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Semantic Security




The notion of semantic security was first put forward by Goldwasser and Micali in 1982 [1]. However, the definition they initially proposed offered no straightforward means to prove the security of practical cryptosystems. Goldwasser/Micali subsequently demonstrated that semantic security is equivalent to the property of Ciphertext Indistinguishability [2]. This equivalence allowed for security proofs of practical cryptosystems, and consequently the indistinguishability definition is used more commonly than the original definition of semantic security.

Indistinguishability under Chosen Plaintext Attack ( IND-CPA ) is commonly defined by the following Game :

# A probabilistic polynomial time-bounded adversary is given a public key, which it may use to generate any number of ciphertexts (within polynomial bounds).
# The adversary generates two equal-length messages m_0 and m_1, and transmits them to a challenge oracle along with the public key.
# The challenge oracle selects one of the messages by flipping a uniformly-weighted coin, encrypts the message under the public key, and returns the resulting ciphertext c to the adversary.

The underlying Cryptosystem is IND-CPA (and thus semantically secure under chosen plaintext attack) if the adversary cannot determine which of the two messages was chosen by the oracle, with probability significantly greater than 1/2 (the success rate of random guessing). Variants of this definition define indistinguishability under Chosen Ciphertext Attack and Adaptive Chosen Ciphertext Attack ( IND-CCA , IND-CCA2 ).

Because the adversary possesses the public encryption key in the above game, a semantically secure encryption scheme must by definition be Probabilistic , possessing a component of Randomness ; if this were not the case, the adversary could simply compute the deterministic encryption of m_0 and m_1 and compare these encryptions with the returned ciphertext c to successfully guess the oracle's choice.

Semantically secure encryption algorithms include Goldwasser-Micali , El Gamal and Paillier . These schemes are considered Provably Secure , as their semantic security can be reduced to solving some hard mathematical problem (e.g., Decisional Diffie-Hellman or the Quadratic Residuosity Problem ). Other, non-semantically-secure algorithms such as RSA , can be made semantically secure (under stronger assumptions) through the use of random encryption padding schemes such as Optimal Asymmetric Encryption Padding (OAEP).


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