [
        jaxite_bool.encrypt(bit, client_key_set, lwe_random_number_generator)
        for bit in byte_array
    ]

def encrypt_hashes(hashes, client_key_set, lwe_random_number_generator):
   return [
        encrypt_bits(hash_bits, client_key_set, lwe_random_number_generator)
        for hash_bits in hashes
    ]

company_emails_encrypted = encrypt_hashes(
    company_emails_hashes, client_key_set, lwe_random_number_generator
)
partner_emails_encrypted = encrypt_hashes(
    partner_emails_hashes, client_key_set, lwe_random_number_generator
)

def are_encrypted_emails_identical(
    company_emails_encrypted,
    partner_emails_encrypted,
    server_keys,
    client_keys,
    encryption_params,
):
  # _xor ?
  pass

are_encrypted_emails_identical(
    company_emails_encrypted[0],
    partner_emails_encrypted[0],
    server_key_set,
    client_key_set,
    parameters,
)

Is there a more efficient way to compare these encrypted email hashes without resorting to bit-by-bit comparison? Am I misunderstanding something, or is there a better approach available within the jaxite library?