Reduction theory of binary forms

二进制多项式模约简的对数反馈深度

Reduction Modulo Binary Polynomials with Logarithmic Feedback Depth

针对二进制多项式模约简的长反馈链问题，提出基于Frobenius因式分解的FFR算法，实现对数级反馈深度，在Rabin不可约性测试中较NTL和Barrett实现显著加速。

二元二次型与Hecke群

On binary quadratic forms and the Hecke groups

本文给出了系数在Z[lambda]中的二元二次型的约化理论，其中lambda是Hecke群中的最小平移.我们将模群Gamma（1）= SL（2，Z）推广到Hecke群，并广泛使用了修正的负连分数.

InfoMod: A visual and computational approach to Gauss' binary quadratic forms

InfoMod is a new software and application devoted to the modular group, PSL2(Z). It has algorithms that deals with the classical correspondences among continued fractions, geodesics on the modular surface and binary quadratic forms. In addition the software implements the recently discovered representation of Gauss' indefinite binary quadratic forms and their classes in terms of certain infinite planar graphs (dessins) called çarks. InfoMod illustrates various aspects of these forms, i.e. Gauss' reduction algorithm, the representation problem of forms, ambiguous and reciprocal forms. It can be…

Constructing minimal periods of quadratic irrationalities in Zagier's reduction theory

Dirichlet's version of Gauss's reduction theory for indefinite binary quadratic forms includes a map from Gauss-reduced forms to strings of natural numbers. It attaches to a form the minimal period of the continued fraction of a quadratic irrationality associated with the form. When Zagier developed his own reduction theory, parallel to Dirichlet's, he omitted an analogue of this map. We define a new map on Zagier-reduced forms that serves as this analogue. We also define a map from the set of Gauss-reduced forms into the set of Zagier-reduced forms that gives a near-embedding of the structure…

On the arithmetic and geometry of binary Hamiltonian forms

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describ…

Binary quadratic forms as dessins

We show that the class of every primitive indefinite binary quadratic form is naturally represented by an infinite graph (named çark) with a unique cycle embedded on a conformal annulus. This cycle is called the spine of the çark. Every choice of an edge of a fixed çark specifies an indefinite binary quadratic form in the class represented by the çark. Reduced forms in the class represented by a çark correspond to some distinguished edges on its spine. Gauss reduction is the process of moving the edge in the direction of the spine of the çark. Ambiguous and reciprocal classes are represented b…

增强的二元二次型整数表示算法：归约为子集和问题

Enhanced Algorithms for the Representation of integers by Binary Quadratic forms: Reduction to Subset Sum

本文提出将二元二次型整数表示问题归约为子集和问题，设计确定性算法，在判别式对数级时多项式时间求解，一般情形较Cornacchia算法二次加速，应用于椭圆曲线与同源密码学。

A new class of irreducible pentanomials for polynomial based multipliers in binary fields

We introduce a new class of irreducible pentanomials over $\mathbb{F}_2$ of the form $f(x) = x^{2b+c} + x^{b+c} + x^b + x^c + 1$. Let $m=2b+c$ and use $f$ to define the finite field extension of degree $m$. We give the exact number of operations required for computing the reduction modulo $f$. We also provide a multiplier based on Karatsuba algorithm in $\mathbb{F}_2[x]$ combined with our reduction process. We give the total cost of the multiplier and found that the bit-parallel multiplier defined by this new class of polynomials has improved XOR and AND complexity. Our multiplier has comparab…

Tabulation of cubic function fields via polynomial binary cubic forms

We present a method for tabulating all cubic function fields over $\mathbb{F}_q(t)$ whose discriminant $D$ has either odd degree or even degree and the leading coefficient of $-3D$ is a non-square in $\mathbb{F}_{q}^*$, up to a given bound $B$ on the degree of $D$. Our method is based on a generalization of Belabas' method for tabulating cubic number fields. The main theoretical ingredient is a generalization of a theorem of Davenport and Heilbronn to cubic function fields, along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of bin…

二元二次型的拓扑图与类数

Topographs for binary quadratic forms and class numbers

本文借助 Conway 拓扑图与新型复数连分数，统一处理各类判别式下二元二次型的约化，并导出新的类数公式、Gauss 三平方和定理简证及无穷级数求值。