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Randomness vs time: Derandomization under a uniform assumption. (English) Zbl 1052.68034
Summary: We prove that if BPP $$\neq$$ EXP, then every problem in BPP can be solved deterministically in subexponential time on almost every input (on every sampleable ensemble for infinitely many input sizes). This is the first deran-domization result for BPP based on uniform, noncryptographic hardness assumptions. It implies the following gap in the deterministic average-case complexities of problems in BPP: either these complexities are always sub-exponential or they contain arbitrarily large exponential functions. We use a construction of a small “pseudorandom” set of strings from a “hard function” in EXP which is identical to that used in the analogous nonuniform results of L. Babai, L. Fortnow, N. Nisan and A. Wigderson [Comput. Complexity 3, 307–318 (1993; Zbl 0802.68054)] and N. Nisan and A. Wigderson [J. Comput. System Sci. 49, 149–167 (1994; Zbl 0821.68057)]. However, previous proofs of correctness assume the “hard function” is not in P/poly. They give a non constructive argument that a circuit distinguishing the pseudorandom strings from truly random strings implies that a similarly sized circuit exists computing the “hard function.” Our main technical contribution is to show that, if the “hard function” has certain properties, then this argument can be made constructive.

##### MSC:
 68P25 Data encryption (aspects in computer science)
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##### References:
 [1] Andreev, A; Clementi, A; Rolim, J, Hitting sets derandomize BPP, XXIII international colloquium on algorithms, logic and programming (ICALP’96), (1996), p. 357-368 · Zbl 1046.68536 [2] Andreev, A; Clementi, A; Rolim, J, Worst-case hardness suffices for derandomization: A new method for hardness – randomness tradeoffs, Theor. comp. sci., 221, 3-18, (1999) · Zbl 0930.68064 [3] Andreev, A; Clementi, A; Rolim, J, A new general derandomization method, J. assoc. comput. Mach., 45, 179-213, (1998) · Zbl 0903.68089 [4] Andreev, A; Clementi, A; Rolim, J; Trevisan, L, Weak random sources, hitting sets, and BPP simulation, SIAM J. comput., 28, 2103-2116, (1999) · Zbl 0943.68064 [5] Bshouty, N; Cleve, R; Gavalda, R; Kannan, S; Tamon, C, Oracles and queries that are sufficient for exact learning, J. comput. system sci., 52, 421-433, (1996) · Zbl 0858.68075 [6] Babai, L; Fortnow, L; Lund, C, Non-deterministic exponential time has two-prover interactive protocols, Comput. complexity, 1, 3-40, (1991) · Zbl 0774.68041 [7] Babai, L; Fortnow, L; Nisan, N; Wigderson, A, BPP has subexponential time simulations unless EXPTIME has publishable proofs, Comput. complexity, 3, 307-318, (1993) · Zbl 0802.68054 [8] Beaver, D; Feigenbaum, J, Hiding instance in multioracle queries, Proc. 7th symposium on theoretical aspects of computer science, Lecture notes on computer science, 415, (1990), p. 37-48 · Zbl 0733.68005 [9] Blum, M; Luby, M; Rubinfeld, R, Self-testing and self-correcting programs with applications to numerical programs, Comput. complexity, 3, 307-318, (1993) [10] Blum, M; Micali, S, How to generate cryptographically strong sequences of pseudo-random bits, SIAM J. comput., 13, 850-864, (1984) · Zbl 0547.68046 [11] Cai, J.-Y; Nerurkar, A; Sivakumar, D, Hardness and hierarchy theorems for probabilistic time, 31st symposium on theory of computing, (1999), p. 726-735 · Zbl 1346.68095 [12] Goldreich, O; Krawcyk, H; Luby, M, On the existence of pseudorandom generators, SIAM J. comput., 22, 1163-1175, (1993) · Zbl 0795.94011 [13] Goldreich, O; Levin, L.A, A hard-core predicate for all one-way functions, ACM symp. on theory of computing, (1989), p. 25-32 [14] Goldreich, O; Nisan, N; Wigderson, A, On Yao’s XOR-lemma, Eccc tr 95-050, (1995) [15] Hastad, J; Impagliazzo, R; Levin, L.A; Luby, M, A pseudorandom generator from any one-way function, SIAM J. comput., 28, 1364-1396, (1999) · Zbl 0940.68048 [16] Impagliazzo, R, Hard-core distributions for somewhat hard problems, 36th foundations of computer science, (1995), p. 538-545 · Zbl 0938.68921 [17] R. Impagliazzo, in preparation. [18] Impaglizzo, R; Shaltiel, R; Wigderson, A, Near-optimal conversion of hardness into pseudo-randomness, 40th foundations of computer science, (1999), p. 181-190 [19] Impaglizzo, R; Shaltiel, R; Wigderson, A, Extractors and pseudo-random generators with optimal seed lengths, 32nd symposium on theory of computing, (2000), p. 1-10 · Zbl 1296.65007 [20] Impagliazzo, R; Wigderson, A, P=BPP unless E has sub-exponential circuits: derandomizing the XOR lemma, Proc. of the 29th symposium on theory of computing, (1997), p. 220-229 · Zbl 0962.68058 [21] Karp, R.M; Lipton, R.J, Turing machines that take advice, L’ensignment math., 28, 191-209, (1982) · Zbl 0529.68025 [22] Levin, L.A, One-way functions and pseudorandom generators, Combinatorica, 7, 357-363, (1987) · Zbl 0641.68061 [23] Luby, M, Pseudorandomness and cryptographic applications, Princeton computer science notes, (1996), Princeton Univ. Press Princeton [24] Levin, L.A, Average case complete problems, SIAM J. comput., 15, 285-286, (1986) · Zbl 0589.68032 [25] Lipton, R, New directions in testing, (), 191-202 [26] Nisan, N, Pseudo-random bits for constant depth circuits, Combinatorica, 11, 63-70, (1991) · Zbl 0732.68056 [27] Nisan, N; Wigderson, A, Hardness vs randomness, J. comput. system sci., 49, 149-167, (1994) · Zbl 0821.68057 [28] Razborov, A.A; Rudich, S, Natural proofs, J. comput. system sci., 55, 24-35, (1997) · Zbl 0884.68055 [29] Shamir, A, On the generation of cryptographically strong pseudo-random sequences, 8th ICALP, Lecture notes in computer science, 62, (1981), Springer-Verlag Berlin, p. 544-550 [30] Toda, S, PP is as hard as the polynomial – time hierarchy, SIAM J. comput., 20, 865-877, (1991) · Zbl 0733.68034 [31] A. C. Yao, Theory and application of trapdoor functions, in 23rd Foundation of Computer Science, pp. 80-91, 1982.
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