## Abstract

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support "cryptography" secure against algebraic circuits. Following a similar result of Williams in the boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the polynomial identity testing problem. That is, whether the coefficient vectors of polylog(N)-degree polylog(N)-size circuits is a hitting set for the class of poly(N)-degree poly(N)-size circuits. Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices. Further, we assess the existing literature constructing hitting sets for restricted classes of algebraic circuits and observe that none of them are succinct as given. Yet, we show how to modify some of these constructions to obtain succinct hitting sets. This constitutes the first evidence supporting the existence of an algebraic natural proofs barrier. Our framework is similar to the Geometric Complexity Theory (GCT) program of Mulmuley and Sohoni, except that here we emphasize constructiveness of the proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.

Original language | English |
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Title of host publication | STOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Pierre McKenzie, Valerie King, Hamed Hatami |

Publisher | Association for Computing Machinery |

Pages | 653-664 |

Number of pages | 12 |

ISBN (Electronic) | 9781450345286 |

DOIs | |

State | Published - 19 Jun 2017 |

Event | 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 - Montreal, Canada Duration: 19 Jun 2017 → 23 Jun 2017 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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Volume | Part F128415 |

ISSN (Print) | 0737-8017 |

### Conference

Conference | 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 |
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Country/Territory | Canada |

City | Montreal |

Period | 19/06/17 → 23/06/17 |

### Funding

Funders | Funder number |
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National Science Foundation | CCF-1617580 |

Defense Advanced Research Projects Agency | |

Stanford University | |

Seventh Framework Programme | 257575 |

Israel Science Foundation | 552/16 |

Seventh Framework Programme |

## Keywords

- Algebraic circuit complexity
- Barriers
- Polynomial identity testing
- Succinct hitting sets