The Diffie-Hellman key exchange protocol as well as the ElGamal signature scheme are based on exponentiation modulo p for some prime p. Thus the security of these schemes is strongly tied to the difficulty of computing discrete logarithms in the finite field Fp. The Diffie-Hellman protocol has been generalized to other finite groups arising in number theory, and even to the sets of reduced prin...

Let K be a finite extension of Q and let S = {ν} denote the collection of normalized absolute values on K. Let V + K denote the additive group of adeles over K and let c : V + K → R≥0 denote the content map defined as c({aν}) = ∏ ν∈S ν(aν) for {aν} ∈ V + K . A classical result of J. W. S. Cassels states that there is a constant c > 0 depending only on the field K with the following property: if...

Using the units appearing in Stark’s conjectures on the values of L-functions at s = 0, we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field. Let k be a real quadratic field of discriminant dk, so that k = Q( √ dk), and let ω denote an algebraic integer such that the ring of integers of k is Ok := Z+ ωZ. An important invariant of ...

(here b and a run over the quadratic nonresidues and quadratic residues, respectively, that lie between 0 and p) is a unit of the real quadratic field R(y/p), and that 77> 1. The fact that -n > 1 is usually deduced from the theory of the classnumber of quadratic fields. We present a short proof independent of the theory of the class-number. As in the paper of Chowla and Mordell [Note on the non...

While the unit group of an imaginary quadratic field is very simple, the unit group of a real quadratic field has nontrivial structure. Its study involves some geometry and analysis, but also it relates to Pell's equation and continued fractions, topics from elementary number theory.

In this paper, we prove that for any positive fundamental discriminant D > 1596, there is always at least one prime p ≤ D0.45 such that the Kronecker symbol (D/p) = −1. This improves a result of Granville, Mollin and Williams, where they showed that the least inert prime p in a real quadratic field of discriminant D > 3705 is at most √ D/2. We use a “smoothed” version of the Pólya–Vinogradov in...

The field Q( √ 5) contains the infinite sequence of uniformly bounded continued fractions [1, 4, 2, 3], [1, 1, 4, 2, 1, 3], [1, 1, 1, 4, 2, 1, 1, 3] . . ., and similar patterns can be found in Q( √ d) for any d > 0. This paper studies the broader structure underlying these patterns, and develops related results and conjectures for closed geodesics on arithmetic manifolds, packing constants of i...

We determine all real quadratic number fields with 2-class field tower of length at most 1.