Obfuscation ==> (IND-CPA Security =/=> Circular Security)

Circular security is an important notion for public-key encryption schemes and is needed by several cryptographic protocols. In circular security the adversary is given an extra “hint” consisting of a cycle of encryption of secret keys i.e., \(\left(E_{pk_1}(sk_2),\ldots, E_{pk_n}(sk_1)\right)\). A natural question is whether every IND-CPA encryption scheme is also circular secure. It is trivial to see that this is not the case when n = 1. In 2010 a separation for n = 2 was shown by [ABBC10,GH10] under standard assumptions in bilinear groups.

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