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PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
The Generation of Random Numbers That Are Probably Prime
 Journal of Cryptology
, 1988
"... In this paper we make two observations on Rabin's probabilistic primality test. The first is a provocative reason why Rabin's test is so good. It turned out that a single iteration has a nonnegligible probability of failing _only_ on composite numbers that can actually be split in expected ..."
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In this paper we make two observations on Rabin's probabilistic primality test. The first is a provocative reason why Rabin's test is so good. It turned out that a single iteration has a nonnegligible probability of failing _only_ on composite numbers that can actually be split in expected polynomial time. Therefore, factoring would be easy if Rabin's test systematically failed with a 25% probability on each composite integer (which, of course, it does not). The second observation is more fundamental because is it _not_ restricted to primality testing: it has consequences for the entire field of probabilistic algorithms. The failure probability when using a probabilistic algorithm for the purpose of testing some property is compared with that when using it for the purpose of obtaining a random element hopefully having this property. More specifically, we investigate the question of how reliable Rabin's test is when used to _generate_ a random integer that is probably prime, rather than to _test_ a specific integer for primality.
Key words: factorization, false witnesses, primality testing, probabilistic algorithms, Rabin's test.
Transparent Proofs and Limits to Approximation
, 1994
"... We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spotchecks. Recent work by a large group of researc ..."
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We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spotchecks. Recent work by a large group of researchers has shown that this seemingly paradoxical concept can be formalized and is feasible in a remarkably strong sense; every formal proof in ZF, say, can be rewritten in transparent format (proving the same theorem in a different proof system) without increasing the length of the proof by too much. This result in turn has surprising implications for the intractability of approximate solutions of a wide range of discrete optimization problems, extending the pessimistic predictions of the PNP theory to approximate solvability. We discuss the main results on transparent proofs and their implications to discrete optimization. We give an account of several links between the two subjects as well ...
Downward Separation Fails Catastrophically For Limited Nondeterminism Classes
, 1994
"... . The fi hierarchy consists of classes fi k = NP[log k n] ` NP. Unlike collapses in the polynomial hierarchy and the Boolean hierarchy, collapses in the fi hierarchy do not seem to translate up, nor does closure under complement seem to cause the hierarchy to collapse. For any consistent set of co ..."
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. The fi hierarchy consists of classes fi k = NP[log k n] ` NP. Unlike collapses in the polynomial hierarchy and the Boolean hierarchy, collapses in the fi hierarchy do not seem to translate up, nor does closure under complement seem to cause the hierarchy to collapse. For any consistent set of collapses and separations of levels of the hierarchy that respects P = fi 1 ` fi 2 ` \Delta \Delta \Delta ` NP, we can construct an oracle relative to which those collapses and separations hold; at the same time we can make distinct levels of the hierarchy closed under computation or not, as we wish. To give two relatively tame examples: For any k 1, we construct an oracle relative to which P = fi k 6= fi k+1 6= fi k+2 6= \Delta \Delta \Delta and another oracle relative to which P = fi k 6= fi k+1 = PSPACE: We also construct an oracle relative to which fi 2k = fi 2k+1 6= fi 2k+2 for all k. Key words. structural complexity theory, limited nondeterminism, hierarchies, oracles AMS subject clas...
The P versus NP problem
 Clay Mathematical Institute; The Millennium Prize Problem
, 2000
"... The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a computer. The standard comp ..."
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The P versus NP problem is to determine whether every language accepted by some nondeterministic algorithm in polynomial time is also accepted by some (deterministic) algorithm in polynomial time. To define the problem precisely it is necessary to give a formal model of a computer. The standard computer model in computability theory is the Turing machine, introduced by Alan Turing in 1936 [37]. Although the model was introduced before physical computers were built, it nevertheless continues to be accepted as the proper computer model for the purpose of defining the notion of computable function. Informally the class P is the class of decision problems solvable by some algorithm within a number of steps bounded by some fixed polynomial in the length of the input. Turing was not concerned with the efficiency of his machines, rather his concern was whether they can simulate arbitrary algorithms given sufficient time. It turns out, however, Turing machines can generally simulate more efficient computer models (for example, machines equipped with many tapes or an unbounded random access memory) by at most squaring or cubing the computation time. Thus P is a
The Power Of Interaction
, 1991
"... : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii Chapter 1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. PRELIMINARIES : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1 Basic Definitions : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.1 Basics : : ..."
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: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii Chapter 1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. PRELIMINARIES : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1 Basic Definitions : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.1 Basics : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.2 Boolean Formulas : : : : : : : : : : : : : : : : : 4 2.1.3 Arithmetic Formulas and Expressions : : : : : : 5 2.2 Computational Models : : : : : : : : : : : : : : : : : : : : 9 2.2.1 Deterministic Computation : : : : : : : : : : : : 9 2.2.2 Probabilistic Computation : : : : : : : : : : : : 11 2.2.3 NonDeterministic Computation : : : : : : : : : 12 2.2.4 Alternating Computations : : : : : : : : : : : : 13 2.2.5 Interactive Proof Systems : : : : : : : : : : : : : 13 2.2.6 Multiple Prover Interactive Proof Systems : : : 15 2.2.7 Computation relative to an Oracle : : : : : : : : 15 2.3 Complexity Classes : : : : : : : : : : : : : : : : : : : : ...
Locating P/poly Optimally in the Extended Low Hierarchy
, 1993
"... The low hierarchy within NP and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes. We relocate P/poly from the third \Sigmalevel EL P;\Sigma 3 (Balc'azar et al., 1986) to the third \Thetalevel EL P;\Theta 3 of the extended low hier ..."
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The low hierarchy within NP and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes. We relocate P/poly from the third \Sigmalevel EL P;\Sigma 3 (Balc'azar et al., 1986) to the third \Thetalevel EL P;\Theta 3 of the extended low hierarchy. The location of P=poly in EL P;\Theta 3 is optimal since, as shown by Allender and Hemachandra (1992), there exist sparse sets that are not contained in the next lower level EL P;\Sigma 2 . As a consequence of our result, all NP sets in P=poly are relocated from the third \Sigmalevel L P;\Sigma 3 (Ko and Schoning, 1985) to the third \Thetalevel L P;\Theta 3 of the low hierarchy.
A Note On Monte Carlo Primality Tests And Algorithmic Information Theory
, 1978
"... Solovay and Strassen, and Miller and Rabin have discovered fast algorithms for testing primality which use coinflipping and whose conclusions are only probably correct. On the other hand, algorithmic information theory provides a precise mathematical definition of the notion of random or patternles ..."
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Solovay and Strassen, and Miller and Rabin have discovered fast algorithms for testing primality which use coinflipping and whose conclusions are only probably correct. On the other hand, algorithmic information theory provides a precise mathematical definition of the notion of random or patternless sequence. In this paper we shall describe conditions under which if the sequence of coin tosses in the SolovayStrassen and MillerRabin algorithms is replaced by a sequence of heads and tails that is of maximal algorithmic information content, i.e., has maximal algorithmic randomness, then one obtains an errorfree test for primality. These results are only of theoretical interest, since it is a manifestation of the Gödel incompleteness phenomenon that it is impossible to "certify" a sequence to be random by means of a proof, even though most sequences have this property. Thus by using certified random sequences one can in principle, but not in practice, convert proba...