## A Lower Bound for Primality (1999)

Citations: | 11 - 5 self |

### BibTeX

@MISC{Allender99alower,

author = {Eric Allender and Michael Saks and Igor Shparlinski},

title = {A Lower Bound for Primality},

year = {1999}

}

### OpenURL

### Abstract

Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the square-free numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by showing that the set of prime numbers (represented in the usual binary notation) is not contained in AC 0 [p] for any prime p. Similar lower bounds are presented for the set of square-free numbers, and for the problem of computing the greatest common divisor of two numbers. 1 Introduction What is the computational complexity of the set of prime numbers? There is a large body of work presenting important upper bounds on the complexity of the set of primes (including [AH87, APR83, Mil76, R80, SS77]), but -- Supported in part by NSF grant CCR-9734918. y Supported in part by NSF grant CCR-9700239. z Supported in part by ARC grant A69700294. as was pointed out recently in [BDS98a, BDS9...

### Citations

225 |
Riemann’s hypothesis and test for primality
- Miller
- 1976
(Show Context)
Citation Context ... does not have. That is, perhaps it is possible to prove that Primes is not complete for any familiar complexity class. Certainly in the above we mean uncoditional proofs. It is well known, thanks to =-=[Mil76]-=-, that Primes belong to P under the Extended Riemann Hypothesis. Additional observations and speculations of this sort pertaining to the factoring problem can be found in [All98]. It would be very int... |

221 | Parity, circuits and the polynomial-time hierarchy - Furst, Saxe, et al. - 1984 |

120 | On distinguishing prime numbers from composite numbers - Adleman, Pomerance, et al. - 1983 |

101 |
Two theorems on random polynomial time
- Adleman
- 1978
(Show Context)
Citation Context ...multiple of p, all of the tests say "no", whereas if x is not a multiple of p, then with probability at least 1=2 2n , at least one of the tests will return "yes". Now, as in the s=-=tandard argument of [Adl78]-=-, there must be at least one sequence of probabilistic inputs for the circuit having the property that, for all n-bit inputs x, the OR of the cn 3 tests is equal to :Mult p (x). This can be seen to be... |

85 | 1 -formulae on finite structures - Ajtai - 1983 |

57 | Recognizing primes in random polynomial time - Adleman, Huang - 1988 |

50 |
bounds on the size of bounded depth networks over a complete basis with logical addition
- Razborov, Lower
- 1987
(Show Context)
Citation Context ... x of length n, the output of C n on input x is f(x). 2.1 Some Known Lower Bounds Our lower bounds follow from the following result of Smolensky [Smo87], which builds on an earlier result of Razborov =-=[Raz87]-=-. Theorem 1 [Smolensky] Let p be a prime, and let m not be a power of p. Then Modm is not in AC 0 [p]. (In fact, [Smo87] provides an exponential lower bound on the size of AC 0 [p] circuits computing ... |

33 | Σ1 1-formulae on finite structures. Annals of Pure and Applied Logic, 24:1–48 - Ajtai - 1983 |

29 | Reducing the complexity of reductions
- Agrawal, Allender, et al.
(Show Context)
Citation Context ...cult to show that there is no AC 0 m reduction from one problem to another (since, for example, the NP 6= NC 1 question can be phrased this way), it is worth noting that a set A in NP is presented in =-=[AAIPR97]-=- such that there is nosAC 0 m reduction from PARITY to A. If Primes were complete for NP (or for any other reasonable complexity class) undersAC 0 m reductions, the isomorphism theorems of [AAR96, AAI... |

29 | Reductions in circuit complexity: An isomorphism theorem and a gap theorem - AGRAWAL, ALLENDER, et al. - 1998 |

29 |
Hardness vs
- Nisan, Wigderson
- 1994
(Show Context)
Citation Context ...dure to build the AC 0 circuits that perform the reduction; we can show only that they exist, via a probabilistic argument.) Although it is possible to use the Nisan-Wigderson pseudorandom generators =-=[NW94]-=- to derandomize this argument, thereby obtaining uniform circuits of size n log O(1) n , this is also not very satisfactory. Surely it is obvious that telling if a number if a multiple of 3 is no hard... |

13 | Circuit and decision tree complexity of some number theoretic problems - BERNASCONI, DAMM, et al. - 2000 |

12 | Probabilistic algorithm for primality testing - Rabin - 1980 |

11 | Circuit complexity of testing square-free numbers - BERNASCONI, SHPARLINSKI |

10 | One-way functions and circuit complexity
- Boppana, Lagarias
- 1987
(Show Context)
Citation Context ...ter to this trend. Almost no number-theoretic problems are known to be complete for natural complexity classes. The following fact is a slight generalization of an observation of Boppana and Lagarias =-=[BL87]-=-. Theorem 3 Let m 2 N be odd. Then Modm AC 0 m Multm . Proof: Note that it is important that m be odd. If m = 2, the conclusion is easily seen to fail. Since m is odd, there is some integer exponent t... |

8 | On the recognition of primes by automata - Hartmanis, Shank - 1968 |

7 | On the average sensitivity of testing square-free numbers - Bernasconi, Damm, et al. - 1627 |

6 |
Lower bounds for arithmetic problems
- Meidânis
- 1991
(Show Context)
Citation Context ...Supported in part by ARC grant A69700294. of the set of primes (including [AH87, APR83, Mil76, R80, SS77]), but -- as was pointed out recently in [BDS98a, BDS98b, BS99, Shp98], other than the work of =-=[Med91]-=- almost nothing has been published regarding lower bounds on the complexity of this set. It was not even known if primality testing could be accomplished by constant-depth, polynomial-size circuits of... |

6 | On tape bounds for single letter alphabet language processing. Theoretical Computer Science 3:213–224 - Hartmanis, Berman - 1976 |

5 |
Infinite sets of primes with fast primality tests and quick generatio of large primes
- Pintz, Steiger, et al.
- 1989
(Show Context)
Citation Context ...would yield a fairly "dense" set of primes in P, by looking at the isomorphic image of f2g \Theta f0; 1g . (Observe that it was shown only fairly recently that there is an infinite set of pr=-=imes in P [PPS89]-=-.) Perhaps the existence of such an isomorphism would bestow Primes with some properties that it provably does not have. That is, perhaps it is possible to prove that Primes is not complete for any fa... |

5 | On the number of primes in an arithmetic progression - Page - 1935 |

4 |
News from the isomorphism front
- Allender
- 1998
(Show Context)
Citation Context ...well known, thanks to [Mil76], that Primes belong to P under the Extended Riemann Hypothesis. Additional observations and speculations of this sort pertaining to the factoring problem can be found in =-=[All98]-=-. It would be very interesting to obtain similar results for other number theoretic problems. For example, it is shown [Shp99] that deciding quadratic residuosity modulo a large prime q is not in AC 0... |

2 | Circuits in bounded arithmetic - Mantzivis - 1992 |

2 | On the circuit complexity of primality, manuscript - Lipton, Viglas |