A Zero-Knowledge Succinct Non-interactive adaptive ARgument of Knowledge is a term coined by Nir Bitansky† Ran Canetti‡ Alessandro Chiesa§ Eran Tromer in 2011 (From extractable collision resistance to succinct non-interactive arguments of knowledge, and back again).
the ZKProof reference documents different types of succintness:
- Fully succinct: Proof length independent of statement size. O(1) crypto elements (fully)
- Polylog succinct: Polylogarithmic number of crypto elements
- Depth-succinct: Depends on depth of a verification circuit representing the statement.
- Sqrt succinct: Proportional to square root of circuit size
- Non succinct: Proof length is larger than circuit size.
Because proofs, in computer science, don't necessarily have to hide all inputs. For example you might want to prove that the result of a computation is correct (e.g. the 64th iteration of the Fibonacci series is 6557470319842) without hiding any of the input. If a prover wants to hide some of the inputs, the zero-knowledgeness part can often be added easily as an extra step in the proof.
- IP: interactive proof
- NIZK: non-interactive zero knowledge proofs
interactive here means that messages are being sent back and forth between the prover and verifier, non-interactive means that the prover can create a proof that can be verified without any further interaction with them (like a signature).