ECCC-Report TR21-009https://eccc.weizmann.ac.il/report/2021/009Comments and Revisions published for TR21-009en-usTue, 09 Nov 2021 17:23:42 +0200
Revision 3
| One-way Functions and a Conditional Variant of MKTP |
Eric Allender,
Mahdi Cheraghchi,
Dimitrios Myrisiotis,
Harsha Tirumala,
Ilya Volkovich
https://eccc.weizmann.ac.il/report/2021/009#revision3One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural $NP$-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows.
1. First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs.
2. Then, we observe that McKTP is $NP$-complete under polynomial-time randomized reductions.
3. Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP.
Thus the existence of OWFs is inextricably linked to the average-case hardness of this $NP$-complete problem. In fact, building on recent results of Ren and Santhanam (CCC 2021), we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs.Tue, 09 Nov 2021 17:23:42 +0200https://eccc.weizmann.ac.il/report/2021/009#revision3
Revision 2
| One-way Functions and a Conditional Variant of MKTP |
Eric Allender,
Mahdi Cheraghchi,
Dimitrios Myrisiotis,
Harsha Tirumala,
Ilya Volkovich
https://eccc.weizmann.ac.il/report/2021/009#revision2One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some natural $NP$-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows.
1. First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs.
2. Then, we observe that McKTP is $NP$-complete under polynomial-time randomized reductions.
3. Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP.
Thus the existence of OWFs is inextricably linked to the average-case hardness of this $NP$-complete problem. In fact, building on recent results of Ren and Santhanam (CCC 2021), we show that McKTP is hard-on-average if and only if there are logspace-computable OWFs.Tue, 19 Oct 2021 23:03:28 +0300https://eccc.weizmann.ac.il/report/2021/009#revision2
Revision 1
| One-way Functions and a Conditional Variant of MKTP |
Eric Allender,
Mahdi Cheraghchi,
Dimitrios Myrisiotis,
Harsha Tirumala,
Ilya Volkovich
https://eccc.weizmann.ac.il/report/2021/009#revision1One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is \emph{equivalent} to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some $\mathsf{NP}$-complete problem. In this paper, we make progress on this question by studying a conditional variant of the Minimum KT-complexity Problem (MKTP), which we call McKTP, as follows.
1. First, we prove that if McKTP is average-case hard on a polynomial fraction of its instances, then there exist OWFs.
2. Then, we observe that McKTP is $\mathsf{NP}$-complete under polynomial-time randomized reductions. That is, there \emph{are} $\mathsf{NP}$-complete problems whose average-case hardness implies the existence of OWFs.
3. Finally, we prove that the existence of OWFs implies the nontrivial average-case hardness of McKTP.
Thus the existence of OWFs is inextricably linked to the average-case hardness of this $\mathsf{NP}$-complete problem.Sun, 18 Apr 2021 06:38:15 +0300https://eccc.weizmann.ac.il/report/2021/009#revision1
Comment 1
| One-way Functions and Partial MCSP |
Eric Allender,
Mahdi Cheraghchi,
Dimitrios Myrisiotis,
Harsha Tirumala,
Ilya Volkovich
https://eccc.weizmann.ac.il/report/2021/009#comment1Mikito Nanashima and Hanlin Ren reported to us bugs in the proofs of Lemma 4.3 and Lemma 4.4, respectively.
Thank you very much! We are now working on fixing these bugs.Sat, 06 Feb 2021 17:16:04 +0200https://eccc.weizmann.ac.il/report/2021/009#comment1
Paper TR21-009
| One-way Functions and Partial MCSP |
Eric Allender,
Mahdi Cheraghchi,
Dimitrios Myrisiotis,
Harsha Tirumala,
Ilya Volkovich
https://eccc.weizmann.ac.il/report/2021/009One-way functions (OWFs) are central objects of study in cryptography and computational complexity theory. In a seminal work, Liu and Pass (FOCS 2020) proved that the average-case hardness of computing time-bounded Kolmogorov complexity is equivalent to the existence of OWFs. It remained an open problem to establish such an equivalence for the average-case hardness of some NP-complete problem. In this paper, we make progress on this question by studying a polynomially-sparse variant of Partial Minimum Circuit Size Problem (Partial MCSP), which we call Sparse Partial MCSP, as follows.
1. First, we prove that if Sparse Partial MCSP is zero-error average-case hard on a polynomial fraction of its instances, then there exist OWFs.
2. Then, we observe that Sparse Partial MCSP is NP-complete under polynomial-time deterministic reductions. That is, there are NP-complete problems whose average-case hardness implies the existence of OWFs.
3. Finally, we prove that the existence of OWFs implies the nontrivial zero-error average-case hardness of Sparse Partial MCSP.
Thus the existence of OWFs is inextricably linked to the average-case hardness of this NP-complete problem.Mon, 01 Feb 2021 23:54:03 +0200https://eccc.weizmann.ac.il/report/2021/009