V. Pratt. Every prime has a succinct certificate. SIAM Journal on Computing, 4:214--220, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....i th bit of the unique answer equal to one . An interesting example comes from the Fellows and Koblitz paper [FK92] which shows how to provide every prime number with a unique certificate that can be used to verify in polynomial time that the number is prime. The certificates provided by Pratt [Pra75] are not unique. The single valued NP search problem coming from Fellows and Koblitz is: Given a number m, list its prime divisors in order, together with their unique certificates. The theorem below shows that none of the type 2 search problems introduced in Section 2 is computationally ....

Vaughan R. Pratt. Every prime has a succinct certificate. SIAM Journal on Computing, 4:214--220, 1975.

.... by a polynomial time reduction from polynomial to integer factorization, which is, however, subject to an old number theoretic conjecture [Adleman and Odlyzko 81] The problem of finding polynomially long irreducibility proofs ( succinct certificates ) was first solved for prime numbers in 1975 [Pratt 75] and has recently also been achieved for densely encoded integral polynomials [Cantor 81] A polynomial time irreducibility test for prime numbers depending on the validity of the generalized Riemann hypothesis (GRH) was discovered in 1976 (cf. Knuth 81, Sec. 4.5.4] P. Weinberger obtained the ....

..... g i ) with n 1 substitutions. P) Guess a number c n and integers p l , 1 l c , not larger than b 1 D f b 2 , b 1 and b 2 from theorem 2.5a) such that the following conditions are verifiable. 50 We show that all p l are prime numbers. For this step we use the prime certificates by [Pratt 75] We also prove that all f mod p l are squarefree. For all p l we perform the following computation: We factor f into irreducibles mod p l , i.e. f 1 . f r l f mod p l . The factors f i can be tested for irreducibility by the distinct degree factorization [Knuth 81, Sec.4.6.2] The proof ....

Pratt, V. R.: Every Prime Has a Succinct Certificate. SIAM J. Comp. 4, 214-220 (1975).

....certificate for PRIMALITY is by no means obvious. In fact, for many decision problems in NP no succinct certificate is kown for the complement, i.e. it is not known whether the complement is also in NP . For PRIMALITY however, one can construct a succinct certificate based on Fermat s Theorem [19]. Hence PRIMALITY2NP . 2.2.4 A first map of complexity Compositeness 2 Coloring Eulerian Circuit Assignment(d) MST(d) 2 SAT 3 Coloring Graph Isomorphism SAT 3 SAT Hamiltonian Cycle TSP(d) Primality P NP Figure 6: A first map of complexity. All problems indicated are defined within ....

V.R. Pratt. Every prime has a succinct certificate. SIAM J. Comput., 4:214--220, 1975. 26

....for tautologies, unless P = NP . We may expect, therefore, that our analogs of co NP complete problems will be harder to support with an efficient checking process. A problem that looks intuitively complementary to the problem of factorization is primality testing. However as shown by Pratt [26], short certificates are possible and so the problem is not only in coNP but also in NP. 4 An especially strong form of this result is due to Pomerance [25] who shows that every prime has an O(log p) certificate, or more precisely that for every prime p there is a proof that it is prime which ....

Vaughan Pratt, `Every prime has a succinct certificate', SIAM Journal of Computing, 4, 214--220, (1975).

....the internet. The complement of an NP complete language is thought not to be in NP; otherwise the complement of every NP language would be in NP. However the complement of the set of composite numbers (essentially the set of primes) was proved to be in NP by an interesting argument due to Pratt [Pra75], and hence is unlikely to be NP complete. There is an NP decision problem with complexity equivalent to that of integer factoring, namely L fact = fha; bi j 9d(1 d a and djb)g Given an integer b 1, the smallest prime divisor of b can be found with about log 2 b queries to L fact , using ....

