Abstract
Ramsey theory assures us that in any graph there is a clique or independent set of a certain size, roughly logarithmic in the graph size. But how difficult is it to find the clique or independent set? If the graph is given explicitly, then it is possible to do so while examining a linear number of edges. If the graph is given by a black-box, where to figure out whether a certain edge exists the box should be queried, then a large number of queries must be issued. But what if one is given a program or circuit for computing the existence of an edge? This problem was raised by Buss and Goldberg and Papadimitriou in the context of TFNP, search problems with a guaranteed solution.
We examine the relationship between black-box complexity and white-box complexity for search problems with guaranteed solution such as the above Ramsey problem. We show that under the assumption that collision-resistant hash function exists (which follows from the hardness of problems such as factoring, discrete-log, and learning with errors) the white-box Ramsey problem is hard and this is true even if one is looking for a much smaller clique or independent set than the theorem guarantees. This is also true for the colorful Ramsey problem where one is looking, say, for a monochromatic triangle.
In general, one cannot hope to translate all black-box hardness for TFNP into white-box hardness: we show this by adapting results concerning the random oracle methodology and the impossibility of instantiating it.
Another model we consider is that of succinct black-box, where the complexity of an algorithm is measured as a function of the description size of the object in the box (and no limitation on the computation time). In this case, we show that for all TFNP problems there is an efficient algorithm with complexity proportional to the description size of the object in the box times the solution size. However, for promise problems this is not the case.
Finally, we consider the complexity of graph property testing in the white-box model. We show a property that is hard to test even when one is given the program for computing the graph (under the appropriate assumptions such as hardness of Decisional Diffie-Hellman). The hard property is whether the graph is a two-source extractor.
- Noga Alon and Joel Spencer. 2008. The Probabilistic Method (3rd ed.). John Wiley.Google Scholar
- Prabhanjan Ananth, Aayush Jain, Moni Naor, Amit Sahai, and Eylon Yogev. 2016. Universal constructions and robust combiners for indistinguishability obfuscation and witness encryption. In Proceedings of the International Cryptology Conference (CRYPTO’16). 491--520. Google ScholarDigital Library
- Boaz Barak, Oded Goldreich, Russell Impagliazzo, Steven Rudich, Amit Sahai, Salil P. Vadhan, and Ke Yang. 2012. On the (im)possibility of obfuscating programs. J. ACM 59, 2 (2012), 6. Google ScholarDigital Library
- Paul Beame, Stephen A. Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi. 1998. The relative complexity of NP search problems. J. Comput. Syst. Sci. 57, 1 (1998), 3--19. Google ScholarDigital Library
- Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma. 2016. Explicit two-source extractors for near-logarithmic min-entropy. Electron. Colloq. Comput. Complex. 23 (2016), 88.Google Scholar
- Itay Berman, Akshay Degwekar, Ron D. Rothblum, and Prashant Nalini Vasudevan. 2018. Multi-collision resistant hash functions and their applications. In Proceedings of the Conference on Advances in Cryptology (EUROCRYPT’18) (Lecture Notes in Computer Science), Vol. 10821. Springer, 133--161.Google ScholarCross Ref
- Nir Bitansky, Yael Tauman Kalai, and Omer Paneth. 2017. Multi-collision resistance: A paradigm for keyless hash functions. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, (STOC’18). 671--684. Google ScholarDigital Library
- Nir Bitansky, Omer Paneth, and Alon Rosen. 2015. On the cryptographic hardness of finding a Nash equilibrium. In Proceedings of the IEEE Annual Symposium on Foundations of Computer Science (FOCS’15). 1480--1498. Google ScholarDigital Library
- Sam Buss. 2009. Introduction to NP Functions and Local Search. Retrieved from: http://www.math.ucsd.edu/sbuss/ResearchWeb/Prague2009/talkslides1.pdf.Google Scholar
- Ran Canetti, Oded Goldreich, and Shai Halevi. 2004. The random oracle methodology, revisited. J. ACM 51, 4 (2004), 557--594. Google ScholarDigital Library
- Eshan Chattopadhyay and David Zuckerman. 2016. Explicit two-source extractors and resilient functions. In Proceedings of the Symposium on the Theory of Computing (STOC’16). ACM, 670--683. Google ScholarDigital Library
