The Complexity of Solving Low Degree Equations over Ring of Integers and Residue Rings

Abstract

It is proved that for an arbitrary polynomial $$f\left( x \right) \in {\mathbb{Z}_{{p^n}}}\left[ X \right]$$ of degree d the Boolean complexity of calculation of one its root (if it exists) equals O(dM(nλ(p))) for a fixed prime p and growing n, where λ(p) = ⌈log2p⌉, and M(n) is the Boolean complexity of multiplication of two binary n-bit numbers. Given the known decomposition of this number into prime factors n = m1...mk, $${m_i} = p_i^{{n_i}}$$ , i = 1,..., k, with fixed k and primes pi, i = 1,..., k, and growing n, the Boolean complexity of calculation of one of solutions to the comparison f(x) = 0 mod n equals O(dM(λ(n))). In particular, the same estimate is obtained for calculation of one root of any given degree in the residue ring Zm. As a corollary, it is proved that the Boolean complexity of calculation of integer roots of a polynomial f(x) is equal to Od(M(n)), where $$f\left( x \right) = {a_d}{x^d} + {a_{d - 1}}{x^{d - 1}} + ... + {a_0},{a_i} \in \mathbb{Z}$$ , |ai| < 2n, i = 0,..., d.