The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions of polynomially bounded degree over fields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor g of a polynomial f in terms of the (approximative) complexity of f and the degree of the factor g. This extends a result by Kaltofen. The concept of approximative complexity allows us to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (two-sided error) decision complexity.
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Bürgisser, P. The Complexity of Factors of Multivariate Polynomials. Found Comput Math 4, 369–396 (2004). https://doi.org/10.1007/s10208-002-0059-5
- Polynomial Time
- Basic Assumption
- Polynomial Function
- Matrix Theory
- Math Application