TY - JOUR

T1 - On Hitting-Set Generators for Polynomials that Vanish Rarely

AU - Doron, Dean

AU - Ta-Shma, Amnon

AU - Tell, Roei

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/12

Y1 - 2022/12

N2 - The problem of constructing pseudorandom generators for polynomials of low degree is fundamental in complexity theory and has numerous well-known applications. We study the following question, which is a relaxation of this problem: Is it easier to construct pseudorandom generators, or even hitting-set generators, for polynomials p: Fn→ F of degree d if we are guaranteed that the polynomial vanishes on at most an ε> 0 fraction of its inputs? We will specifically be interested in tiny values of ε≪ d/ | F|. This question was first considered by Goldreich and Wigderson (STOC 2014), who studied a specific setting geared for a particular application, and another specific setting was later studied by the third author (CCC 2017). In this work, our main interest is a systematic study of the relaxed problem,in its general form, and we prove results that significantly improve and extend the two previously known results. Our contributions are of two types: ∘ Over fields of size 2 ≤ | F| ≤ poly (n) we show that the seed length of any hitting-set generator for polynomials of degree d≤ n. 49 that vanish on at most ε= | F| -t of their inputs is at least Ω ((d/ t) · log (n)). ∘ Over F2, we show that there exists a (non-explicit) hitting-set generator for polynomials of degree d≤ n. 99 that vanish on at most ε= | F| -t of their inputs with seed length O((d- t) · log (n)). We also show a polynomial-time computable hitting-set generator with seed length O((d- t) · (2 d-t+ log (n))). In addition, we prove that the problem we study is closely related to the following question: “Does there exist a small set S⊆ Fn whose degree-d closure is very large?”, where the degree-d closure of S is the variety induced by the set of degree-d polynomials that vanish on S.

AB - The problem of constructing pseudorandom generators for polynomials of low degree is fundamental in complexity theory and has numerous well-known applications. We study the following question, which is a relaxation of this problem: Is it easier to construct pseudorandom generators, or even hitting-set generators, for polynomials p: Fn→ F of degree d if we are guaranteed that the polynomial vanishes on at most an ε> 0 fraction of its inputs? We will specifically be interested in tiny values of ε≪ d/ | F|. This question was first considered by Goldreich and Wigderson (STOC 2014), who studied a specific setting geared for a particular application, and another specific setting was later studied by the third author (CCC 2017). In this work, our main interest is a systematic study of the relaxed problem,in its general form, and we prove results that significantly improve and extend the two previously known results. Our contributions are of two types: ∘ Over fields of size 2 ≤ | F| ≤ poly (n) we show that the seed length of any hitting-set generator for polynomials of degree d≤ n. 49 that vanish on at most ε= | F| -t of their inputs is at least Ω ((d/ t) · log (n)). ∘ Over F2, we show that there exists a (non-explicit) hitting-set generator for polynomials of degree d≤ n. 99 that vanish on at most ε= | F| -t of their inputs with seed length O((d- t) · log (n)). We also show a polynomial-time computable hitting-set generator with seed length O((d- t) · (2 d-t+ log (n))). In addition, we prove that the problem we study is closely related to the following question: “Does there exist a small set S⊆ Fn whose degree-d closure is very large?”, where the degree-d closure of S is the variety induced by the set of degree-d polynomials that vanish on S.

KW - 11T06 Polynomials over finite fields

KW - 68Q87 Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)

KW - Bounded-Degree Closure

KW - Hitting-Set Generators

KW - Polynomials

KW - Pseudorandom Generators

KW - Quantified Derandomization

UR - http://www.scopus.com/inward/record.url?scp=85140634131&partnerID=8YFLogxK

U2 - 10.1007/s00037-022-00229-2

DO - 10.1007/s00037-022-00229-2

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AN - SCOPUS:85140634131

SN - 1016-3328

VL - 31

JO - Computational Complexity

JF - Computational Complexity

IS - 2

M1 - 16

ER -