/pdf/handbook-of-magma-functions-5anhrdxxj3.pdf

Handbook of magma functions

This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.

Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers

In this paper we show how using a representation of an elliptic curve as the intersection of two quadrics in P3 can provide a defence against Simple and Differental Power Analysis (SPA/DPA) style attacks. We combine this with a 'random window' method of point multiplication and point blinding. The proposed method offers considerable advantages over standard algorithmic techniques of preventing SPA and DPA which usually require a significant increased computational cost, usually more than double. Our method requires roughly a seventy percent increase in computational cost of the basic cryptographic operation, although we give some indication as to how this can be reduced. In addition we show that the Jacobi form is also more efficient than the standard Weierstrass form for elliptic curves in the situation where SPA and DPA are not a concern.

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Preventing SPA/DPA in ECC Systems Using the Jacobi Form

If P(Xl,… xn) is a polynomial with integer coefficients, the Mahler measureM(P) of P is defined to be the geometric mean of |P| over the n-torus Tn. For n = 1, M(P) is an algebraic integer, but for n > 1, there is reason to believe that M(P) is usually transcendental. For example, Smyth showed that log M(l + x +y) = L'(X–3, −1), where X–3 is the odd Dirichlet character of conductor 3. Here we will describe some examples for which it appearsthat log M(P(x, y)) = rL'(E, 0), where E is an elliptic curve and r is a rational number, often either an integer or the reciprocal of an integer. Most of the formulas we discover have been verified numerically to high accuracy but not rigorously proved.

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Mahler's Measure and Special Values of L-functions

This paper presents an elementary introduction to some of the theory of hyperelliptic curves over finite fields of arbitrary characteristic that has cryptographic relevance. Cantor’s algorithm for adding in the jacobian of a hyperelliptic curve and a proof of correctness of the algorithm are presented.

An elementary introduction to hyperelliptic curves

The number theoretic properties of curves of genus 2 are attracting increasing attention. This book provides new insights into this subject; much of the material here is entirely new, and none has appeared in book form before. Included is an explicit treatment of the Jacobian, which throws new light onto the geometry of the Kummer surface. The Mordell–Weil group can then be determined for many curves, and in many non-trivial cases all rational points can be found. The results exemplify the power of computer algebra in diophantine contexts, but computer expertise is not assumed in the main text. Number theorists, algebraic geometers and workers in related areas will find that this book offers unique insights into the arithmetic of curves of genus 2.

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Prolegomena to a middlebrow arithmetic of curves of genus 2

Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2: Weddle's surface

It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N=4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X$_1$(16), whose rational points had been previously computed. We prove there are none with N=5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Galois-stable 5-cycles, and show that there exist Galois-stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N=6.

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Cycles of quadratic polynomials and rational points on a genus-$2$ curve

It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\bold Q[z]$ with rational periodic points of period $N$. Morton proved there were none with $N=4$, by showing that the genus~$2$ algebraic curve that classifies periodic points of period~4 is birational to $X_1(16)$, whose rational points had been previously computed. We prove there are none with $N=5$. Here the relevant curve has genus~$14$, but it has a genus~$2$ quotient, whose rational points we compute by performing a~$2$-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal$_{\bold Q}$-stable $5$-cycles, and show that there exist Gal$_{\bold Q}$-stable $N$-cycles for infinitely many $N$. Furthermore, we answer a question of Morton by showing that the genus~$14$ curve and its quotient are not modular. Finally, we mention some partial results for $N=6$.

Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve

A strategy is proposed for applying Chabauty's Theoremto hyperelliptic curves of genus > 1. In the genus 2case, it shown how recent developments on the formal group of the Jacobiancan be used to give a flexible and computationallyviable method for applying this strategy. The details are described for a general curveof genus 2, and are then applied to find C(ℚ) fora selection of curves. A fringe benefit is a more explicit proof of a result of Coleman.

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E. V. Flynn

Papers

Prolegomena to a middlebrow arithmetic of curves of genus 2

Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2: Weddle's surface

Cycles of quadratic polynomials and rational points on a genus-$2$ curve

Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve

A flexible method for applying Chabauty's Theorem