The Goldreich-Goldwasser-Micali (GGM) construction and IND-CPA encryption

Note: This problem may or may not be directly related to CPAs, but just gets you thinking about cryptography and the associated cryptoanalysis.

Consider the following cryptographic scheme that combines encryption and digital signatures:

  1. Key Generation: Alice generates two pairs of keys - (kenc,kdec)(k_{enc}, k_{dec}) for encryption/decryption and (ksign,kverify)(k_{sign}, k_{verify}) for digital signature.

  2. Encryption: To encrypt a message mm, Alice computes c=Enckenc(m)c = Enc_{k_{enc}}(m).

  3. Digital Signature: To sign a message mm, Alice computes s=Signksign(m)s = Sign_{k_{sign}}(m).

  4. Combined Ciphertext: The combined ciphertext for a message mm is the concatenation of the encryption and digital signature: ccombined=cβˆ₯sc_{\text{combined}} = c \| s.

  5. Verification and Decryption: To verify the message and decrypt, Bob receives the combined ciphertext ccombinedc_{\text{combined}} and performs the following steps:

    • Verify the signature: Verifykverify(m,s)Verify_{k_{verify}}(m, s)
    • If the signature verification fails, reject the message.
    • If the signature verification succeeds, decrypt the message: m=Deckdec(c)m = Dec_{k_{dec}}(c).

Task: Explain whether this cryptographic scheme provides confidentiality and integrity. If it provides both, justify how. If not, explain the vulnerabilities and potential attacks against this scheme. Additionally, suggest possible improvements to address any identified weaknesses.