V. R. Pratt. Every prime has a succinct certificate. SIAM Journal on Computing, 4(3):214--220, 1975.

....bounded versions of some classical undecidable problems except for the distributional graph edge coloring problem and the distributional matrix transformation problem. The task of bounding a witness size to put a problem in NP is not always trivial. A well known example is the proof of Primes 2 NP [Pra75]. It is interesting to note that the use of randomized reductions has opened a door for us to find average case NP complete problems that do not belong to such kind of bounded problems. It is an important and challenging task to systematically investigate the average case complexity of the ....

V. Pratt. Every prime has a succinct certificate. SIAM Journal on Computing, 4:214--220, 1975.

....i th bit of the unique answer equal to one . An interesting example comes from the Fellows and Koblitz paper [FK92] which shows how to provide every prime number with a unique certificate that can be used to verify in polynomial time that the number is prime. The certificates provided by Pratt [Pra75] are not unique. The single valued NP search problem coming from Fellows and Koblitz is: Given a number m, list its prime divisors in order, together with their unique certificates. The theorem below shows that none of the type 2 search problems introduced in Section 2 is computationally ....

Vaughan R. Pratt. Every prime has a succinct certificate. SIAM Journal on Computing, 4:214--220, 1975.

....standard nondeterministic algorithms solve many NP complete problems with O(n) nondeterministic moves, there are other problems that seem to require very different amounts of nondeterminism. For instance, clique can be solved with only O( p n) nondeterministic moves, and Pratt s algorithm [16] solves primality, which is not believed to be NP complete, with O(n 2 ) nondeterministic moves. Motivated by the different amounts of nondeterminism apparently needed to solve problems in NP, Kintala and Fischer [9, 10, 11] defined limited nondeterminism classes within NP, including the classes ....

V. R. Pratt. Every prime has a succinct certificate. SICOMP, 4(3):214--220, 1975.

....[20] 64] see also [49] According to [66] the current record in primality testing is held by F. Morain [65] who proved the primality of a 1505 digit number of a general form using massive parallel computational resources. The history of theoretical results on primality testing is long. Pratt [75] showed that the primes are recognizable in non deterministic polynomial time, Miller [62] proved that the Riemann hypothesis for Dirichlet L functions implied that the primes were recognizable in deterministic polynomial time, Adleman, Pomerance and Rumely [2] showed that the primes were ....

....10 16 . Jaeschke [42] has also derived correctness bounds for the Miller Rabin test when applied for several bases. In this paper we consider the problem of generating random primes together with a certificate of primality. Our results draw on Pocklington s, Pratt s and on Bach s work [69] [75], 4] the certificate for a prime p contains a partial factorization of p Gamma 1. However, in contrast to Bach s algorithm [4] for generating (truly) random factored integers, our algorithm does not make use of a general primality test. Of course, if such a general primality test were ....

V.R. Pratt, Every prime has a succinct certificate, SIAM Journal on Computing, Vol. 4, No. 3, pp. 214-220, 1975.

.... is true, then C1 is in P [Mil76] There exists a constant c 2 N and a deterministic algorithm for C1 with running time O( log n) c log log log n ) APR83] If Cram er s conjecture on the gaps between consecutive primes is true, then C1 is recognized in R [GK86] C1 is recognized in NP [Pra75]. Furer [Fur85] has shown that the problem of distinguishing between products of two primes that are 6j 1 (mod 24) and primes that are 6j 1 (mod 24) is in R. Rem1 94 Problem O1b has been settled in the affirmative by Adleman and Huang [AH92] As a result of the work of H. Maier on gaps between ....

Vaughn Pratt. Every prime has a succinct certificate. SIAM Journal of Computing, 4:214--220, 1975.

....Very recently, Sheu and Long [SL92] proved that the extended low hierarchy is an infinite hierarchy. All sets that could be located within the low hierarchies roughly fall into two categories. Either they have low complexity, i.e. they are close to the class P, as for example the primality [Pr75] and the graph isomorphism [Sc88] problems, or they have low information content like sparse sets or sets reducible to (or equivalent to) sparse or tally sets via different kinds of reducibilities. In the past, a variety of language classes has been shown to be included in the low hierarchies (for ....

