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6.7 Summary and Bibliographic Remarks
V. Arvind and J. Köbler. New lowness results for ZPP(NP) and other complexity classes. Journal of Computer and System Sciences, 65(2):257–277, 2002.
V. Arvind and P. Kurur. Graph isomorphism is in SPP. In Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, pages 743–750. IEEE Computer Society Press, November 2002.
L. Babai. Trading group theory for randomness. In Proceedings of the 17th ACM Symposium on Theory of Computing, pages 421–429. ACM Press, April 1985.
J. Balcázar. Simplicity, relativizations and nondeterminism. SIAM Journal on Computing, 14(1):148–157, 1985.
J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity II. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1990.
R. Beigel. Relativized counting classes: Relations among thresholds, parity, and mods. Journal of Computer and System Sciences, 42(1):76–96, 1991.
A. Beutelspacher. Kryptologie. Vieweg, 6th edition, 2002. In German.
R. Beigel and J. Gill. Counting classes: Thresholds, parity, mods, and fewness. Theoretical Computer Science, 103(1):3–23, 1992.
B. Borchert, L. Hemaspaandra, and J. Rothe. Restrictive acceptance suffices for equivalence problems. London Mathematical Society Journal of Computation and Mathematics, 3:86–95, March 2000.
R. Beigel, L. Hemachandra, and G. Wechsung. Probabilistic polynomial time is closed under parity reductions. Information Processing Letters, 37(2):91–94, 1991.
R. Boppana, J. Håstad, and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25(2):127–132, 1987.
L. Babai and S. Moran. Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Sciences, 36(2):254–276, 1988.
J. Balcázar and D. Russo. Immunity and simplicity in relativizations of probabilistic complexity classes. R.A.I.R.O. Theoretical Informatics and Applications, 22(2):227–244, 1988.
R. Beigel, N. Reingold, and D. Spielman. PP is closed under intersection. In Proceedings of the 23rd ACM Symposium on Theory of Computing, pages 1–9. ACM Press, May 1991.
D. Bruschi. Strong separations of the polynomial hierarchy with oracles: Constructive separations by immune and simple sets. Theoretical Computer Science, 102(2):215–252, 1992.
J. Buchmann. Introduction to Cryptography. Undergraduate Texts in Mathematics. Springer-Verlag, 2001.
J. Carter and M. Wegman. Universal classes of hash functions. Journal of Computer and System Sciences, 18:143–154, 1979.
[DGH+02]_E. Dantsin, A. Goerdt, E. Hirsch, R. Kannan, J. Kleinberg, C. Papadimitriou, P. Raghavan, and U. Schöning. A deterministic (2 − 2/(k + 1))n algorithm for k-SAT based on local search. Theoretical Computer Science, 289(1):69–83, October 2002.
S. Even, A. Selman, and Y. Yacobi. The complexity of promise problems with applications to public-key cryptography. Information and Control, 61(2):159–173, 1984.
S. Even and Y. Yacobi. Cryptocomplexity and NP-completeness. In Proceedings of the 7th International Colloquium on Automata, Languages, and Programming, pages 195–207. Springer-Verlag Lecture Notes in Computer Science, 1980.
S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48(1):116–148, 1994.
L. Fortnow and N. Reingold. PP is closed under truth-table reductions. Information and Computation, 124(1):1–6, 1996.
M. Furst, J. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13–27, 1984.
J. Gill. Computational complexity of probabilistic Turing machines. SIAM Journal on Computing, 6(4):675–695, 1977.
S. Goldwasser, S. Micali, and C. Rackoff. The knowledge complexity of interactive proof systems. SIAM Journal on Computing, 18(1):186–208, February 1989.
O. Goldreich, S. Micali, and A. Wigderson. Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. Journal of the ACM, 38(3):691–729, July 1991.
O. Goldreich. Foundations of Cryptography. Cambridge University Press, 2001.
O. Goldreich. Randomness, interactive proofs, and zero-knowledge—A survey. In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pages 377–405. Oxford University Press, Oxford, 1988.
S. Goldwasser. Interactive proof systems. In J. Hartmanis, editor, Computational Complexity Theory, pages 108–128. AMS Short Course Lecture Notes: Introductory Survey Lectures, Proceedings of Symposia in Applied Mathematics, Volume 38, American Mathematical Society, 1989.
L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43(1):43–58, 1986.
F. Green. An oracle separating ⊕P from PPPH. Information Processing Letters, 37(3):149–153, 1991.
J. Grollmann and A. Selman. Complexity measures for public-key cryptosystems. SIAM Journal on Computing, 17(2):309–335, 1988.
