Searching for Elements in Black Box Fields and Applications

We introduce the notion of a black box eld and discuss the problem of explicitly exposing eld elements given in a black box form. We present several sub-exponential algorithms for this problem using a technique due to Maurer. These algorithms make use of elliptic curves over nite elds in a crucial way. We present three applications for our results: (1) We show that any algebraically homomorphic encryption scheme can be broken in expected sub-exponential time. The existence of such schemes has been open for a number of years. (2) We give an expected sub-exponential time reduction from the problem of nding roots of polynomials over nite elds with low straight line complexity (e.g. sparse polynomials) to the problem of testing whether such polynomials have a root in the eld. (3) We show that the hardness of computing discrete-log over elliptic curves implies the security of the Diie-Hellman protocol over elliptic curves. Finally in the last section of the paper we prove the hardness of exposing black box eld elements in a eld of characteristic zero.

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