The notion of universal re-encryption is an established primitive used in the design of many anonymity protocols. It allows anyone to randomize a ciphertext without changing its size, without first decrypting it, and without knowing who the receiver is (i.e., not knowing the public key used to create it). By design it prevents the randomized cipher-text from being correlated with the original ciphertext. We revisit and analyze the security foundation of universal re-encryption and show a subtlety in it, namely, that it does not require that the encryption function achieve key anonymity. Recall that the encryption function is different from the re-encryption function. We demonstrate this subtlety by constructing a cryptosystem that satisfies the established definition of a universal cryptosystem but that has an encryption function that does not achieve key anonymity, thereby instantiating the gap in the definition of security of universal re-encryption. We note that the gap in the definition carries over to a set of applications that rely on universal re-encryption, applications in the original paper on universal re-encryption and also follow-on work. This shows that the original definition needs to be corrected and it shows that it had a knock-on effect that negatively impacted security in later work. We then introduce a new definition that includes the properties that are needed for a re-encryption cryptosystem to achieve key anonymity in both the encryption function and the re-encryption function, building on Goldwasser and Micali's "semantic security" and the original "key anonymity" notion of Bellare, Boldyreva, Desai, and Pointcheval. Omitting any of the properties in our definition leads to a problem. We also introduce a new generalization of the Decision Diffie-Hellman (DDH) random self-reduction and use it, in turn, to prove that the original ElGamal-based universal cryptosystem of Golle et al. is secure under our revised security definition.
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