- Benny Chor and Oded Goldreich. 1988. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput. 17, 2 (1988), 230--261. Google ScholarDigital Library
- Gil Cohen. 2016a. Two-source dispersers for polylogarithmic entropy and improved Ramsey graphs. In Proceedings of the Symposium on the Theory of Computing (STOC’16), Daniel Wichs and Yishay Mansour (Eds.). ACM, 278--284. Google ScholarDigital Library
- Gil Cohen. 2016b. Two-source extractors for quasi-logarithmic min-entropy and improved privacy amplification protocols. Electron. Colloq. Comput. Complex. 23 (2016), 114.Google Scholar
- David Conlon. 2008. A new upper bound for the bipartite Ramsey problem. J. Graph Theor. 58, 4 (2008), 351--356. Google ScholarDigital Library
- Paul Erdös. 1947. Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53, 4 (1947), 292--294.Google ScholarCross Ref
- Paul Erdös and Richard Rado. 1960. Intersection theorems for systems of sets. J. London Math. Soc. 35 (1960), 85--90.Google ScholarCross Ref
- David Mandell Freeman, Oded Goldreich, Eike Kiltz, Alon Rosen, and Gil Segev. 2013. More constructions of lossy and correlation-secure trapdoor functions. J. Cryptology 26, 1 (2013), 39--74. Google ScholarDigital Library
- Sanjam Garg, Craig Gentry, Shai Halevi, Mariana Raykova, Amit Sahai, and Brent Waters. 2013. Candidate indistinguishability obfuscation and functional encryption for all circuits. In Proceedings of the IEEE Annual Symposium on Foundations of Computer Science (FOCS’13). Google ScholarDigital Library
- Sanjam Garg, Omkant Pandey, and Akshayaram Srinivasan. 2016. Revisiting the cryptographic hardness of finding a Nash equilibrium. In Proceedings of the International Cryptology Conference (CRYPTO’16). 579--604. Google ScholarDigital Library
- Paul W. Goldberg and Christos H. Papadimitriou. 2018. Towards a unified complexity theory of total functions. J. Comput. Syst. Sci. 94 (2018), 167--192.Google ScholarCross Ref
- Paul W. Goldberg and Aaron Roth. 2016. Bounds for the query complexity of approximate equilibria. ACM Trans. Econ. Comput. 4, 4 (2016), 24:1--24:25. Google ScholarDigital Library
- Oded Goldreich. 2011. Introduction to testing graph properties. In Studies in Complexity and Cryptography. Vol. 6650. 470--506. Google ScholarDigital Library
- Shafi Goldwasser and Yael Tauman Kalai. 2003. On the (in)security of the Fiat-Shamir paradigm. In Proceedings of the IEEE Annual Symposium on Foundations of Computer Science (FOCS’03). 102--113. Google ScholarDigital Library
- Mika Göös, Toniann Pitassi, and Thomas Watson. 2015. Deterministic communication vs. partition number. In Proceedings of the IEEE Annual Symposium on Foundations of Computer Science (FOCS’15). 1077--1088. Google ScholarDigital Library
- Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer. 1990. Ramsey Theory (2nd ed.). John Wiley. Google ScholarDigital Library
- Satoshi Hada. 2000. Zero-knowledge and code obfuscation. In Proceedings of the International Conference on the Theory and Application of Cryptology and Information Security (ASIACRYPT’00). 443--457. Google ScholarDigital Library
- Brett Hemenway and Rafail Ostrovsky. 2012. Extended-DDH and lossy trapdoor functions. In Proceedings of the International Conference on Theory and Practice of Public Key Cryptography (PKC’12). 627--643. Google ScholarDigital Library
- Michael D. Hirsch, Christos H. Papadimitriou, and Stephen A. Vavasis. 1989. Exponential lower bounds for finding Brouwer fixed points. J. Complexity 5, 4 (1989), 379--416. Google ScholarDigital Library
- Pavel Hubácek, Moni Naor, and Eylon Yogev. 2017. The journey from NP to TFNP hardness. In Proceedings of the 8th Innovations in Theoretical Computer Science Conference (ITCS’17) (LIPIcs), Vol. 67. Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik, 60:1--60:21.Google Scholar
- Pavel Hubácek and Eylon Yogev. 2017. Hardness of continuous local search: Query complexity and cryptographic lower bounds. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA’17). 1352--1371. Google ScholarDigital Library
- Russell Impagliazzo and Moni Naor. 1988. Decision trees and downward closures. In Proceedings of the 3rd Structure in Complexity Theory Conference. 29--38.Google ScholarCross Ref
- Emil Jerábek. 2009. Approximate counting by hashing in bounded arithmetic. J. Symb. Log. 74, 3 (2009), 829--860.Google ScholarCross Ref
- Emil Jerábek. 2016. Integer factoring and modular square roots. J. Comput. Syst. Sci. 82, 2 (2016), 380--394. Google ScholarDigital Library
