Derandomization from Algebraic Hardness: Treading the Borders

Abstract

A hitting-set generator (HSG) is a polynomial map Gen : F k → F n such that for all n-variate polynomials C of small enough circuit size and degree, if C is nonzero, then C • Gen is nonzero. In this paper, we give a new construction of such an HSG assuming that we have an explicit polynomial of sufficient hardness. Formally, we prove the following result over any field F of characteristic zero:Suppose P(z 1 , . . . , z k ) is an explicit k-variate degree d polynomial that is not computable by circuits of size s. Then, there is an explicit hitting-set generator Gen P :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. Unlike the prior constructions of such maps [NW94, KI04, AGS19, KST19], our construction is purely algebraic and does not rely on the notion of combinatorial designs.As a direct consequence, we show that even saving a single point from the "trivial" explicit, exponential sized hitting sets for constant-variate polynomials of low individual-degree which are computable by small circuits, implies a deterministic polynomial time algorithm for PIT. More precisely, we show the following:Let k be a large enough constant. Suppose for every s large enough, there is an explicit hitting set of size at most ((s + 1) k − 1) for the class of k-variate polynomials of individual degree s that are computable by size s circuits. Then there is an explicit hitting set of size s O(k 2 ) for the class of s-variate polynomials, of degree s, that are computable by size s circuits.

Algebraic Hardness versus Randomness in Low Characteristic

“…With a more careful instantiation of the Kabanets-Impagliazzo result, we are able to derandomize PIT in a way that suffices for the bootstrapping results of Agrawal, Ghosh, and Saxena [AGS19] and Kumar, Saptharishi, and Tengse [KST19] to take effect. This allows us to prove nearly-optimal hardness-randomness tradeoffs for PIT over fields of positive characteristic, which comes close to matching the characteristic zero result of Guo, Kumar, Saptharishi, and Solomon [GKSS19]. More concretely, we prove the following.…”

“…Returning to the setting of unrestricted circuits, recent work of Guo, Kumar, Saptharishi, and Solomon [GKSS19] uses a stronger hardness assumption than that of Kabanets and Impagliazzo [KI04] and obtains a stronger derandomization of PIT. Specifically, Guo, Kumar, Saptharishi, and Solomon [GKSS19] obtain a polynomial-time derandomization of PIT using lower bounds against an explicit family of constant-variate polynomials. For comparison, Kabanets and Impagliazzo [KI04] only obtain quasipolynomial-time algorithms for PIT under multivariate hardness assumptions.…”

“…As the assumption of a hard univariate family seems strong, it raises the question of whether or not one can obtain a stronger derandomization of PIT over fields of positive characteristic under a univariate hardness assumption. There is evidence this can be done, as Guo, Kumar, Saptharishi, and Solomon [GKSS19] use univariate lower bounds to obtain a complete derandomization of PIT over fields of characteristic zero. With a more careful instantiation of the Kabanets-Impagliazzo result, we are able to derandomize PIT in a way that suffices for the bootstrapping results of Agrawal, Ghosh, and Saxena [AGS19] and Kumar, Saptharishi, and Tengse [KST19] to take effect.…”

“…Over fields of characteristic zero, the recent work of Guo, Kumar, Saptharishi, and Solomon [GKSS19] obtained what is currently the best-known derandomization of polynomial identity testing for C F (s, s, s) under a hardness assumption. From an explicit family of k-variate degree d polynomials of hardness d Ω(1) , they obtain an explicit hitting set for C F (s, s, s) of size s O(1) .…”

“…We remark that Guo, Kumar, Saptharishi, and Solomon [GKSS19] do not define the notion of explicitness they use in their result, but it is enough for P k,d to be computable by a uniform algorithm which runs in time d O(k) . This is slightly different from our notion of strong explicitness, where we require the coefficients of P k,d to be computable in d O(k) time.…”