Composition Of Binary Quadratic Forms Research Articles

Secure Groups for Threshold Cryptography and Number-Theoretic Multiparty Computation

In this paper, we introduce secure groups as a cryptographic scheme representing finite groups together with a range of operations, including the group operation, inversion, random sampling, and encoding/decoding maps. We construct secure groups from oblivious group representations combined with cryptographic protocols, implementing the operations securely. We present both generic and specific constructions, in the latter case specifically for number-theoretic groups commonly used in cryptography. These include Schnorr groups (with quadratic residues as a special case), Weierstrass and Edwards elliptic curve groups, and class groups of imaginary quadratic number fields. For concreteness, we develop our protocols in the setting of secure multiparty computation based on Shamir secret sharing over a finite field, abstracted away by formulating our solutions in terms of an arithmetic black box for secure finite field arithmetic or for secure integer arithmetic. Secure finite field arithmetic suffices for many groups, including Schnorr groups and elliptic curve groups. For class groups, we need secure integer arithmetic to implement Shanks’ classical algorithms for the composition of binary quadratic forms, which we will combine with our adaptation of a particular form reduction algorithm due to Agarwal and Frandsen. As a main result of independent interest, we also present an efficient protocol for the secure computation of the extended greatest common divisor. The protocol is based on Bernstein and Yang’s constant-time 2-adic algorithm, which we adapt to work purely over the integers. This yields a much better approach for multiparty computation but raises a new concern about the growth of the Bézout coefficients. By a careful analysis, we are able to prove that the Bézout coefficients in our protocol will never exceed 3max(a,b) in absolute value for inputs a and b. We have integrated secure groups in the Python package MPyC and have implemented threshold ElGamal and threshold DSA in terms of secure groups. We also mention how our results support verifiable multiparty computation, allowing parties to jointly create a publicly verifiable proof of correctness for the results accompanying the results of a secure computation.

Composition of binary quadratic forms over number fields

AbstractIn this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

Ideal composition in quadratic fields: from Bhargava to Gauss

Recent work of Bhargava has described higher composition laws that, among other things, subsume the composition of binary quadratic forms first given by Gauss. Bhargava’s presentation, however, does not provide an algorithmic method for compounding forms in the manner of Arndt’s classical composition method. In this paper, we will show, given two binary quadratic forms, how to find the Bhargava cube that represents their composition, and we will show that this can be done by what is essentially Arndt’s classical method.

Rings and ideals parameterized by binary n -ic forms

The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied composition of binary quadratic forms, which we now understand via Dedekind's association of ideal classes of quadratic rings to integral binary quadratic forms. Delone and Faddeev, in 1940, showed that cubic rings are parameterized by equivalence classes of integral binary cubic forms. Birch, Merriman, Nakagawa, del Corso, Dvornicich, and Simon have all studied rings associated to binary forms of degree n for any n, but it has not previously been known which rings, and with what additional structure, are associated to binary forms. In this paper, we show exactly what algebraic structures are parameterized by binary n-ic forms, for all n. The algebraic data associated to an integral binary n-ic form includes a ring isomorphic to ℤn as a ℤ-module, an ideal class for that ring, and a condition on the ring and ideal class that comes naturally from geometry. In fact, we prove these parameterizations when any base scheme replaces the integers, and show that the correspondences between forms and the algebraic data are functorial in the base scheme. We give geometric constructions of the rings and ideals from the forms that parameterize them and a simple construction of the form from an appropriate ring and ideal.

Reading Gauss in the Computer Age: On the U.S. Reception of Gauss’s Number Theoretical Work (1938–1989)

C.F Gauss’s computational work in number theory attracted renewed interest in the twentieth century due to, on the one hand, the edition of Gauss’s Werke, and, on the other hand, the birth of the digital electronic computer. The involvement of the U.S. American mathematicians Derrick Henry Lehmer and Daniel Shanks with Gauss’s work is analysed, especially their continuation of work on topics as arccotangents, factors of n2 + a2, composition of binary quadratic forms. In general, this strand in Gauss’s reception is part of a more general phenomenon, i.e. the influence of the computer on mathematics and one of its effects, the reappraisal of mathematical exploration.

Action of Hecke operators on Maass theta series and zeta functions

The introductory part contains definitions and basic properties of harmonic theta series, Siegel modular forms, and Hecke operators. Then the transformation formulas are recalled, related to the action of modular substitutions and regular Hecke operators on general harmonic theta series, including specialization to the case of Maass theta series. The following new results are obtained: construction of infinite sequences of eigenfunctions for all regular Hecke operators on spaces of Maass theta series; in the case of Maass theta series of genus $2$, all the eigenfunctions are constructed and the corresponding Andrianov zeta functions are expressed in the form of products of two $L$-functions of the relevant imaginary quadratic rings. The proofs are based on a combination of explicit formulas for the action of Hecke operators on theta series with Gauss composition of binary quadratic forms.

