$(1)$ Use the tool mobilefish to generate your public key for Paillier encryption and determine your private key for Paillier decryption. The size of the modulo is required to be $1024$ bits.

$(2)$ Use your public key to encrypt your monthly incomes from January to March and upload the ciphertexts to the cloud. Assume that your monthly income is MD$5(1234567||$the month$)($mod $10000).$ What are the three ciphertexts?

$(3)$ How does the cloud server compute the encryption of the sum of your monthly salaries for these $3$ months $($assume that the cloud server returns one encrypted result to you$)?$ What is the ciphertext computed by the cloud server?

$(4)$ How do you decrypt the encrypted result to get the sum of your monthly salaries for these $3$ months? Show the steps in detail.

$(5)$ Implement Paillier encryption algorithm $($submit your code$)$ and verify your encryption results in $(2).$