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If Enc is a PKE(Public Key Encryption) with key generation algorithm KG and message space $M = \{0, 1\}^n$, and SE is a Symmetric Key Encryption(SKE) with key space $\{0, 1\}^n$ and message space $M' = \{0, 1\}^*$ then the hybrid encryption uses the same key generation KG, but its message space is $M' = \{0, 1\}^*$ and its encryption procedure $Enc'(p_k, m)$ picks a random symmetric key $k \gets \{0, 1\}^n$ and outputs ciphertext $(c, c')$ where $c = Enc(p_k, k)$ and $c' = SE(k, m)$. The decryption given $(c, c')$ decrypts $k$ from $c$ using the private key corresponding to pk and then decrypts $m$ from $c'$ using $k$.

If Enc is a CPA secure PKE on $M = \{0, 1\}^n$ and SE is IND secure SKE on$ M' = \{0, 1\}^*$, is Enc' CPA secure PKE on $M'$?

If Enc is CPA secure PKE and SE is CCA secure SKE, is Enc' CCA secure?

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  • $\begingroup$ Not quite sure, but intuition tells me that Enc should be CCA secure (at least when considering active adversaries). $\endgroup$ – VincBreaker Jun 2 at 12:49
  • $\begingroup$ So "Introduction to Modern Cryptography" talks a bit about hybrid encryption and KEMs. It would appear that plain CPA-secure encryption doesn't yield a CCA-secure KEM (because eg ElGamal is CPA secure but you can win the KEM-CCA game using the homomorphism) and the book requires a CCA secure KEM for CCA secure hybrid encryption. $\endgroup$ – SEJPM Jun 4 at 9:32
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If the KEM is only CPA-secure, then the resulting scheme might not be CCA-secure. For example, consider a CPA-secure KEM that ignores the last bit of its ciphertext. Then it is possible to modify a KEM ciphertext in a way that doesn't affect its payload, which is enough to break CCA security of the hybrid scheme.

If the KEM and DEM are both CCA-secure, then the overall hybrid scheme is also CCA-secure. This is proved formally in Section 7 of:

Cramer, Shoup: Design and Analysis of Practical Public-Key Encryption Schemes Secure against Adaptive Chosen Ciphertext Attack

However, it is possible to have a CCA-secure hybrid encryption scheme where the KEM is (slightly) weaker than CCA secure. This was first demonstrated in:

Kurosawa, Desmedt: A New Paradigm of Hybrid Encryption Scheme

The weakened KEM security definition and composition theorem for hybrid encryption were formalized/generalized in:

Hofheinz, Kiltz: Secure Hybrid Encryption from Weakened Key Encapsulation

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