On the Classification of Knowledge-of-exponent Assumptions in Cyclic Groups

Inspired by the work of Ghadafi and Groth (ASIACRYPT 2017) on a certain type of computational hardness assumptions in cyclic groups (which they call ‘‘target assumptions’’), we initiate an analogous work on another type of hardness assumptions, namely the ‘‘knowledge-of-exponent’’ assumptions (KEAs). Originally introduced by Damga˚rd to construct practical encryption schemes secure against chosen ciphertext attacks, KEAs have subsequently been used primarily to construct succinct non-interactive arguments of knowledge (SNARKs), and proved to be inherent to such constructions. Since SNARKs (and their zero-knowledge variant, zk-SNARKs) are already used in practice in such systems as the Zcash digital currency, it can be expected that the use of KEAs will increase in the future, which makes it important to have a good understanding of those assumptions. Using a proof technique first introduced by Bellare and Palacio (but acknowledged by them as being due to Halevi), we first investigate the internal structure of the q -power knowledge-of-exponent ( q -PKE) family of assumptions introduced by Groth, which is thus far the most general variant of KEAs. We then introduce a generalisation of the q -PKE family, and show that it can be simplified.