- Antoine Joux. 2004. Multicollisions in iterated hash functions. Application to cascaded constructions. In Proceedings of the International Cryptology Conference (CRYPTO’04), Vol. 3152. 306--316.Google ScholarCross Ref
- Stasys Jukna. 2011. Extremal Combinatorics—With Applications in Computer Science (2nd ed.). Springer. Google ScholarDigital Library
- Joe Kilian. 1992. A note on efficient zero-knowledge proofs and arguments (extended abstract). In Proceedings of the Symposium on the Theory of Computing (STOC’92). ACM, 723--732. Google ScholarDigital Library
- Eike Kiltz, Adam O’Neill, and Adam D. Smith. 2010. Instantiability of RSA-OAEP under chosen-plaintext attack. In Proceedings of the International Cryptology Conference (CRYPTO’10). Springer, 295--313. Google ScholarDigital Library
- Ilan Komargodski, Moni Naor, and Eylon Yogev. 2017. White-box vs. black-box complexity of search problems: Ramsey and graph property testing. In Proceedings of the 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS’17). IEEE Computer Society, 622--632.Google ScholarCross Ref
- Ilan Komargodski, Moni Naor, and Eylon Yogev. 2018. Collision resistant hashing for paranoids: Dealing with multiple collisions. In Proceedings of the Conference on Advances in Cryptology (EUROCRYPT’18) (Lecture Notes in Computer Science), Vol. 10821. Springer, 162--194.Google ScholarCross Ref
- Ilan Komargodski and Gil Segev. 2017. From minicrypt to obfustopia via private-key functional encryption. In Proceedings of the International Conference on the Theory and Application of Cryptographic Techniques (EUROCRYPT’17). 122--151.Google ScholarCross Ref
- Ilan Komargodski and Eylon Yogev. 2018. On distributional collision resistant hashing. In Proceedings of the 38th Annual International Cryptology Conference—Advances in Cryptology (CRYPTO’18). 303--327.Google ScholarCross Ref
- Jan Krajícek. 2001. On the weak pigeonhole principle. Fundam. Math. 170 (2001), 123--140.Google ScholarCross Ref
- Jan Krajícek. 2005. Structured pigeonhole principle, search problems and hard tautologies. J. Symb. Log. 70, 2 (2005), 619--630.Google ScholarCross Ref
- Xin Li. 2016. Improved non-malleable extractors, non-malleable codes and independent source extractors. Electron. Colloq. Comput. Complex. 23 (2016), 115.Google Scholar
- László Lovász, Moni Naor, Ilan Newman, and Avi Wigderson. 1995. Search problems in the decision tree model. SIAM J. Discrete Math. 8, 1 (1995), 119--132. Google ScholarDigital Library
- Nimrod Megiddo and Christos H. Papadimitriou. 1991. On total functions, existence theorems and computational complexity. Theor. Comput. Sci. 81, 2 (1991), 317--324. Google ScholarDigital Library
- Joseph Naor and Moni Naor. 1993. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput. 22, 4 (1993), 838--856. Google ScholarDigital Library
- Moni Naor. 1992. Constructing Ramsey Graphs from Small Probability Spaces. IBM Research Report RJ 8810 (1992). Retrieved from: http://www.wisdom.weizmann.ac.il/naor/PAPERS/ramsey.ps.Google Scholar
- Christos H. Papadimitriou. 1994. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci. 48, 3 (1994), 498--532. Google ScholarDigital Library
- Chris Peikert and Brent Waters. 2011. Lossy trapdoor functions and their applications. SIAM J. Comput. 40, 6 (2011), 1803--1844. Google ScholarDigital Library
- David Pointcheval and Jacques Stern. 1996. Security proofs for signature schemes. In Proceedings of the International Conference on the Theory and Application of Cryptographic Techniques (EUROCRYPT’96). 387--398. Google ScholarDigital Library
- Pavel Pudlák. 1990. Ramsey’s theorem in bounded arithmetic. In Proceedings of the 4th Workshop on Computer Science Logic (CSL’90). 308--317. Google ScholarDigital Library
- Ran Raz and Pierre McKenzie. 1999. Separation of the monotone NC hierarchy. Combinatorica 19, 3 (1999), 403--435.Google ScholarDigital Library
- Amit Sahai and Brent Waters. 2014. How to use indistinguishability obfuscation: Deniable encryption, and more. In Proceedings of the Symposium on the Theory of Computing (STOC’14). Google ScholarDigital Library
- Alexander A. Sherstov. 2011. The pattern matrix method. SIAM J. Comput. 40, 6 (2011), 1969--2000. Google ScholarDigital Library
- Daniel R. Simon. 1998. Finding collisions on a one-way street: Can secure hash functions be based on general assumptions? In Proceedings of the International Conference on the Theory and Application of Cryptographic Techniques (EUROCRYPT’98). 334--345.Google ScholarCross Ref
Index Terms
- White-Box vs. Black-Box Complexity of Search Problems: Ramsey and Graph Property Testing
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