Abstract.We propose and rigorously analyze two randomized algorithms to factor univariate polynomials over finite fields using rank 2 Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from Euler-Poincare characteristics of random Drinfeld modules. Knowledge of a factor degree allows one to rapidly extract all factors of that degree. As a consequence, the problem of factoring polynomials over finite fields in time nearly linear in the degree is reduced to finding Euler-Poincare characteristics of random Drinfeld modules with high probability. The second algorithm is a random Drinfeld module analogue of Berlekamp's algorithm. During the course of its analysis, we prove a new bound on degree distributions in factorization patterns of polynomials over finite fields in certain short intervals.
We describe a deterministic algorithm for finding a generating element of the multiplicative group of the finite field with p n elements. In time polynomial in p and n, the algorithm either outputs an element that is provably a generator or declares that it has failed in finding one. Under a heuristic assumption, we argue that the algorithm does always succeed in finding a generator. The algorithm relies on a relation generation technique in a recent breakthrough by Antoine Joux's for discrete logarithm computation in small characteristic finite fields in L(1/4, o(1)) time. For the special case when the order of p in (Z/nZ) × is small (bounded by (log p (n)) O(1) ), we present a modified algorithm which is reliant on weaker heuristic assumptions.
We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. These codes are obtained as a result of folding the set of rational places of a function field using certain elements (automorphisms) from the Galois group of the extension. This work is an extension of Folded Reed Solomon codes to the setting of Algebraic Geometric codes. We describe two constructions based on this framework depending on if the order of the automorphism used to fold the code is large or small compared to the block length. When the automorphism is of large order, the codes have polynomially bounded list size in the worst case. This construction gives codes of rate R over an alphabet of size independent of block length that can correct a fraction of 1−R −ǫ errors subject to the existence of asymptotically good towers of function fields with large automorphisms. The second construction addresses the case when the order of the element used to fold is small compared to the block length. In this case a heuristic analysis shows that for a random received word, the expected list size and the running time of the decoding algorithm are bounded by a polynomial in the block length. When applied to the GarciaStichtenoth tower, this yields codes of rate R over an alphabet of size ( , that can correct a fraction of 1 − R − ǫ errors.
We propose a randomized algorithm to compute isomorphisms between finite fields using elliptic curves. To compute an isomorphism between two fields of cardinality q n , our algorithm takestime, where ℓ runs through primes dividing n but not q(q − 1) and n ℓ denotes the highest power of ℓ dividing n. Prior to this work, the best known run time dependence on n was quadratic. Our run time dependence on n is at worst quadratic but is subquadratic if n has no large prime factor. In particular, the n for which our run time is nearly linear in n have natural density at least 3/10. The crux of our approach is finding a point on an elliptic curve of a prescribed prime power order or equivalently finding preimages under the Lang map on elliptic curves over finite fields. We formulate this as an open problem whose resolution would solve the finite field isomorphism problem with run time nearly linear in n.
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