Summary

Today we show how to construct an efficient CCA-secure public-key encryption scheme in the random oracle model using RSA.

As we discussed in the previous lecture, a cryptographic scheme defined in the random oracle model is allowed to use a random function {H: \{ 0,1 \}^n \rightarrow \{ 0,1 \}^m} which is known to all the parties. In an implementation, usually a cryptographic hash function replaces the random oracle. In general, the fact that a scheme is proved secure in the random oracle model does not imply that it is secure when the random oracle is replaced by a hash function; the proof of security in the random oracle model gives, however, at least some heuristic confidence in the soundness of the design.

1. Hybrid Encryption with a Random Oracle

We describe a public-key encryption scheme {(\overline G, \overline E, \overline D)} which is based on: (1) a family of trapdoor permutations (for concreteness, we shall refer specifically to RSA below); (2) a CCA-secure private-key encryption scheme {(E,D)}; (3) a random oracle {H} mapping elements in the domain and range of the trapdoor permutation into keys for the private-key encryption scheme {(E,D)}.

  1. Key generation: {\overline G} picks an RSA public-key / private-key pair {(N,e),(N,d)};
  2. Encryption: given a public key {N,e} and a plaintext {M}, {\overline E} picks at random {R \in {\mathbb Z}_N}, and outputs

    \displaystyle  R^d \bmod N, E(H(R),M)

  3. Decryption: given a private key {N,d} and a cyphertext {C_1,C_2}, {\overline D} decrypts the plaintext by computing {R := C_1^d \bmod N} and {M := D(H(R),C_2)}.

This is a hybrid encryption scheme in which RSA is used to encrypt a “session key” which is then used to encrypt the plaintext via a private-key scheme. The important difference from hybrid schemes we discussed before is that the random string encrypted with RSA is “hashed” with the random oracle before being used as a session key.

2. Security Analysis

Theorem 1 Suppose that, for the key size used in {(\overline G, \overline E, \overline D)}, RSA is a {(t,\epsilon)}-secure family of trapdoor permutations, and that exponentiation can be computed in time {\leq r}; assume also that {(E,D)} is a {(t,\epsilon)} CCA-secure private-key encryption scheme and that {E,D} can be computed in time {\leq r}.

Then {(\overline G, \overline E, \overline D)} is {(t/O(r),2\epsilon)} CCA-secure in the random oracle model.

We sketch an outline of the proof. We assume that there is an algorithm {A} showing that {(\overline G, \overline E, \overline D)} is not {(t',\epsilon')} CCA-secure. From {A} we derive an algorithm {A'} running in time {O(t'\cdot r)} which is a CCA attack on {(E,D)}. If {A'} succeeds with probability at least {\epsilon} as a distinguishing CCA attack against {(E,D)}, then we have violated the assumption on the security of {(E,D)}. But if {A'} succeeds with probability less than {\epsilon}, then we can devise an algorithm {A''}, also of running time {O(t'\cdot r)}, which inverts RSA with probability at least {\epsilon}.

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