Binary quadratic forms of determinant −pq

Factoring pq with Quadratic Forms: Nice Cryptanalyses

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...

Composition of Binary Quadratic Forms

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

Division and Binary Quadratic Forms

has only three elements, written h(−23) = 3. There is an binary operation called composition that takes two primitive forms of the same discriminant to a third. Composition is commutative and associative, and makes the set of forms into a group, with identity 〈1, 0,−∆/4〉 for even discriminant and 〈1, 1, (1−∆)/4〉 for odd. From page 49 of Buell [1]: if a form 〈α, β, γ〉 represents a number r primi...

On Perfect Binary Quadratic Forms

A quadratic form f is said to be perfect if its values at points of the integer lattice form a semigroup under multiplication. A problem of V. Arnold is to describe all perfect binary integer quadratic forms. If there is an integer bilinear map s such that f(s(x, y)) = f(x)f(y) for all vectors x and y from the integer 2-dimensional lattice, then the form f is perfect. We give an explicit descri...