The Cramer–Shoup system is an asymmetric key encryption algorithm, and was the first efficient scheme proven to be secure against adaptive chosen ciphertext attack using standard cryptographic assumptions. Its security is based on the computational intractability (widely assumed, but not proved) of the decisional Diffie–Hellman assumption. Developed by Ronald Cramer and Victor Shoup in 1998, it is an extension of the Elgamal cryptosystem. In contrast to Elgamal, which is extremely malleable, Cramer–Shoup adds other elements to ensure non-malleability even against a resourceful attacker. This non-malleability is achieved through the use of a universal one-way hash function and additional computations, resulting in a ciphertext which is twice as large as in Elgamal.
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... ciphertext, since , and If the space of possible messages is larger than the size of, then Cramer–Shoup may be used in a hybrid cryptosystem to improve efficiency on ...