TY - GEN

T1 - Testing equivalence of polynomials under shifts

AU - Dvir, Zeev

AU - De Oliveira, Rafael Mendes

AU - Shpilka, Amir

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Two polynomials f, g ∈ double-struck F[x1...,xn] are called shift-equivalent if there exists a vector (a1..., a n ∈ double-struck Fn such that the polynomial identity f(x1+a1,...,xn+an) ≡g(x1,...,xn) holds. Our main result is a new randomized algorithm that tests whether two given polynomials are shift equivalent. Our algorithm runs in time polynomial in the circuit size of the polynomials, to which it is given black box access. This complements a previous work of Grigoriev [Gri97 who gave a deterministic algorithm running in time nO(d) for degree d polynomials. Our algorithm uses randomness only to solve instances of the Polynomial Identity Testing (PIT) problem. Hence, if one could de-randomize PIT (a long-standing open problem in complexity) a de-randomization of our algorithm would follow. This establishes an equivalence between de-randomizing shift-equivalence testing and de-randomizing PIT (both in the black-box and the white-box setting). For certain restricted models, such as Read Once Branching Programs, we already obtain a deterministic algorithm using existing PIT results.

AB - Two polynomials f, g ∈ double-struck F[x1...,xn] are called shift-equivalent if there exists a vector (a1..., a n ∈ double-struck Fn such that the polynomial identity f(x1+a1,...,xn+an) ≡g(x1,...,xn) holds. Our main result is a new randomized algorithm that tests whether two given polynomials are shift equivalent. Our algorithm runs in time polynomial in the circuit size of the polynomials, to which it is given black box access. This complements a previous work of Grigoriev [Gri97 who gave a deterministic algorithm running in time nO(d) for degree d polynomials. Our algorithm uses randomness only to solve instances of the Polynomial Identity Testing (PIT) problem. Hence, if one could de-randomize PIT (a long-standing open problem in complexity) a de-randomization of our algorithm would follow. This establishes an equivalence between de-randomizing shift-equivalence testing and de-randomizing PIT (both in the black-box and the white-box setting). For certain restricted models, such as Read Once Branching Programs, we already obtain a deterministic algorithm using existing PIT results.

UR - http://www.scopus.com/inward/record.url?scp=84904176159&partnerID=8YFLogxK

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U2 - 10.1007/978-3-662-43948-7_35

DO - 10.1007/978-3-662-43948-7_35

M3 - Conference contribution

AN - SCOPUS:84904176159

SN - 9783662439470

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 417

EP - 428

BT - Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings

PB - Springer Verlag

T2 - 41st International Colloquium on Automata, Languages, and Programming, ICALP 2014

Y2 - 8 July 2014 through 11 July 2014

ER -