On Solving Univariate Polynomial Equations over Finite Fields and Some Related Problems

We show deterministic polynomial time algorithms over some family of finite fields for solving univariate polynomial equations and some related problems such as taking nth roots, constructing nth nonresidues, constructing primitive elements and computing elliptic curve “nth roots”. In additional, we present a deterministic polynomial time primality test for some family of integers. All algorithms can be proved by elementary means (without assuming any unproven hypothesis). The problem of solving polynomial equations over finite fields is a generalization of the following problems over finite fields • constructing primitive nth roots of unity, • taking nth roots, • constructing nth nonresidues, • constructing primitive elements (generators of the multiplicative group) for any positive n dividing the number of elements of the underlying field. By the Tonelli-Shanks square root algorithm [21, 19] and its generalization for taking nth roots, constructing nth nonresidues and taking nth roots are polynomial time equivalent for all n. It is clear that primitive nth roots of unity can be computed efficiently from any nth nonresidue when n is prime. It is obvious that a primitive element is also an nth nonresidue. In [20], we showed that, for some families of finite fields, once we can compute a primitive nth root of unity for some suitably chosen n, we can take square roots. The problem of solving polynomial equations is a special case of the problem of polynomial factoring. There is a deterministic polynomial time algorithm, Lenstra-Lenstra-Lovász [15], for factoring polynomials over rational numbers. For polynomial factoring over finite fields, we have Berlekamp’s algorithm [4], which is deterministic but exponential time. So it only works well in small finite fields. There is a probabilistic version of Berlekamp’s algorithm [5] for large finite fields. We also have Cantor-Zassenhaus [7], which is a probabilistic algorithm, for polynomial factoring over finite fields. For a survey of polynomial factoring, see [10]. The problem of solving polynomial equations is to find the solutions of f(x) = 0 over Fq, where Fq is a finite field with q elements and f(x) ∈ Fq[x] is a polynomial with deg f = O(poly(log q)). We may assume f is a product of distinct linear factors since squarefree factorization1 (see [26] and Suppose the input polynomial is a product of some irreducible factors with multiplicity ≥ 1. Squarefree factorization is the process finding the product of the same set of irreducible factors with multiplicity equal 1.