V. Pratt. Every prime has a succinct certificate. SIAM Journal on Computing 4 (1975) 214--220.

....we can certainly compute an integer at least as large as S( E) in polynomial time via remark 8. Thus the number of necessary random bits, the number of bits of any integer in [x j ; x j 1 ) and the time needed to compute fx j ; x j 1 g are all polynomial in the size of F . Finally, via [Pra75] any prime p2 [x j ; x j 1 ) can be certified (as being a prime) within NP. Also, via [Coh93] a putative root of F mod p can indeed be verified in polynomial time. So we really need just one call to an NP oracle. Our algorithm is thus indeed an AM algorithm. Xi Remark 9 Results weaker than Main ....

Pratt, Vaughan R., "Every Prime has a Succinct Certificate," SIAM J. Comput. 4 (1975), pp. 327--340.

....A 0 from A 1 . This yields the second part of the theorem. 2 The advantage of this pair is that we do not have to mention primes as it is the case with other examples based on conjectured one way functions. The problem with primes is that it is unlikely that the NP definition of primes of Pratt [24] is provably in S 1 2 equivalent to the natural coNP definition. If it were so, we would immediately get a polynomial time algorithm for factoring. Let Composite(a) 9u; v a; u Delta v = a. Proposition2. Suppose S 1 2 proves :A(a) Composite(a) where A(a) is a Sigma b 1 formula ....

....b 1 formula saying that w = g; p; q 1 ; e 1 ; q t ; e t ; w 1 ; w t ) such that: 1. g 2 Z p and g p Gamma1 j 1 mod p 2. p Gamma 1 = Pi it q e i i 3. g p Gamma1 q i 6j 1 mod p, for all i t 4. C(q i ; w i ) for all i t. The NP definition of primes by Pratt [24] is 9w t(a)C(a; w) for a suitable term t(a) We shall denote the definition by P ratt(a) Denote by Phi the following 8 Pi b 1 formula: Phi : 8x; P ratt(x) Composite(x) Theorem 5. The theory S 1 2 Phi implicitly defines factoring, i.e. it proves the sequent: p; q; p 0 ; q 0 ....

Pratt, V.R. (1975) Every prime has a succinct certificate, SIAM J. Computing, 4:214-220.

....hu; ni in which n = pq, where p and q are distinct primes congruent to 3 mod 4 and u is a quadratic residue mod n. SQRP is clearly in NP, because the following algorithm determines membership of hu; ni: guess the factorization of n, verify that both factors are prime, using Pratt s algorithm ([Pra]) guess a square root a of u, and verify that a 2 j u mod n. Similarly, SQRP is in NP: given a pair hu; ni, we can show in nondeterministic polynomial time either that the real prime factorization of n has the wrong form, that u is not relatively prime to n, or that u is of the form ( Gamma1) ....

Pratt, Vaughan. "Every Prime Has a Succinct Certificate," SIAM J. on Comput., 4, 1975, 214--220.

....at 9 MacKenzie[Mac] notes that formalist, constructive, and modal logics, as well as many particular specialized logics, have all been used in proofs of computer system correctness. 10 A simple system of unconventional deductions in which the theorems are the prime numbers, is given by Pratt [Pr]. our definition of proof systems. 3. The definition can still be interpreted as referring to deductive systems (in a broader sense) An ID (instantaneous description) of a state of a Turing machine comprises the contents of each of its tapes, the positions of the read write heads, and the ....

....a strengthening of the P 6= NP conjecture since clearly P NP coNP. No NP complete language belongs to coNP unless NP = coNP. We remark that while Comp and Fact trivially belong to NP, they also happen to belong to coNP: primality and therefore prime factorization can be certified [Pr]. 3.2 Randomized algorithms: Monte Carlo and Las Vegas The term Monte Carlo algorithm is used as a synonym for randomized algorithm , an algorithm executed by a randomizing machine. In addition to its input string x, a randomizing machine has access to a string ae of random bits. Such a ....

