Derandomization from Algebraic Hardness: Treading the Borders
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A hitting-set generator (HSG) is a polynomial map Gen:F^k →F^n such that for all n-variate polynomials Q of small enough circuit size and degree, if Q is non-zero, then Q∘ Gen is non-zero. In this paper, we give a new construction of such a HSG assuming that we have an explicit polynomial of sufficient hardness in the sense of approximative or border complexity. Formally, we prove the following result over any characteristic zero field F: Suppose P(z_1,..., z_k) is an explicit k-variate degree d polynomial that is not in the border of circuits of size s. Then, there is an explicit hitting-set generator Gen_P:F^2k→F^n such that every non-zero n-variate degree D polynomial Q(x_1, x_2, ..., x_n) in the border of size s' circuits satisfies Q ≠ 0 Q ∘Gen_P ≠ 0, provided n^10kd^2 D s'< s. This is the first HSG in the algebraic setting that yields a complete derandomization of polynomial identity testing (PIT) for general circuits from a suitable algebraic hardness assumption. As a direct consequence, we prove the following bootstrapping result for PIT: Let δ > 0 be any constant and k be a large enough constant. Suppose, for every s ≥ k, there is an explicit hitting set of size s^k-δ for all degree s polynomials in the border of k-variate size s algebraic circuits. Then, there is an explicit hitting set of size poly(s) for the border s-variate algebraic circuits of size s and degree s. Unlike the prior constructions of such maps [NW94, KI04, AGS18, KST19], our construction is purely algebraic and does not rely on the notion of combinatorial designs.
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