Higher composition laws II: On cubic analogues of Gauss composition

In our first article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number fields, while the spaces underlying those composition laws were closely related to certain exceptional Lie groups. In this paper, our aim is to develop analogous laws of composition on certain spaces of forms so that the resulting groups yield information on the class groups of orders in fields; that is, we wish to obtain genuine cubic analogues of Gauss composition. The fundamental object in our treatment of quadratic composition [2] was the space of 2 x 2 x 2 cubes of integers. In particular, Gauss composition arose from the three different ways of slicing a cube A into two 2 x 2 matrices Mi, Ni (i = 1, 2, 3). Each such pair (Mi, Ni) gives rise to a binary quadratic form QA(x, y) = Qi (x, y), defined by Qi(x, y) = -Det(Mix + Niy). The Cube Law of [2] declares that as A ranges over all cubes, the sum of [Qi], [Q2], [Q3] is zero. It was shown in [2] that the Cube Law gives a law of addition on binary quadratic forms that is equivalent to Gauss composition. Various other invariant-theoretic constructions using the space of 2 x 2 x 2 cubes led to several new composition laws on other spaces of forms. Furthermore, we showed that each of these composition laws gave rise to groups that are closely related to the class groups of orders in quadratic fields. Based on the quadratic case described above, our first inclination for the case might be to examine 3 x 3 x 3 cubes of integers. A 3 x 3 x 3 cube C can be sliced (in three different ways) into three 3 x 3 matrices Li, Mi, Ni (i = 1,2,3). We may therefore obtain from C three ternary forms fl(x, y, Z), f2(x, y,z), f3(x, y,z), defined by

Composition and genera of norm-type forms

M. Kneser has recently discovered a way to define a composition of binary quadratic forms in general [5]. His basic idea can be described as expanding the structure to include a specified action of a ring of similitudes. This approach avoids the traditional problem of orienting the forms, since the proper equivalences can be defined simply as the isometries that preserve the action of the similitude ring. But more is true: when we view his idea in this way, we can extend it to norm-type forms of higher degrees. Besides throwing a new light on the quadratic case, this extension reveals a natural concept of underlying the genus fields already known in number theory. Fix a base ring R (commutative with unit). If P is a free i?-module, then a of degree m supported by P is of course a homogeneous polynomial ƒ of degree m in the coordinates on P; technically, this means that ƒ is an element of the symmetric power S(P*), and in this version we can (and do) extend the definition to projective P of finite rank. Carrying over the usual terminology for quadratic forms, we call (P, ƒ) primitive if ƒ is not identically zero modulo any maximal ideal of R. Now fix an extension C of R, and assume that C is projective of rank m as an i?-module. A form of type C/R will be a pair (P, ƒ) where (1) P is an invertible C-module (and hence projective of rank m over R), (2) ƒ is a primitive form of degree m on the i?-module P, and (3) there is a formal identity f(cp) = N(c)f(p), where N is the norm from CtoR. Two such forms are equivalent if there is a form isometry preserving the C-module structure. Let F(C/R) be the set of equivalence classes. If R —• S is any ring homomorphism, then ® H S induces a map F(C/R) —• F(C ® S/S). There is always at least one form of type C/R, the trivial form (C, iV), and in a sense this is the basic one:

Composition of binary quadratic forms

Composition of binary quadratic forms over an arbitrary commutative base ring is shown to be closely related to homomorphisms (and in particular isomorphisms) of the corresponding even Clifford algebras.

Analysis of an algorithm for composition of binary quadratic forms

This note presents an algorithm which composes two reduced properly primitive binary quadratic forms of the same nonquadratic determinant D in O( M(log∥ D∥)log log∥ D∥) elementary operations.

Class Numbers of Real Quadratic Number Fields

This article is a study of congruence conditions, modulo powers of two, on class number of real quadratic number fields $Q(\sqrt d )$, for which d has at most three distinct prime divisors. Techniques used are those associated with Gaussian composition of binary quadratic forms.

Class numbers of real quadratic number fields

This article is a study of congruence conditions, modulo powers of two, on class number of real quadratic number fields Q ( d ) Q(\sqrt d ) , for which d has at most three distinct prime divisors. Techniques used are those associated with Gaussian composition of binary quadratic forms.

Composition of binary quadratic forms over integral domains

Composition of binary quadratic forms over integral domains

Quaternions and Binary Quadratic Forms

Methods are discussed for studying binary quadratic forms by use of quaternions derived from the ternary quadratic form $f = {x^2} - yz$. In particular, Gauss composition of binary quadratic forms may be achieved by factoring and multiplying quaternions in a natural way.

Quaternions and binary quadratic forms

Methods are discussed for studying binary quadratic forms by use of quaternions derived from the ternary quadratic form f = x 2 − y z f = {x^2} - yz . In particular, Gauss composition of binary quadratic forms may be achieved by factoring and multiplying quaternions in a natural way.

Composition of binary quadratic forms

Composition of binary quadratic forms

Composition of binary quadratic forms

Composition of binary quadratic forms