S. Goldwasser and M. Sipser. Private coins versus public coins in interactive proof systems. In S. Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 73–90. JAI Press, Greenwich, 1989. A preliminary version appeared in Proc. 18th Ann. ACM Symp. on Theory of Computing, 1986, pp. 59–68.
S. Gupta. The power of witness reduction. In Proceedings of the 6th Structure in Complexity Theory Conference, pages 43–59. IEEE Computer Society Press, June/July 1991.
S. Gupta. On bounded-probability operators and C=P. Information Processing Letters, 48:93–98, 1993.
F. Harary. Graph Theory. Addison-Wesley, 1969.
F. Harary. A survey of the reconstruction conjecture. In Graphs and Combinatorics, pages 18–28. Springer-Verlag Lecture Notes in Mathematics #406, 1974.
U. Hertrampf. Relations among MOD-classes. Theoretical Computer Science, 74(3):325–328, 1990.
J. Hartmanis and L. Hemachandra. Complexity classes without machines: On complete languages for UP. Theoretical Computer Science, 58(1–3):129–142, 1988.
E. Hemaspaandra, L. Hemaspaandra, S. Radziszowski, and R. Tripathi. Complexity results in graph reconstruction. In Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science, pages 287–297. Springer-Verlag Lecture Notes in Computer Science #3153, 2004.
Y. Han, L. Hemaspaandra, and T. Thierauf. Threshold computation and cryptographic security. SIAM Journal on Computing, 26(1):59–78, February 1997.
L. Hemachandra and M. Ogiwara. Is #P closed under subtraction? In G. Rozenberg and A. Salomaa, editors, Current Trends in Theoretical Computer Science: Essays and Tutorials, pages 523–536. World Scientific Press, 1993.
C. Hoffman. Group-Theoretic Algorithms and Graph Isomorphism. Lecture Notes in Computer Science #136. Springer-Verlag, 1982.
C. Homan. Tight lower bounds on the ambiguity in strong, total, associative, one-way functions. Journal of Computer and System Sciences, 68(3):657–674, 2004.
L. Hemaspaandra and J. Rothe. Unambiguous computation: Boolean hierarchies and sparse Turing-complete sets. SIAM Journal on Computing, 26(3):634–653, June 1997.
L. Hemaspaandra, J. Rothe, and G. Wechsung. On sets with easy certificates and the existence of one-way permutations. In Proceedings of the Third Italian Conference on Algorithms and Complexity, pages 264–275. Springer-Verlag Lecture Notes in Computer Science #1203, March 1997.
C. Homan and M. Thakur. One-way permutations and self-witnessing languages. Journal of Computer and System Sciences, 67(3):608–622, 2003.
K. Iwama and S. Tamaki. Improved upper bounds for 3-SAT. Technical Report TR03-053, Electronic Colloquium on Computational Complexity, July 2003. 3 pages.
D. Kratsch and L. Hemaspaandra. On the complexity of graph reconstruction. Mathematical Systems Theory, 27(3):257–273, 1994.
K. Ko. Some observations on the probabilistic algorithms and NP-hard problems. Information Processing Letters, 14(1):39–43, 1982.
K. Ko. A note on separating the relativized polynomial time hierarchy by immune sets. R.A.I.R.O. Theoretical Informatics and Applications, 24(3):229–240, 1990.
J. Köbler, U. Schöning, and J. Torán. Graph isomorphism is low for PP. Computational Complexity, 2:301–330, 1992.
J. Köbler, U. Schöning, and J. Torán. The Graph Isomorphism Problem: Its Structural Complexity. Birkhäuser, 1993.
J. Köbler, U. Schöning, S. Toda, and J. Torán. Turing machines with few accepting computations and low sets for PP. Journal of Computer and System Sciences, 44(2):272–286, 1992.
C. Lautemann. BPP and the polynomial hierarchy. Information Processing Letters, 17(4):215–217, 1983.
M. Ogiwara and L. Hemachandra. A complexity theory for feasible closure properties. Journal of Computer and System Sciences, 46(3):295–325, 1993.
C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.
R. Paturi, P. Pudlák, M. Saks, and F. Zane. An improved exponential-time algorithm for k-SAT. In Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, pages 628–637. IEEE Computer Society Press, November 1998.
C. Papadimitriou and S. Zachos. Two remarks on the power of counting. In Proceedings of the 6th GI Conference on Theoretical Computer Science, pages 269–276. Springer-Verlag Lecture Notes in Computer Science #145, 1983.
A. Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mat. Zametki, 41(4):598–607, 1987. In Russian. English Translation in Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333–338, 1987.
