Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = n O(1) . Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p) assuming the ERH. Appeared in Mathematics of Computation 58, pp. 369--380, 1992. An earlier version of this paper appeared in the 22nd Annual ACM Symposium on Theory of Computing (1990), pp. 546-554. 1980 Mathematics Subject Classification (1985 revision): 11T06. 1. Introduction Consider the problem of finding a primitive ...

Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree-1 prime p of K such that p = σ, satis-

Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomial-time bound holds even if k is exponentially large. More generally, assuming the ERH, in time (n log p) O(n) we can construct a set of elements

In recent years, many probabilistic algorithms (i.e., algorithms that can toss coins) that run in polynomial time have been discovered for problems with no known deterministic polynomial time algorithms. Perhaps the most famous example is the problem of testing large (say, 100 digit) numbers for primality. Even for problems which are known to have deterministic polynomial time algorithms, these algorithms are often not as fast as some probabilistic algorithms for the same problem. Even though probabilistic algorithms are useful in practice, we would like to know, for both theoretical and practical reasons, if randomization is really necessary to obtain the most efficient algorithms for certain problems. That is, we would like to know for which problems there is an inherent gap between the deterministic and probabilistic complexities of these problems. In this research, we consider two problems of a number theoretic nature: factoring polynomials over finite fields and constructing irred...

Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a powerof2. Next, we define an x-pseudopower of the base 2tobeapositiveintegern that is not a power of 2, but looks like a power of 2 modulo all primes p ≤ x. Let P2(x) denote the least such n. We give an unconditional upper bound on P2(x), a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that P2(x)isaboutexp(c2x/log x) for a certain constant c2. We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than 2 are also given. 1.

We describe a randomized algorithm that, given an integer a, produces a certificate that the integer is not a pure power of an integer in expected (log a) 1+o(1) bit operations under the assumption of the Generalized Riemann Hypothesis. The certificate can then be verified in deterministic (log a) 1+o(1) time. The certificate constitutes for each possible prime exponent p a prime number qp, such that a mod qp is a p-th non-residue. We use an effective version of the Chebotarev density theorem to estimate the density of such prime numbers qp.

Let π and π ′ be two unitary cuspidal representations of GLm(AQ) with restricted tensor product decompositions π = ⊗νπν and π ′ = ⊗νπ ′ ν. The strong multiplicity one theorem asserts that if πν ≃ π ′ ν for all but finitely many non-archimedean places ν, then π = π ′. This theorem was proved in the 1970’s for GL2 by Casselman [C], and

Abstract. We give a parametrization of the possible Serre invariants (N, k, ν) of modular mod ℓ Galois representations of the exceptional types A4, S4, A5, in terms of local data attached to the fields cut out by the associated projective representations. We show how this result combined with certain global considerations leads to an effective procedure that will determine for a given eigenform f and prime ℓ whether a mod ℓ representation attached to f is exceptional. We illustrate with numerical examples. 1. Introduction. Suppose that f is a eigenform of weight ≥ 2 for some Γ1(M), and let ℓ be a prime number. If λ is a prime of Q above ℓ then by a construction [12] of Deligne followed by reduction mod λ and semisimplification, there is attached to f a mod λ representation ρ: GQ − → GL2(Fℓ) of the absolute Galois group of Q. If ρ is

statistics for a family of elliptic curves