Minimal polynomials of algebraic numbers with rational parameters

AbstractThis paper presents an algorithm to construct cryptographically strong genus $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ curves and their Kummer surfaces via Rosenhain invariants and related Kummer parameters. The most common version of the complex multiplication (CM) algorithm for constructing cryptographic curves in genus 2 relies on the well-studied Igusa invariants and Mestre’s algorithm for reconstructing the curve. On the other hand, the Rosenhain invariants typically have much smaller height, so computing them requires less precision, and in addition, the Rosenhain model for the curve can be written down directly given the Rosenhain invariants. Similarly, the parameters for a Kummer surface can be expressed directly in terms of rational functions of theta constants. CM-values of these functions are algebraic numbers, and when computed to high enough precision, LLL can recognize their minimal polynomials. Motivated by fast cryptography on Kummer surfaces, we investigate a variant of the CM method for computing cryptographically strong Rosenhain models of curves (as well as their associated Kummer surfaces) and use it to generate several example curves at different security levels that are suitable for use in cryptography.

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On approximations of solutions of the equation P(z,lnz) = 0 by algebraic numbers

Moscow Journal of Combinatorics and Number Theory ◽

10.2140/moscow.2020.9.435 ◽

2020 ◽

Vol 9 (4) ◽

pp. 435-440

Author(s):

Alexander Galochkin ◽

Anastasia Godunova

Keyword(s):

Algebraic Numbers

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ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

International Journal of Number Theory ◽

10.1142/s1793042110002818 ◽

2010 ◽

Vol 06 (01) ◽

pp. 69-87 ◽

Cited By ~ 11

Author(s):

ALISON MILLER ◽

AARON PIXTON

Keyword(s):

Modular Form ◽

Modular Forms ◽

Fourier Coefficients ◽

Algebraic Numbers ◽

Heegner Points ◽

Positive Weight ◽

Poincare Series ◽

Half Integral Weight ◽

Integral Weight ◽

Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

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