J. Rothe and L. Hemaspaandra. On characterizing the existence of partial one-way permutations. Information Processing Letters, 82(3):165–171, May 2002.
J. Rothe. Some facets of complexity theory and cryptography: A five-lecture tutorial. ACM Computing Surveys, 34(4):504–549, December 2002.
J. Rothe. A promise class at least as hard as the polynomial hierarchy. Journal of Computing and Information, 1(1):92–107, April 1995. Special Issue: Proceedings of the Sixth International Conference on Computing and Information, CD-ROM ISSN 1201-8511, Trent University Press.
J. Rothe. Immunity and simplicity for exact counting and other counting classes. R.A.I.R.O. Theoretical Informatics and Applications, 33(2):159–176, March/April 1999.
R. Rao, J. Rothe, and O. Watanabe. Upward separation for FewP and related classes. Information Processing Letters, 52(4):175–180, April 1994. Corrigendum appears in the same journal, 74(1–2):89, 2000.
A. Salomaa. Public-Key Cryptography, volume 23 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, second edition, 1996.
U. Schöning. Algorithmik. Spektrum Akademischer Verlag, Heidelberg, Berlin, 2001. In German.
U. Schöning. A probabilistic algorithm for k-SAT based on limited local search and restart. Algorithmica, 32(4):615–623, 2002.
U. Schöning. Algorithmics in exponential time. In Proceedings of the 22nd Annual Symposium on Theoretical Aspects of Computer Science, pages 36–43. Springer-Verlag Lecture Notes in Computer Science #3404, 2005.
U. Schöning. Graph isomorphism is in the low hierarchy. Journal of Computer and System Sciences, 37(3):312–323, 1988.
U. Schöoning. Probabilistic complexity classes and lowness. Journal of Computer and System Sciences, 39(1):84–100, 1989.
U. Schöning. A probabilistic algorithm for k-SAT and constraint satisfaction problems. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, pages 410–414. IEEE Computer Society Press, October 1999.
A. Selman. Promise problems complete for complexity classes. Information and Computation, 78:87–98, 1988.
A. Selman. A survey of one-way functions in complexity theory. Mathematical Systems Theory, 25(3):203–221, 1992.
A. Shamir. IP = PSPACE. Journal of the ACM, 39(4):869–877, 1992.
A. Shamir. RSA for paranoids. CryptoBytes, 1(3):1–4, 1995.
J. Simon. On Some Central Problems in Computational Complexity. PhD thesis, Cornell University, Ithaca, NY, January 1975. Available as Cornell Department of Computer Science Technical Report TR75-224.
M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th ACM Symposium on Theory of Computing, pages 330–335. ACM Press, 1983.
R. Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In Proceedings of the 19th ACM Symposium on Theory of Computing, pages 77–82. ACM Press, May 1987.
D. Stinson. Cryptography: Theory and Practice. CRC Press, Boca Raton, second edition, 2002.
J. Tarui. Probabilistic polynomials, AC0 functions and the polynomial-time hierarchy. Theoretical Computer Science, 113:167–183, 1993.
S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316–328, 1992.
S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20(5):865–877, 1991.
J. Torán. Complexity classes defined by counting quantifiers. Journal of the ACM, 38(3):753–774, 1991.
L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5(1):20–23, 1976.
L. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189–201, 1979.
L. Valiant. The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8(3):410–421, 1979.
N. Vereshchagin. On the power of PP. In Proceedings of the 7th Structure in Complexity Theory Conference, pages 138–143. IEEE Computer Society Press, June 1992.
L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85–93, 1986.
K. Wagner. The complexity of combinatorial problems with succinct input representations. Acta Informatica, 23:325–356, 1986.
G. Wechsung. Vorlesungen zur Komplexitätstheorie, volume 32 of Teubner-Texte zur Informatik. Teubner, Stuttgart, 2000. In German.
G. Woeginger. Exact algorithms for NP-hard problems. In M. Jünger, G. Reinelt, and G. Rinaldi, editors, Combinatorical Optimization: “Eureka, you shrink!”, pages 185–207. Springer-Verlag Lecture Notes in Computer Science #2570, 2003.
S. Zachos. Probabilistic quantifiers and games. Journal of Computer and System Sciences, 36:433–451, 1988.
S. Zachos and H. Heller. A decisive characterization of BPP. Information and Control, 69:125–135, 1986.
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(2005). Randomized Algorithms and Complexity Classes. In: Complexity Theory and Cryptology. Texts in Theoretical Computer Science An EATCS Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28520-2_6
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