TY - GEN
T1 - Hard Languages in NP ∩ coNP and NIZK Proofs from Unstructured Hardness
AU - Ghosal, Riddhi
AU - Ishai, Yuval
AU - Korb, Alexis
AU - Kushilevitz, Eyal
AU - Lou, Paul
AU - Sahai, Amit
N1 - Publisher Copyright:
© 2023 Owner/Author.
PY - 2023/6/2
Y1 - 2023/6/2
N2 - The existence of “unstructured” hard languages in NP ∩ coNP is an intriguing open question. Bennett and Gill (SICOMP, 1981) asked whether P is separated from NP∩coNP relative to a random oracle, a question that remained open ever since. While a hard language in NP ∩ coNP can be constructed in a black-box way from a oneway permutation, for which only few (structured) candidates exist, Bitansky et al. (SICOMP, 2021) ruled out such a construction based on an injective one-way function, an unstructured primitive that is easy to instantiate heuristically. In fact, the latter holds even with a black-box use of indistinguishability obfuscation. We give the first evidence for the existence of unstructured hard languages in NP ∩ coNP by showing that if UP ⊈ RP, which follows from the existence of injective one-way functions, the answer to Bennett and Gill’s question is armative: with probability 1 over a random oracle O, we have that P O ≠ NPO ∩ coNPO. Our proof gives a constructive non-black-box approach for obtaining candidate hard languages in NP ∩ coNP from cryptographic hash functions. The above conditional separation builds on a new construction of non-interactive zero-knowledge (NIZK) proofs, with a computationally unbounded prover, to convert a hard promise problem into a hard language. We obtain such NIZK proofs for NP, with a uniformly random reference string, from a special kind of hash function which is implied by (an unstructured) random oracle. This should be contrasted with previous constructions of such NIZK proofs that are based on one-way permutations or other structured primitives, as well as with (computationally sound) NIZK arguments in the random oracle model.
AB - The existence of “unstructured” hard languages in NP ∩ coNP is an intriguing open question. Bennett and Gill (SICOMP, 1981) asked whether P is separated from NP∩coNP relative to a random oracle, a question that remained open ever since. While a hard language in NP ∩ coNP can be constructed in a black-box way from a oneway permutation, for which only few (structured) candidates exist, Bitansky et al. (SICOMP, 2021) ruled out such a construction based on an injective one-way function, an unstructured primitive that is easy to instantiate heuristically. In fact, the latter holds even with a black-box use of indistinguishability obfuscation. We give the first evidence for the existence of unstructured hard languages in NP ∩ coNP by showing that if UP ⊈ RP, which follows from the existence of injective one-way functions, the answer to Bennett and Gill’s question is armative: with probability 1 over a random oracle O, we have that P O ≠ NPO ∩ coNPO. Our proof gives a constructive non-black-box approach for obtaining candidate hard languages in NP ∩ coNP from cryptographic hash functions. The above conditional separation builds on a new construction of non-interactive zero-knowledge (NIZK) proofs, with a computationally unbounded prover, to convert a hard promise problem into a hard language. We obtain such NIZK proofs for NP, with a uniformly random reference string, from a special kind of hash function which is implied by (an unstructured) random oracle. This should be contrasted with previous constructions of such NIZK proofs that are based on one-way permutations or other structured primitives, as well as with (computationally sound) NIZK arguments in the random oracle model.
KW - Complexity Theory Separation
KW - Non-Interactive Zero Knowledge Proofs
KW - Random Oracles
UR - http://www.scopus.com/inward/record.url?scp=85163063472&partnerID=8YFLogxK
U2 - 10.1145/3564246.3585119
DO - 10.1145/3564246.3585119
M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???
AN - SCOPUS:85163063472
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 1243
EP - 1256
BT - STOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing
A2 - Saha, Barna
A2 - Servedio, Rocco A.
T2 - 55th Annual ACM Symposium on Theory of Computing, STOC 2023
Y2 - 20 June 2023 through 23 June 2023
ER -