Pratt, V.: Every prime has a succinct certificate. SIAM J. Computing 4 (1975), 214--220.

....k distinct primes. Alice is convinced of the truth of her claim because she randomly selected k distinct integers p 1 , p 2 , p k that passed some probabilistic primality test [Rab76, SoSt77] to her satisfaction. Although proofs of primality for these factors exist since PRIMES NP [Pr75], there is no known feasible algorithm for Alice to get these proofs 1 . In other words, Alice knows (with an arbitrary small probability of error) that m is in the proper form, she knows there exists a short proof of this statement, but she cannot find the proof. Using our protocol, she can ....

Pratt, V., "Every prime has a succinct certificate", SIAM Journal on Computing , Vol. 4, 1975, pp. 214-220.

....given x, to find a y or a z, as appropriate. Notice that TFNP coincides with F(NP coNP) One inclusion takes R = R 1 [ R 2 , and the other takes R 1 = R and R 2 = Clearly, factoring is in TFNP, as each integer possesses a unique decomposition into primes, each with a certificate a la Pratt [13]. A related problem is that of the discrete logarithm modulo a (certified) prime p of a (certified) primitive root x of p. Notice that both of these problems are in fact both in the class TFNP and in the class FUP of unambiguous functions in NP [18] the subset of FNP that consists of all those ....

V. R. Pratt, "Every prime has a succinct certificate," SIAM J. Comput. 4 (1975) 214--220.

....here an exhaustive review of its long history. Let us only mention some of the most outstanding modern steps. It has been known for several years that the problem of recognizing prime numbers belongs to P under the Extended Riemann s Hypothesis [15] and that it belongs to Co RP [21, 23] and NP [19] without any assumptions. It can also be solved in almost polynomial time by a deterministic algorithm that runs for a number of steps in O (m O (loglogm ) where m is the size of the number to be tested [2, 17, 7] More recently, it was found to lie in RP [11, 1] and therefore in ZPP [10] as ....

....[22] Although it is possible to generate such primes with certainty using the algorithms of [2, 1] their running time is currently too high to be used in practice. It is also possible to efficiently generate large certified primes by a variation on Pratt s non deterministic algorithm [19] (generate the NP certificate and the resulting prime hand in hand) or by more sophisticated techniques [8] but the resulting distribution would not be uniform. Again, the most attractive solution in practice is to use Rabin s test as follows : function GenPrime (l , k ) l is the size of the ....

Pratt, V., "Every prime has a succinct certificate", SIAM Journal on Computing , pp. 214-220, 1975.

....i th bit of the unique answer equal to one . An interesting example comes from the Fellows and Koblitz paper [FK92] which shows how to provide every prime number with a unique certificate that can be used to verify in polynomial time that the number is prime. The certificates provided by Pratt [Pra75] are not unique. The single valued NP search problem coming from Fellows and Koblitz is: Given a number m, list its prime divisors in order, together with their unique certificates. The theorem below shows that none of the type 2 search problems introduced in Section 2 is computationally ....

Vaughan R. Pratt. Every prime has a succinct certificate. SIAM Journal on Computing, 4:214--220, 1975.

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V. Pratt. Every prime has a succinct certificate. SIAM Journal on Computing, 4:214--220, 1975.

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V. Pratt. "Every prime has a succinct certificate". SIAM J. Comput. 4 (1975), 214-220.

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V. R. Pratt. Every prime has a succinct certificate, SIAM J. Comput. 4:214--220, 1975.

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V. R. Pratt, "Every prime has a succinct certificate," SIAM J. Comput. 4 (1975) 214--220.

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Pratt, V. R., Every prime has a succinct certificate, SIAM J. Comput. 4,

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V. Pratt, "Every prime has a succinct certificate", SIAM J. Computing 4 (1975), 214--220.

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