We describe implementations for solving the discrete logarithm problem in the class group of an imaginary quadratic field and in the infrastructure of a real quadratic field. The algorithms used incorporate improvements over previously-used algorithms, and extensive numerical results are presented demonstrating their efficiency. This data is used as the basis for extrapolations, used to provide...

A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor’...

Motivated by the conjectures of “Mazur-Tate-Teitelbaum type” formulated in [BD1] and by the main result of [BD3], we describe a conjectural construction of a global point PK ∈ E(K), where E is a (modular) elliptic curve over Q of prime conductor p, and K is a real quadratic field satisfying suitable conditions. The point PK is constructed by applying the Tate p-adic uniformization of E to an ex...

Noncommutative tori with real multiplication are the irrational rotation algebras that have special equivalence bimodules. Y. Manin proposed the use of noncommutative tori with real multiplication as a geometric framework for the study of abelian class field theory of real quadratic fields. In this paper, we consider the Cuntz-Pimsner algebras constructed by special equivalence bimodules of irr...

We present a new algorithm based on binary quadratic forms to factor integers of the form N = pq. Its heuristic running time is exponential in the general case, but becomes polynomial when special (arithmetic) hints are available, which is exactly the case for the so-called NICE family of public-key cryptosystems based on quadratic fields introduced in the late 90s. Such cryptosystems come in t...

We present a new algorithm based on binary quadratic forms to factor integers of the form N = pq. Its heuristic running time is exponential in the general case, but becomes polynomial when special (arithmetic) hints are available, which is exactly the case for the so-called NICE family of public-key cryptosystems based on quadratic fields introduced in the late 90s. Such cryptosystems come in t...

We consider upper and lower bounds on the minimal height of an irrational number lying in a particular real quadratic field.